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CT1 – P C – 09
Combined Materials Pack

ActEd Study Materials: 2009 Examinations
Subject CT1
Contents
Study Guide for the 2009 exams
Course Notes
Question and Answer Bank
Series X Assignments*
*Note: The Series X Assignment Solutions should also be supplied with this pack unless you chose not to receive them with your study material.
If you think that any pages are missing from this pack, please contact ActEd’s admin team by email at ActEd@bpp.com or by phone on 01235 550005.
How to use the Combined Materials Pack
Guidance on how and when to use the Combined Materials Pack is set out in the Study Guide for the 2009 exams.

Important: Copyright Agreement
This study material is copyright and is sold for the exclusive use of the purchaser. You may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of it. You must take care of your material to ensure that it is not used or copied by anybody else. By opening this pack you agree to these conditions.

The Actuarial Education Company

© IFE: 2009 Examinations

All study material produced by ActEd is copyright and is sold for the exclusive use of the purchaser. The copyright is owned by Institute and Faculty Education Limited, a subsidiary of the Faculty and Institute of Actuaries.

You may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of the study material.

You must take care of your study material to ensure that it is not used or copied by anybody else.

Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the profession or through your employer.

These conditions remain in force after you have finished using the course.

© IFE: 2009 Examinations

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CT1: Study Guide

Page 1

2009 Study Guide
Subject CT1
Introduction
This Study Guide contains all the information that you will need before starting to study
Subject CT1 for the 2009 exams. Please read this Study Guide carefully before reading the Course Notes, even if you have studied for some actuarial exams before.
The study guide includes:


information about the course structure and how it links to the tutorials



information about how the notes are written



advice on how to study efficiently and prepare for the exam



a summary of the Assignment and Mock Exam deadlines



details of what to do if you have a query



the full Syllabus.

Contents:
Section 1
Section 2
Section 3
Section 4
Section 5
Section 6
Section 7
Section 8
Section 9

The Subject CT1 course structure
The course
ActEd study support
Information from the profession
Study skills
Frequently asked questions
Syllabus
Summary of useful information
File tabs

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CT1: Study Guide

The Subject CT1 course structure
There are four parts to the Subject CT1 course. This should help you plan your progress across the study session. The parts cover related topics and have broadly equal lengths.
The parts are broken down into chapters.
The following table shows how the parts, the chapters and the syllabus items relate to each other. The end columns show how the chapters relate to the days of the regular tutorials. Part

Chapter

Title

No of pages 18

Syllabus objectives (i)

1
2

34

4

Real and money interest rates

12

(iv)

Discounting and accumulating

35

(v)

Level annuities

32

(vi) 1

Deferred and increasing annuities

37

(vi) 2&3

8

Equations of value

23

(vii)

9

Loan schedules

30

(viii)

10

Project appraisal

51

(ix)

11

Investments

40

(x)

12

Elementary compound interest problems 50

(xi)

13

Arbitrage and forward contracts

31

(xii)

14

Term structure of interest rates

50

(xiii)

15

Stochastic interest rate models

32

(xiv)

3 full days (iii)

7

4

Interest rates

2 full days (ii)

6

3

16

5

2

The time value of money

3
1

Cashflow models

Half day © IFE: 2009 Examinations

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1
1

2
2
3
2
4

3

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The Course
The Course consists of Course Notes, a Question and Answer Bank and Series X
Assignments. Collectively, these are referred to as the Combined Materials Pack
(CMP).

2.1

Course notes
Each chapter of the Course Notes includes the Syllabus, a chapter summary and, where appropriate, a page of important formulae or definitions.
The Syllabus for Subject CT1 has been written by the profession to state the requirements of the examiners. The relevant individual Syllabus Objectives are included at the start of each course chapter and a complete copy of the Syllabus is included in Section 7 of this Study Guide. We recommend that you use the Syllabus as an important part of your study. The Syllabus is supplemented by Core Reading, which has also been written by the profession. The purpose of Core Reading is to give the examiners, tutors and students a clear, shared understanding of the depth and breadth of treatment required by the Syllabus. In examinations students are expected to demonstrate their understanding of the concepts in Core Reading. Examiners have the
Core Reading available when setting papers.
Core Reading deals with each Syllabus objective. However, the Core Reading in isolation is not ideal to pass the exam. Core Reading is supplemented by tuition material that has been written by ActEd to help you prepare for the exam. The Subject CT1
Course Notes include the Core Reading in full, integrated throughout the course. Here is an excerpt from some ActEd Course Notes to show you how to identify Core Reading and the ActEd material. Core Reading is shown in this bold font.
Note that in the example given above, the index will fall if the actual share price goes below the theoretical ex-rights share price. Again, this is consistent with what would happen to an underlying portfolio.
This is
ActEd
After allowing for chain-linking, the formula for the investment index then text becomes:
This is Core
Reading

∑ Ni ,t Pi ,t

I (t ) = i

B(t )

where Ni ,t is the number of shares issued for the ith constituent at time t;

B(t ) is the base value, or divisor, at time t.

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CT1: Study Guide

You should not simply try to memorise the Core Reading (or, for that matter, the tuition material). Rather, you should understand the principles stated in the Core Reading and developed in the tuition material.

2.2

Question and Answer Bank
The Question and Answer Bank provides a comprehensive and balanced bank of examstyle questions with solutions and comments, which will be helpful as part of your preparation for the Subject CT1 exam. Some students like to attempt the questions from the relevant part of the Question and Answer Bank before attempting the corresponding assignment. Others use the Question and Answer Bank at the revision stage. Our advice is to try out various approaches and pick the one that best suits you.
The Question and Answer Bank is divided into five parts. The first four parts of the
Question and Answer Bank include a range of short and long questions to test your understanding of the corresponding part of the Course Notes. Part five of the Question and Answer Bank consists of 100 marks of exam-style questions. You may wish to use this as part of your revision closer to the examinations.

2.3

Assignments
The four Series X Assignments (X1 to X4) cover the material in Parts 1 to 4 respectively. Assignments X1 and X2 are 80-mark tests and should take you two and a half hours to complete. Assignments X3 and X4 are 100–mark tests and should take you three hours to complete. The actual Subject CT1 examination will have a total of
100 marks.
If you order Series X Marking at the same time as you order the assignments, you can choose whether or not to receive a copy of the solutions in advance – see the back of the order form for details. If you choose not to receive the solutions with the study material, we will send the assignment solutions to you when we mark your script (or following the deadline date if you don’t submit the assignment).

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ActEd study support
This section lists the study support available from ActEd for Subject CT1.

3.1

Combined Materials Pack
The Combined Materials Pack (CMP) comprises the Course Notes, the Question and
Answer Bank and the Series X Assignments, as already described.

3.2

Series Y Assignments
The Series Y Assignments comprise two 100-mark assignments (Y1 & Y2), each covering the whole course. Series Y is suitable for retakers who have previously used the Series X Assignments and for first-time sitters who want additional question practice. 3.3

Mock Exam
Two 100-mark mock exam papers are available for students as a realistic test of their exam preparation – Mock Exam 2008 and Mock Exam 2009. Both are issued with full marking schedules and are available with or without marking.
Mock Exam 2009 is completely new. Mock Exam 2008 is also still available for students wanting extra practice. It has been updated to reflect any changes to the
Syllabus and Core Reading over the last year.

3.4

ActEd Solutions with Exam Technique (ASET)
The ActEd Solutions with Exam Technique (ASET) contains ActEd’s solutions to the:


Subject CT1 papers from April 2006 to September 2008, ie six papers

plus comment and explanation. In particular it will highlight how questions might have been analysed and interpreted so as to produce a good solution with a wide range of relevant points. This will be valuable in approaching questions in subsequent examinations. A “Mini-ASET” will also be available (from July 2009) covering the April 2009 Exam only. The Actuarial Education Company

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CT1: Study Guide

In addition, a discount is available if you have already purchased a full ASET in a previous year and wish to purchase this year’s full ASET in the same subject. Please email ActEd@bpp.com for details.

3.5

CMP Upgrade
The CMP Upgrade lists all significant changes to the Core Reading and ActEd material so that you can manually amend your 2008 study material to make it suitable for study for the 2009 exams. The Upgrade includes replacement pages and additional pages where appropriate. If a large proportion of the material has changed significantly, making it inappropriate to include all changes, the upgrade will still explain what has changed and if necessary recommend that students purchase a replacement CMP or
Course Notes at a significantly reduced price. The CMP Upgrades can be downloaded free of charge from our website at www.acted.co.uk.

3.6

Revision Notes
ActEd’s Revision Notes have been designed with input from students to help you revise efficiently. They are suitable for first-time sitters who have worked through the ActEd
Course Notes or for retakers (who should find them much more useful and challenging than simply reading through the course again). The Revision Notes are a set of six A5 spiral-bound booklets – perfect for revising on the train or tube to work. Each booklet covers one main theme of the course and includes Core Reading (with a set of integrated short questions to develop your bookwork knowledge), relevant past exam questions (with concise solutions) since April 2000, detailed analysis of key past exam questions and other useful revision aids.

3.7

Flashcards
Flashcards are a set of A6-sized cards that cover the key points of the subject that most students want to commit to memory. Each flashcard has questions on one side and the answers on the reverse. We recommend that you use the cards actively and test yourself as you go.
Flashcards may be used to complement your other study and revision materials. They are not a substitute for question practice but they should help you learn the essential material required.

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CT1: Study Guide

3.8

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Marking
You can have your attempts at Assignments or Mock Exams marked using Series
Marking, Mock Exam marking or Marking Vouchers. These are described below.
Marking is not included with the CMP and you need to order it separately.
On every assignment or mock exam that you submit for marking, you must write your
ActEd Student Number clearly in the box on the cover sheet. Your ActEd Student
Number is printed on all personal correspondence from ActEd. This is not the same as your ARN (Actuarial Reference Number), which is used by the profession. By filling in your ActEd Student Number, you will help us to process and return your script more quickly. If you do not supply this information, your script may be delayed.
When completing an assignment which you are having marked, please also remember the following guidelines:


Leave plenty of space for markers to write their comments – the less room you leave, the fewer helpful comments the marker will be able to add.



Photocopy your assignment before posting it to ActEd – we’ll be happy to mark your copy if your original script should get lost in the post.



Only use current versions of assignments. Do not submit any assignments from the 2008 exam session. We will return old versions of assignments unmarked. •

Identify how much you completed in the recommended time – this will help the marker to advise you on your chances of passing the exam. Do not stop after the allotted allocation of time however. Use the marker to get feedback on all questions, even if the assignment takes you longer to complete than the time allocated.



Grade and comment on your previous assignment marker (where applicable)
– this helps us to improve the quality of future marking.

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CT1: Study Guide

Submission of assignments and mock exams by fax or email
Many students submit assignments and mock exams by post. However, in order to reduce the turnaround time for marking, we are happy to accept scripts submitted by fax or email.
If you are planning to submit your assignments by fax, please read the instructions on the assignment coversheet before completing your assignment. When submitting scripts by fax, you will need to complete the cover sheet in the usual way and fax this, together with a Marking Voucher (if applicable) and your attempt at the assignment or mock exam. Alternatively, you may wish to scan your script and submit a pdf file by email. Please read the instructions on the assignment coversheet if you are interested in using this approach. Series Marking and Mock Exam marking
You may buy marking for a specified series of assignments (eg Subject CT1 Series X) or
Mock Exam Marking for a specified subject (eg Subject CT1). By submitting your scripts in line with our published recommended submission dates, you will make steady progress through the course. We also publish a set of final deadline dates – if you miss the final deadline date for an assignment or mock exam, your script will not be marked.
Recommended submission dates and final deadline dates are set out on the summary pages at the end of this document. Once you have booked Series Marking or Mock Exam
Marking, you will not be able to defer the marking to a future study period.
If you order Series X Marking at the same time as you order the assignments, you can choose whether or not to receive a copy of the solutions in advance – see the back of the order form for details. If you choose not to receive the solutions in advance, we will send the assignment solutions to you when we mark your script (or following the deadline date if you don’t submit the assignment).

Marking Vouchers
If you would prefer not to be restricted to deadlines during the session, or a particular series of assignments in any one study session, you may buy Marking Vouchers instead.
Each Marking Voucher gives the holder the right to submit an attempt at any assignment or mock exam for marking at any time, irrespective of the individual assignment deadlines, study session, subject or person. Marking Vouchers are valid for four years from the date of purchase. Expired Marking Vouchers cannot be refunded.

© IFE: 2009 Examinations

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Important information
Although you may submit your script with a Marking Voucher at any time, you will need to adhere to the explicit Marking Voucher deadline dates to ensure that your script is returned before the date of the exam. The deadline dates are given on pages 34 and 35 of this study guide.
If you live outside the UK you must ensure that your last script reaches the ActEd office earlier than this to allow the extra time needed to return your marked script.

Submission dates and deadlines
Your preparation for the exam should be based on a programme of sustained study over the whole session. If you are having your attempts at the assignments marked by ActEd, you should submit your scripts regularly throughout the session.
If you are attempting the assignments or mock exams you should aim to submit scripts according to the schedule of recommended dates set out in the summary at the end of this document. This will help you to pace your study throughout the session and leave an adequate amount of time for revision and question practice. Scripts submitted after the deadline date will not be marked. It is your responsibility to ensure that scripts are posted in good time.

Important information
The recommended deadline dates are realistic targets for the majority of students. Your scripts will be returned more quickly if you submit them well before the final deadline dates. 3.9

Tutorials
ActEd tutorials are specifically designed to develop the knowledge that you will acquire from the course material into the higher level understanding that is needed to pass the exam. We expect you to have read the relevant part of the Course Notes before attending the tutorial so that the group can spend time on exam questions and discussion to develop understanding rather than basic bookwork.
ActEd run a range of different tutorials at various locations. Full details are set out in
ActEd’s Tuition Bulletin, which is sent regularly to all students based in the UK, Eire and South Africa and is also available from the ActEd website at www.acted.co.uk.

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CT1: Study Guide

Taught Courses
We offer 4-day Taught Courses in some subjects. Each day will cover one part of the course. You will not be expected to have read the relevant part of the notes before attending as the tutor will introduce the material, key concepts and principles that you will need to master for the exam. The course will not replace the need to read the notes, but it will enable you to work through the material at a far quicker pace following the tutorial than would otherwise be the case.
Although the courses will involve more tutor-led sessions, simple examples, exercises and questions will be used to develop your understanding. Therefore, some active student participation will still be required. If you attend a taught course, you may also like to consider a Block Tutorial or a Revision Day closer to the exams to provide some guided exam-style question practice and additional support.
See the Tuition Bulletin for further details.

Regular and Block Tutorials
You can choose one of the following types of tutorial:


Regular Tutorials (four half days, two full days or three full days) spread over the session.



A Block Tutorial (two or three full days) held 2 to 8 weeks before the exam.

The Regular Tutorials provide an even progression through the course. Block Tutorials cover the whole course.

Revision Days
Revision Days are intensive one-day tutorials in the final run-up to the exam. They are particularly suitable for first-time sitters who attended Regular Tutorials and would like to spend a day close to the exam focusing on the course or retakers who have already attended ActEd tutorials. Revision Days give you the opportunity to practise interpreting and answering past exam questions and to raise any outstanding queries with an ActEd tutor. These courses are most suitable if you have previously attended
Regular Tutorials or a Block Tutorial in the same subject.
Details of how to apply for ActEd’s tutorials are set out in our Tuition Bulletin, which is sent regularly to all students based in the UK, Eire and South Africa and is also available from the ActEd website at www.acted.co.uk.

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3.10 Queries and feedback
From time to time you may come across something in the study material that is unclear to you. The easiest way to solve such problems is often through discussion with friends, colleagues and peers – they will probably have had similar experiences whilst studying.
If there’s no-one at work to talk to then use ActEd’s discussion forum at www.acted.co.uk/forums (or use the link from our home page at www.acted.co.uk).
Our online forum is dedicated to actuarial students so that you can get help from fellow students on any aspect of your studies from technical issues to study advice. You could also use it to get ideas for revision or for further reading around the subject that you are studying. ActEd Tutors will visit the site from time to time to ensure that you are not being led astray and we also post other frequently asked questions from students on the forum as they arise.
If you are still stuck, then you can send queries by email to CT1@bpp.com or by fax to
01235 550085 (but we recommend that you try the forum first). We will endeavour to contact you as soon as possible after receiving your query but you should be aware that it may take some time to reply to queries, particularly when tutors are away from the office running tutorials. At the busiest teaching times of year, it may take us more than a week to get back to you.
If you have many queries on the course material, you should raise them at a tutorial or book a personal tuition session with an ActEd Tutor. Information about personal tuition is set out in our current brochure. Please email ActEd@bpp.com for more details.
Each year ActEd Tutors work hard to improve the quality of the study material and to ensure that the courses are as clear as possible and free from errors. We are always happy to receive feedback from students, particularly details concerning any errors, contradictions or unclear statements in the courses. If you have any comments on this course please email them to CT1@bpp.com or fax them to 01235 550085.
The ActEd Tutors also work with the profession to suggest developments and improvements to the Syllabus and Core Reading. If you have any comments or concerns about the Syllabus or Core Reading, these can be passed on via ActEd.
Alternatively, you can address them directly to the Profession’s Examination Team at
Napier House, 4 Worcester Street, Oxford, OX1 2AW or by email to examinations@actuaries.org.uk. The Actuarial Education Company

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CT1: Study Guide

3.11 The ActEd website
The ActEd website at www.acted.co.uk contains much useful information on all aspects of ActEd’s products and services, including:


copies of the Study Guide in every subject



the current Student Brochure and application forms



a link to ActEd’s online store



the latest version of the Tuition Bulletin, including details of finalisation dates and all new courses



details of how to organise tutorials at a convenient location for you



a link to the ActEd discussion forum



details of any minor corrections to the study material



advice on study and a blank personal study plan



details of recommended assignment submission dates and deadline dates.

© IFE: 2009 Examinations

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Information from the profession

4.1

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The Profession’s Copyright
All of the course material is copyright. The copyright belongs to Institute and Faculty
Education Ltd, a subsidiary of the Faculty of Actuaries and the Institute of Actuaries.
The material is sold to you for your own exclusive use. You may not hire out, lend, give, sell, transmit electronically, store electronically or photocopy any part of it. You must take care of your material to ensure it is not used or copied by anyone at any time.
Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the profession or through your employer.
These conditions remain in force after you have finished using the course.

4.2

Changes to the Syllabus and Core Reading
The Syllabus and Core Reading are updated as at 31 May each year. The exams in
April and September 2009 will be based on the Syllabus and Core Reading as at
31 May 2008.
We recommend that you always use the up-to-date Core Reading to prepare for the exams. 4.3

Core Reading accreditation
The Faculty and Institute of Actuaries would like to thank the numerous people who have helped in the development of this material and in the previous versions of Core
Reading.
The following book has been used as the basis for several units:
An introduction to the mathematics of finance.
McCutcheon, J. J.; Scott, W. F.
Heinemann, 1986. ISBN: 043491228X, by permission of the authors who are the holders of copyright of the book. All rights reserved.

4.4

Past exam papers
You can download some past papers and reports from the profession’s website at www.actuaries.org.uk. The Actuarial Education Company

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4.5

CT1: Study Guide

The profession’s libraries and website
All students are entitled to use the profession’s libraries in Edinburgh, London and
Oxford. The following services are offered:

Loans
You can borrow books by post if you do not live or work near one of the libraries. The libraries stock most publications on the list of suggested further reading for the CT
Subjects and have multiple copies of the popular titles.

Photocopies
The library staff can post photocopies to you. The cost is 20p per sheet (£5.00 postage and packing added for destinations outside the UK).

Enquiries
The library staff will always help with enquiries. They will try to obtain items not held in stock and can advise on access to other libraries. They compile lists of references on specialist topics on request.

Study facilities
All three libraries have comfortable quiet study space. Past exam papers, Examiners’
Reports, ActEd Course Notes and Core Reading are all available for reference.

Website
You can search the library catalogue on the website and order items online. Many catalogue records include links to full text documents for downloading. You can also access the Publications Shop and order items for purchase. The website is a free information resource for the latest thinking from the profession. You will find briefing statements, press releases, responses to consultations, CMI reports, conference papers, sessional meeting papers and the latest news.
The actuarial education section contains practical information such as exam dates, past papers and reports, syllabuses, the Student Handbook, guidance on study and exam techniques and the lists of suggested additional reading.

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Bookmark these pages:
Information for students

http://www.actuaries.org.uk/students
Exams information

http://www.actuaries.org.uk/students/exams
Library services

http://www.actuaries.org.uk/knowledge/library
Publications shop

http://www.actuaries.org.uk/members/transactions/publications_shop

4.6

Further reading
The exam will be based on the relevant Syllabus and Core Reading and the ActEd course material will be the main source of tuition for students.
However, some students may find it useful to obtain a different viewpoint on a particular topic covered in Subject CT1. The following list of further reading for
Subject CT1 has been prepared by the Institute and Faculty. This list is not exhaustive and other useful material may be available.
Actuarial mathematics. Bowers, Newton L et al. – 2nd ed. – Society of Actuaries, 1997. xxvi, 753 pages. ISBN: 0 938959 46 8.
An introduction to the mathematics of finance. McCutcheon, John J; Scott, William F.
London: Heinemann, 1986. 463 pages. ISBN: 0 434 91228 x. Available from the publications unit.
Mathematics of compound interest. Butcher, M V; Nesbitt, Cecil J. Ulrich's Books,
1971. 324 pages.
Theory of financial decision making. Ingersoll, Jonathan E. Rowman & Littlefield,
1987. 474 pages. ISBN: 0 8476 7359 6.
The theory of interest. Kellison, Stephen G. 2nd ed. Irwin, 1991. 446 pages. ISBN: 0
256 09150 1. Available from the publications unit.

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Life contingencies. Neill, Alistair. Heinemann, 1977. vii, 452 pages. ISBN: 0 434
91440 1.
(This book is no longer in print, but has been used as a textbook under earlier education strategies. You should find it relatively easy to borrow a copy from a colleague. Alternatively you can borrow it from the libraries.)

Life insurance mathematics. Gerber, Hans U. 3rd ed. Springer. Swiss Association of
Actuaries, 1997. 217 pages. ISBN: 3 540 62242 x. Available from the publications unit. 4.7

Calculators
The profession has issued the following advice. However, you are strongly recommended to consult www.actuaries.org.uk for the latest position.
Candidates may use electronic calculators in all the examinations subject to the following conditions:


Candidates must provide their own calculators.



Under no circumstances should hand-held personal computers, of any description, be taken into the examination room.



Calculators must be silent, have visual display only and be battery- or solaroperated.



Any stored data and / or stored program facilities must be cleared before the calculator is taken into the examination room.



Candidates are advised that in all calculations intermediate results should normally be shown to gain full marks.

The following calculators ONLY are permitted:


Casio FX85 (with or without any suffix)



Hewlett Packard HP9S



Hewlett Packard HP 12C (with or without any suffix)



Sharp EL531 (with or without any suffix)



Texas Instruments BA II Plus (with or without any suffix)



Texas Instruments TI-30 (with or without any suffix).

The list of permitted calculators will be reviewed each year by the Education
Committee. Student comments are considered and should be forwarded to the
Examinations Team for submission.

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Calculators which have been discontinued by the manufacturer, or which the Education
Committee has decided to remove for any reason, will remain on the list for one year to give students time to become familiar with an alternative.
Candidates are advised that invigilators will be asked to report the use of calculators not on the permitted list and the Board of Examiners will decide how to treat such cases at the results meetings.
No extra time will be allowed for candidates who do not use calculators or whose calculators break down in the course of the examination. Exam supervisors will not have extra batteries or calculators.

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5

Study skills

5.1

CT1: Study Guide

The CT Subject exams
The Core Reading and exam papers for these subjects tend to be very technical. The exams themselves have many calculation and manipulation questions. The emphasis in the exam will therefore be on understanding the mathematical techniques and applying them to various, frequently unfamiliar, situations. It is important to have a feel for what the numerical answer should be by having a deep understanding of the material and by doing reasonableness checks.
Subjects CT2 and CT7 are more “wordy” than the other subjects, including an “essaystyle” question in Subject CT7.
Since there will be a high level of mathematics required in the courses it is important that your mathematical skills are extremely good. If you are a little rusty you may wish to consider buying the Foundation ActEd Course (FAC) available from ActEd. This covers all of the mathematical techniques that are required for the CT Subjects, some of which are beyond A-Level (or Higher Level) standard. It is a reference document to which you can refer when you need help on a particular topic.
You will have sat many exams before and will have mastered the exam and revision techniques that suit you. However it is important to note that due to the high volume of work involved in the CT Subjects it is not possible to leave all your revision to the last minute. Students who prepare well in advance have a better chance of passing their exams on the first sitting.
Many of the exam questions are lengthy and most students find it difficult to complete the paper in 3 hours. Therefore it is important to find ways of maximising your score in the shortest possible time.
We recommend that you prepare for the exam by practising a large number of examstyle questions under exam conditions. This will:


help you to develop the necessary understanding of the techniques required



highlight which are the key topics which crop up regularly in many different contexts and questions



help you to practise the specific skills that you will need to pass the exam.

There are many sources of exam-style questions. You can use past exam papers, the
Question and Answer Bank (which includes many past exam questions), assignments, mock exams, the Revision Notes and ASET.

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5.2

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Overall study plan
We suggest that you develop a realistic study plan, building in time for relaxation and allowing some time for contingencies. Be aware of busy times at work, when you may not be able to take as much study leave as you would like. Once you have set your plan, be determined to stick to it. You don’t have to be too prescriptive at this stage about what precisely you do on each study day. The main thing is to be clear that you will cover all the important activities in an appropriate manner and leave plenty of time for revision and question practice.
Aim to manage your study so as to allow plenty of time for the concepts you meet in this course to “bed down” in your mind. Most successful students will probably aim to complete the course at least a month before the exam, thereby leaving a sufficient amount of time for revision. By finishing the course as quickly as possible, you will have a much clearer view of the big picture. It will also allow you to structure your revision so that you can concentrate on the important and difficult areas of the course.
A sample CT subject study plan is available on our website at www.acted.co.uk. It includes details of useful dates, including assignment deadlines and tutorial finalisation dates. 5.3

Study sessions
Only do activities that will increase your chance of passing. Try to avoid including activities for the sake of it and don’t spend time reviewing material that you already understand. You will only improve your chances of passing the exam by getting on top of the material that you currently find difficult.
Ideally, each study session should have a specific purpose and be based on a specific task, eg “Finish reading Chapter 3 and attempt Questions 1.4, 1.7 and 1.12 from the
Question and Answer Bank”, as opposed to a specific amount of time, eg “Three hours studying the material in Chapter 3”.
Try to study somewhere quiet and free from distractions (eg a library or a desk at home dedicated to study). Find out when you operate at your peak, and endeavour to study at those times of the day. This might be between 8am and 10am or could be in the evening. Take short breaks during your study to remain focused – it’s definitely time for a short break if you find that your brain is tired and that your concentration has started to drift from the information in front of you.

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5.4

CT1: Study Guide

Order of study
We suggest that you work through each of the chapters in turn. To get the maximum benefit from each chapter you should proceed in the following order:
1.

Read the Syllabus Objectives. These are set out in the box on Page 1 of each chapter. 2.

Read the Chapter Summary at the end of each chapter. This will give you a useful overview of the material that you are about to study and help you to appreciate the context of the ideas that you meet.

3.

Study the Course Notes in detail, annotating them and possibly making your own notes. Try the self-assessment questions as you come to them. Our suggested solutions are at the end of each chapter. As you study, pay particular attention to the listing of the Syllabus Objectives and to the Core Reading.

4.

Read the Chapter Summary again carefully. If there are any ideas that you can’t remember covering in the Course Notes, read the relevant section of the notes again to refresh your memory.

You may like to attempt some questions from the Question and Answer Bank when you have completed a part of the course. It’s a good idea to annotate the questions with details of when you attempted each one. This makes it easier to ensure that you try all of the questions as part of your revision without repeating any that you got right first time.
Once you’ve read the relevant part of the notes, tried a selection of exam-style questions from the Question and Answer Bank (and attended a tutorial, if appropriate), you should attempt the corresponding assignment. If you submit your assignment for marking, spend some time looking through it carefully when it is returned. It can seem a bit depressing to analyse the errors you made, but you will increase your chances of passing the exam by learning from your mistakes. The markers will try their best to provide practical comments to help you to improve.
It’s a fact that people are more likely to remember something if they review it from time to time. So, do look over the chapters you have studied so far from time to time. It is useful to re-read the Chapter Summaries or to try the self-assessment questions again a few days after reading the chapter itself.
To be really prepared for the exam, you should not only know and understand the Core
Reading but also be aware of what the examiners will expect. Your revision programme should include plenty of question practice so that you are aware of the typical style, content and marking structure of exam questions. You should attempt as many questions as you can from the Question and Answer Bank and past exam papers.

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5.5

Page 21

Active study
Here are some techniques that may help you to study actively.
1.

Don’t believe everything you read! Good students tend to question everything that they read. They will ask “why, how, what for, when?” when confronted with a new concept, and they will apply their own judgement. This contrasts with those who unquestioningly believe what they are told, learn it thoroughly, and reproduce it (unquestioningly?) in response to exam questions.

2.

Another useful technique as you read the Course Notes is to think of possible questions that the examiners could ask. This will help you to understand the examiners’ point of view and should mean that there are fewer nasty surprises in the exam room! Use the Syllabus to help you make up questions.

3.

Annotate your notes with your own ideas and questions. This will make you study more actively and will help when you come to review and revise the material. Do not simply copy out the notes without thinking about the issues.

4.

Attempt the questions in the notes as you work through the course. Write down your answer before you check against the solution.

5.

Attempt other questions and assignments on a similar basis, ie write down your answer before looking at the solution provided. Attempting the assignments under exam conditions has some particular benefits:



When you have your assignments marked it is much more useful if the marker’s comments can show you how to improve your performance under exam conditions than your performance when you have access to the notes and are under no time pressure.



The knowledge that you are going to do an assignment under exam conditions and then submit it (however good or bad) for marking can act as a powerful incentive to make you study each part as well as possible.


6.

It forces you to think and act in a way that is similar to how you will behave in the exam.

It is also quicker than trying to write perfect answers.

Sit a mock exam 4 to 6 weeks before the real exam to identify your weaknesses and work to improve them. You could use the mock exam written by ActEd or a past exam paper.

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6

CT1: Study Guide

Frequently asked questions
Q:

What knowledge of earlier subjects should I have?

A:

The Course Notes are written on the assumption that students have studied
Subject CT3. However, you should be able to study Subjects CT3 and CT1 in parallel, ie you do not need to complete the whole of Subject CT3 before starting
Subject CT1.

Q:

What level of mathematics is required?

A:

The level of maths you need for this course is broadly A-level standard.
However, there may be some symbols (eg the gamma function) that are not usually included on A-level syllabuses. You will find the course (and the exam!) much easier if you feel comfortable with the mathematical techniques used in the course and you feel confident in applying them yourself. If you feel that you need to brush up on your mathematical skills before starting the course, you may find it useful to study the Foundation ActEd Course (FAC) or read an appropriate textbook. The full Syllabus for FAC, a sample of the Course Notes and an Initial Assessment to test your mathematical skills can be found on our website at www.acted.co.uk.

Q:

What should I do if I discover an error in the course?

A:

If you find an error in the course, please check our website at: www.acted.co.uk/Html/paper_corrections.htm to see if the correction has already been dealt with. Otherwise please send details via email to CT1@bpp.com or send a fax to 01235 550085.

Q:

What calculators am I allowed to use in the exam?

A:

You are allowed to use the following calculators:


Casio FX85 (with or without any suffix)



Hewlett Packard HP9S



Hewlett Packard HP 12C (with or without any suffix)



Sharp EL531 (with or without any suffix)



Texas Instruments BA II Plus (with or without any suffix)



Texas Instruments TI-30 (with or without any suffix).

© IFE: 2009 Examinations

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CT1: Study Guide

Page 23

This is the current advice from the profession. However, we strongly recommend that you check on the profession’s website for the latest details.

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© IFE: 2009 Examinations

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7

CT1: Study Guide

Syllabus
The full Syllabus for Subject CT1 is given here. The numbers to the right of each objective are the chapter numbers in which the objective is covered in the ActEd course.
Aim
The aim of the Financial Mathematics course is to provide a grounding in financial mathematics and their simple applications.
Links to other subjects


Subject CT2 – Finance and Financial Reporting: develops the use of the asset types introduced in this subject.



Subject CT4 – Models: develops the idea of stochastic interest rates.



Subject CT5 – Contingencies: develops some of the techniques introduced in this subject in situations where cashflows are dependent on survival.



Subject CT7 – Economics: develops the behaviour of interest rates.



Subject CT8 – Financial Economics: develops the principles further.



Subjects CA1 – Actuarial Risk Management, CA2 – Modelling and the
Specialist Technical and Specialist Applications subjects: use the principles introduced in this subject.

© IFE: 2009 Examinations

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CT1: Study Guide

Page 25

Objectives
On completion of the course the trainee actuary will be able to:
(i)

Describe how to use a generalised cash-flow model to describe financial transactions (Chapter 1)
1.

2.

(ii)

For a given cash flow process, state the inflows and outflows in each future time period and discuss whether the amount or the timing (or both) is fixed or uncertain.
Describe in the form of a cash flow model the operation of a zero coupon bond, a fixed interest security, an index-linked security, cash on deposit, an equity, an “interest only” loan, a repayment loan, and an annuity certain. Describe how to take into account the time value of money using the concepts of compound interest and discounting.
(Chapter 2)
1.

Accumulate a single investment at a constant rate of interest under the operation of:


simple interest



compound interest.

2.

Define the present value of a future payment.

3.

Discount a single investment under the operation of simple (commercial) discount at a constant rate of discount.

4.

Describe how a compound interest model can be used to represent the effect of investing a sum of money over a period.

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(iii)

CT1: Study Guide

Show how interest rates or discount rates may be expressed in terms of different time periods.
(Chapter 3)
1.

Derive the relationship between the rates of interest and discount over one effective period arithmetically and by general reasoning.

2.

Derive the relationships between the rate of interest payable once per effective period and the rate of interest payable p times per time period and the force of interest.

3.

Explain the difference between nominal and effective rates of interest and derive effective rates from nominal rates.

4.

Calculate the equivalent annual rate of interest implied by the accumulation of a sum of money over a specified period where the force of interest is a function of time.

(iv)

Demonstrate a knowledge and understanding of real and money interest rates.
(Chapter 4)

(v)

Calculate the present value and the accumulated value of a stream of equal or unequal payments using specified rates of interest and the net present value at a real rate of interest, assuming a constant rate of inflation.
(Chapter 5)
1.

Discount and accumulate a sum of money or a series (possibly infinite) of cash flows to any point in time where:



the rate of interest or discount varies with time but is not a continuous function of time



2.

the rate of interest or discount is constant

either or both the rate of cashflow and the force of interest are continuous functions of time.

Calculate the present value and accumulated value of a series of equal or unequal payments made at regular intervals under the operation of specified rates of interest where the first payment is:


deferred for a period of time



not deferred.

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CT1: Study Guide

(vi)

Page 27

Define and use the more important compound interest functions, including annuities certain.
(Chapters 6 and 7)
1.

Derive formulae in terms of i, v, n, d, δ , i ( p ) and d ( p ) for
(
(
(
( an , sn , an p ) , sn p ) , an , sn , an p ) , sn p ) , an and sn .

2.

Derive formulae in terms of i, v, n, d, δ , i ( p ) and d ( p ) for m 3.

an ,

a ( p ) , m an m n

,

a ( p ) and m m n

an .

Derive formulae in terms of i, v, n, δ , an and an for ( Ia ) n , ( Ia ) n ,
( Ia ) n , ( Ia ) n , and the respective deferred annuities.

(vii)

Define an equation of value.

(Chapter 8)

1.

Define an equation of value, where payment or receipt is certain.

2.

Describe how an equation of value can be adjusted to allow for uncertain receipts or payments.

3.

Understand the two conditions required for there to be an exact solution to an equation of value.

(viii) Describe how a loan may be repaid by regular instalments of interest and capital.
(Chapter 9)
1.

Describe flat rates and annual effective rates.

2.

Calculate a schedule of repayments under a loan and identify the interest and capital components of annuity payments where the annuity is used to repay a loan for the case where annuity payments are made once per effective time period or p times per effective time period and identify the capital outstanding at any time.

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(ix)

CT1: Study Guide

Show how discounted cashflow techniques can be used in investment project appraisal. (Chapter 10)
1.

2.

Calculate the internal rate of return implied by the receipts and payments from an investment project.

3.

Describe payback period and discounted payback period and discuss their suitability for assessing the suitability of an investment project.

4.

Determine the payback period and discounted payback period implied by the receipts and payments from an investment project.

5.

(x)

Calculate the net present value and accumulated profit of the receipts and payments from an investment project at given rates of interest.

Calculate the money-weighted rate of return, the time-weighted rate of return and the linked internal rate of return on an investment or a fund.

Describe the investment and risk characteristics of the following types of asset available for investment purposes:
(Chapter 11)


fixed-interest government borrowings



fixed-interest borrowing by other bodies



shares and other equity-type finance



derivatives.

© IFE: 2009 Examinations

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CT1: Study Guide

(xi)

Analyse elementary compound interest problems.

Page 29

(Chapter 12)

1.

Calculate the present value of payments from a fixed interest security where the coupon rate is constant and the security is redeemed in one instalment. 2.

Calculate upper and lower bounds for the present value of a fixed interest security that is redeemable on a single date within a given range at the option of the borrower.

3.

Calculate the running yield and the redemption yield from a fixed interest security (as in 1.), given the price.

4.

Calculate the present value or yield from an ordinary share and a property, given simple (but not necessarily constant) assumptions about the growth of dividends and rents.

5.

Solve an equation of value for the real rate of interest implied by the equation in the presence of specified inflationary growth.

6.

Calculate the present value or real yield from an index-linked bond, given assumptions about the rate of inflation.

7.

Calculate the price of, or yield from, a fixed interest security where the investor is subject to deduction of income tax on coupon payments and redemption payments are subject to the deduction of capital gains tax.

8.

Calculate the value of an investment where capital gains tax is payable, in simple situations, where the rate of tax is constant, indexation allowance is taken into account using specified index movements and allowance is made for the case where an investor can offset capital losses against capital gains.

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(xii)

CT1: Study Guide

Calculate the delivery price and the value of a forward contract using arbitrage free pricing methods.
(Chapter 13)
1.

Define “arbitrage” and explain why arbitrage may be considered impossible in many markets.

2.

Calculate the price of a forward contract in the absence of arbitrage assuming: •

no income or expenditure associated with the underlying asset during the term of the contract



a fixed income from the asset during the term



a fixed dividend yield from the asset during the term.

3.

Explain what is meant by “hedging” in the case of a forward contract.

4.

Calculate the value of a forward contract at any time during the term of the contract, in the absence of arbitrage, in the situations listed in 2 above. (xiii) Show an understanding of the term structure of interest rates.

(Chapter 14)

1.

Describe the main factors influencing the term structure of interest rates.

2.

Explain what is meant by the par yield and yield to maturity.

3.

Explain what is meant by, derive the relationships between and evaluate:


discrete spot rates and forward rates



continuous spot rates and forward rates.

4.

Define the duration and convexity of a cashflow sequence, and illustrate how these may be used to estimate the sensitivity of the value of the cashflow sequence to a shift in interest rates.

5.

Evaluate the duration and convexity of a cashflow sequence.

6.

Explain how duration and convexity are used in the (Redington) immunisation of a portfolio of liabilities.

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CT1: Study Guide

(xiv)

Page 31

Show an understanding of simple stochastic interest models for investment returns. (Chapter 15)
1.

Describe the concept of a stochastic interest rate model and the fundamental distinction between this and a deterministic model.

2.

Derive algebraically, for the model in which the annual rates of return are independently and identically distributed and for other simple models, expressions for the mean value and the variance of the accumulated amount of a single premium.

3.

Derive algebraically, for the model in which the annual rates of return are independently and identically distributed, recursive relationships which permit the evaluation of the mean value and the variance of the accumulated amount of an annual premium.

4.

Derive analytically, for the model in which each year the random variable (1 + i ) has an independent log-normal distribution, the distribution functions for the accumulated amount of a single premium and for the present value of a sum due at a given specified future time.

5.

Apply the above results to the calculation of the probability that a simple sequence of payments will accumulate to a given amount at a specific future time.

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CT1: Study Guide

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8

Summary of useful information – Subject CT1

8.1

Structure of the course
Part

Chapter

Title

No of pages Syllabus objectives 1

Interest rates

34

(iii)

4

Real and money interest rates

12

(iv)

Discounting and accumulating

35

(v)

Level annuities

32

(vi) 1

7

Deferred and increasing annuities

37

(vi) 2&3

8

Equations of value

23

(vii)

9

Loan schedules

30

(viii)

10

Project appraisal

51

(ix)

11

Investments

40

(x)

12

Elementary compound interest problems 50

(xi)

13

Arbitrage and forward contracts

31

(xii)

14

Term structure of interest rates

50

(xiii)

15

Stochastic interest rate models

32

(xiv)

3 full days (ii)

6

4

16

5

3

The time value of money

2 full days (i)

3

2

18

2
1

Cashflow models

Half day 1

1
1

2
2
3
2
4

3

Send queries or feedback on Subject CT1 by email to CT1@bpp.com.
ActEd’s website: www.acted.co.uk
The profession’s website: www.actuaries.org.uk

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© IFE: 2009 Examinations

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8.2

CT1: Study Guide

Assignment Deadlines – CT Subjects
For the session leading to the April 2009 exams – CT Subjects
Marking vouchers
Subjects

Assignments

Mocks

CTs with exams in week beginning 20 April 2009

25 March 2009

1 April 2009

Other CT subjects

1 April 2009

8 April 2009

Recommended submission date

Final deadline date 19 November 2008

21 January 2009

26 November 2008

28 January 2009

3 December 2008

11 February 2009

10 December 2008

18 February 2009

28 January 2009

4 March 2009

4 February 2009

11 March 2009

18 February 2009

18 March 2009

25 February 2009

25 March 2009

Series X and Y Assignments
Subjects
CTs with exams in week beginning
20 April 2009

Assignment

X1

Other CT subjects
CTs with exams in week beginning
20 April 2009

X2

Other CT subjects
CTs with exams in week beginning
20 April 2009

X3

Other CT subjects
CTs with exams in week beginning
20 April 2009

X4

Other CT subjects
CT1 – CT8

Y1

11 March 2009

18 March 2009

CT1 – CT8

Y2

18 March 2009

25 March 2009

Subjects

Recommended submission date

Final deadline date CTs with exams in week beginning 20 April 2009

25 March 2009

1 April 2009

Other CT subjects

1 April 2009

8 April 2009

Mock Exams

We suggest that you work to the recommended submission dates where possible. Please remember that the turnaround of your script is likely to be quicker if you submit it well before the final deadline date.
At the time of going to print, the Profession had not confirmed the order of the 2009 exams. An up-todate version of these Assignment deadlines, showing the specific subjects in each sub-group, is available on our website at www.ActEd.co.uk.

© IFE: 2009 Examinations

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CT1: Study Guide

Page 35

For the session leading to the September 2009 exams – CT Subjects
Marking vouchers
Subjects

Assignments

Mocks

CTs with exams in week beginning 28 Sept 2009

2 September 2009

9 September 2009

Other CT subjects

9 September 2009

16 September 2009

Recommended submission date

Final deadline date

1 July 2009

15 July 2009

Other CT subjects

8 July 2009

22 July 2009

CTs with exams in week beginning
28 Sept 2009

15 July 2009

29 July 2009

Other CT subjects

22 July 2009

5 August 2009

CTs with exams in week beginning
28 Sept 2009

29 July 2009

12 August 2009

Other CT subjects

5 August 2009

19 August 2009

CTs with exams in week beginning
28 Sept 2009

12 August 2009

26 August 2009

19 August 2009

2 September 2009

Series X and Y Assignments
Subjects
CTs with exams in week beginning
28 Sept 2009

Assignment

X1
X2
X3
X4

Other CT subjects
CT1 – CT8

Y1

5 August 2009

19 August 2009

CT1 – CT8

Y2

19 August 2009

2 September 2009

Subjects

Recommended submission date

Final deadline date

CTs with exams in week beginning 28 Sept 2009

26 August 2009

9 September 2009

Other CT subjects

2 September 2009

16 September 2009

Mock Exams

We suggest that you work to the recommended submission dates where possible. Please remember that the turnaround of your script is likely to be quicker if you submit it well before the final deadline date.
At the time of going to print, the Profession had not confirmed the order of the 2009 exams. An up-todate version of these Assignment deadlines, showing the specific subjects in each sub-group, is available on our website at www.ActEd.co.uk.

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CT1: Study Guide

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9

Page 37

File tabs
You might want to use the tabs printed below to label your course files. You will only need two course files for the Combined Materials Pack but we have given you nine tabs so that you can choose which ones to cut out in order to label your files.

CT1

CT1

CT1

Financial
Mathematics

Financial
Mathematics

Financial
Mathematics

2009 Exams

2009 Exams

2009 Exams
Questions

CT1

CT1

CT1

Financial
Mathematics

Financial
Mathematics

Financial
Mathematics

2009 Exams
File 1

2009 Exams
File 2

2009 Exams
Notes

CT1

CT1

CT1

Financial
Mathematics

Financial
Mathematics

Financial
Mathematics

2009 Exams
Q&A Bank

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n

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CT1: Study Guide

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CT1: Index

Page 1

CT1 Index
Accumulated profit.............................................................. Ch10
Accumulation factor ............................................................ Ch3
Annual percentage rate (APR)............................................. Ch9
Annuities
Accumulated value .................................................. Ch6
Certain ..................................................................... Ch1
Ch6
In advance................................................................ Ch6
In arrears .................................................................. Ch6
Non-integer n ......................................................... Ch6
Payable continuously............................................... Ch6
Payable forever ........................................................ Ch6
Payable pthly, accumulated value ........................... Ch6
Payable pthly, present value, p < 1 ......................... Ch6
Payable pthly, present value, p > 1 ......................... Ch6
Arbitrage.............................................................................. Ch13

p5 p4 p19 p8 p10 p1 p6 p3 p18 p10 p22 p16 p18 p12 p3

Bond future.......................................................................... Ch11 p25
Bonds ................................................................................. Ch11 p2
Borrowing and lending at different rates of interest............ Ch10 p15
Capital gains, indexation ..................................................... Ch12
Capital gains tax .................................................................. Ch12
Capital outstanding
Loans ....................................................................... Ch9
Loans, prospective ................................................... Ch9
Loans, retrospective................................................. Ch9
Capital repaid, loans ............................................................ Ch9
Cashflow.............................................................................. Ch1
Cash on deposit.................................................................... Ch1
Certificates of deposit.......................................................... Ch11
Commercial discount........................................................... Ch2
Compound increasing annuities .......................................... Ch7
Compound interest............................................................... Ch2
Continuously payable cashflows ......................................... Ch5
Continuous time forward rates ............................................ Ch14
Continuous time spot rates .................................................. Ch14
Convertible pthly
Rates of interest ....................................................... Ch3
Ch3
Rates of discount ..................................................... Ch3

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Page 2

CT1: Index

Convertibles......................................................................... Ch11
Corporate bonds................................................................... Ch11
Convexity............................................................................. Ch14
Coupon................................................................................. Ch11
Currency future.................................................................... Ch11
Currency swap ..................................................................... Ch11

p17 p9 p24 p2 p27 p30 Debentures ........................................................................... Ch11
Decreasing annuity .............................................................. Ch7
Deferred annuity .................................................................. Ch7
Deferred income tax ........................................................... Ch12
Deflation .............................................................................. Ch4
Derivatives........................................................................... Ch11
Discounted mean term ......................................................... Ch14
Discounted payback period, DPP ........................................ Ch10
Dividends ............................................................................ Ch11
Duration .............................................................................. Ch14

p10 p13 p2 p12 p5 p22 p21 p17 p14 p21 Effective duration ................................................................ Ch14
Effective rates of interest, conversion ................................. Ch2
Equation of value................................................................. Ch8
Equity ................................................................................. Ch1
Equity, pricing ..................................................................... Ch12
Eurobonds ............................................................................ Ch11
Ex-dividend ......................................................................... Ch12
Expectations theory ............................................................. Ch14

p20 p11 p1 p8 p13 p11 p14 p15 Fixed interest government bond .......................................... Ch11
Fixed interest rate model ..................................................... Ch15
Fixed interest security
Definition ................................................................. Ch1
Price ......................................................................... Ch12
Flat rate of interest ............................................................... Ch9
Flat yield .............................................................................. Ch12
Force of interest
Constant ................................................................... Ch3
Dependent on t ........................................................ Ch3
Forward contract.................................................................. Ch13
Forward contract, value ....................................................... Ch13
Forward price
Security with fixed cash income .............................. Ch13
Security with known dividend yield ........................ Ch13
Security with no income .......................................... Ch13
Forward rates, instantaneous ............................................... Ch14

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CT1: Index

Page 3

Forward rates of interest...................................................... Ch14
Forward rates and spot rates, connection ............................ Ch14
Fund value .......................................................................... Ch10
Ch10
Futures, definition................................................................ Ch11

p5 p6 p27 p30 p22

Government bills ................................................................. Ch11
Government bonds............................................................... Ch11
Gross redemption yield........................................................ Ch12
Gross yield........................................................................... Ch12

p7 p2 p5 p5 Hedging ............................................................................... Ch13 p14
Immunisation....................................................................... Ch14
Increasing annuity ............................................................... Ch7
Increasing annuity, continuously payable ........................... Ch7
Index linked bond ................................................................ Ch11
Index linked bond, calculations........................................... Ch12
Index linked security ........................................................... Ch1
Inflation ............................................................................... Ch1
Ch12
Instantaneous forward rate ................................................. Ch14
Interest only loan ................................................................. Ch1
Interest payable on a loan .................................................... Ch9
Interest rate swap................................................................. Ch11
Internal rate of return........................................................... Ch10
Interpolation ........................................................................ Ch8

p27 p6 p10 p5 p30 p6 p6 p30 p10 p11 p10 p29 p9 p10 Law of one price.................................................................. Ch13
Lending and borrowing at different rates of interest ........... Ch10
Linked internal rate of return............................................... Ch10
Liquidity preference theory ................................................. Ch14
Loans ................................................................................. Ch9
Loans repaid pthly ............................................................... Ch9
Loan schedule...................................................................... Ch9
Lognormal model
Fixed rate model ...................................................... Ch15
Variable rate model ................................................. Ch15

p4 p15 p32 p15 Macauley duration ............................................................... Ch14
Market segmentation theory ................................................ Ch14
Money rate of interest.......................................................... Ch4
Money weighted rate of return (MWRR)............................ Ch10
Mortgage.............................................................................. Ch1

p21 p16 p2 p27 p11

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CT1: Index

Net cashflow ........................................................................ Ch10
Net present value (NPV)...................................................... Ch10
Net redemption yield ........................................................... Ch12
No arbitrage assumption ..................................................... Ch13
Nominal amount of bond .................................................... Ch11
Nominal rates of interest...................................................... Ch3

p3 p7 p5 p4 p2 p2 Optional redemption dates................................................... Ch12 p9
Options................................................................................. Ch11 p28
Ordinary shares.................................................................... Ch11 p14
Par yield ............................................................................... Ch14
Payback period .................................................................... Ch10
Payment streams .................................................................. Ch5
Perpetuity............................................................................. Ch12
Ch6
Preference shares ................................................................. Ch11
Present value........................................................................ Ch2
Ch5
Price of bond ....................................................................... Ch12
Profit (see accumulated profit)
Project appraisal ................................................................. Ch10
Property ............................................................................... Ch11
Property, valuation............................................................... Ch12

p17 p17 p15 p17 p23 p16 p6 p7 p3

Rate of discount ................................................................... Ch3
Real rate of interest
Calculation ............................................................... Ch12
Definition ................................................................. Ch4
Real yield
Constant inflation..................................................... Ch12
Inflation index.......................................................... Ch12
Redemption ......................................................................... Ch11
Ch12
Redemption at par ............................................................... Ch11
Redemption yield................................................................. Ch12
Redington’s conditions ........................................................ Ch14
Repayment loan ................................................................... Ch1
Risk discount rate ................................................................ Ch10
Risk free force of interest .................................................... Ch13
Running yield
Property.................................................................... Ch11
Securities.................................................................. Ch12

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Short interest rate futures..................................................... Ch11
Simple discount ................................................................... Ch2
Simple interest ..................................................................... Ch2
Spot rate of interest.............................................................. Ch14
Spot rates and forward rates, connection............................. Ch14
Stock index futures .............................................................. Ch11
Swaps ................................................................................. Ch11

p26 p8 p3 p3 p6 p27 p29

Tax (income) ....................................................................... Ch11
Ch12
Tax (capital gains) .............................................................. Ch12
Tax on bonds ....................................................................... Ch12
Term structure of interest rates............................................ Ch14
Term to redemption (effect on yield) ................................. Ch12
Time weighted rate of return (TWRR)................................ Ch10

p4 p4 p33 p4 p4 p8 p29

Uncertain payments ............................................................. Ch8 p14
Unsecured loan stock........................................................... Ch11 p11
Value of a forward contract................................................. Ch13
Varying interest rate model
Annual payment....................................................... Ch15
Single payment ........................................................ Ch15
Volatility.............................................................................. Ch14

p10 p7 p20

Yield ................................................................................. Ch8
Yield curve .......................................................................... Ch14
Yield on bonds .................................................................... Ch12
Yield to redemption............................................................. Ch12

p3 p4 p5 p5 Zero coupon bond................................................................ Ch1

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CT1-01: Cashflow models

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Chapter 1
Cashflow models
Syllabus objective
(i)

Describe how to use a generalised cashflow model to describe financial transactions. 1.

2.

0

For a given cashflow process, state the inflows and outflows in each future time period and discuss whether the amount or the timing (or both) is fixed or uncertain.
Describe in the form of a cashflow model the operation of a zero-coupon bond, a fixed-interest security, an index-linked security, cash on deposit, an equity, an “interest only” loan, a repayment loan, and an annuity certain. Introduction
A cashflow model is a mathematical projection of the payments arising from a financial transaction, eg a loan, a share or a capital project. Payments received are referred to as income and are shown as positive cashflows. Payments made are referred to as outgo and are shown as negative cashflows. The difference at a single point in time (income less outgo) is called the net cashflow at that point in time.
This chapter considers the cashflows that emerge in a number of practical situations that you will come across in the actuarial field.

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CT1-01: Cashflow models

Cashflow process
The practical work of the actuary often involves the management of various cashflows. These are simply sums of money, which are paid or received at different times. The timing of the cashflows may be known or uncertain. The amount of the individual cashflows may also be known or unknown in advance.
For example, a company operating a privately owned bridge, road or tunnel will receive toll payments. The company will pay out money for maintenance, debt repayment and for other management expenses. From the company’s viewpoint the toll payments are positive cashflows (ie money received) while the maintenance, debt repayments and other expenses are negative cashflows (ie money paid out). Similar cashflows arise in all businesses.
From a theoretical viewpoint one may also consider a continuously payable cashflow. The theory of continuously payable cashflows is often used when cashflows are paid very frequently, eg daily or weekly. The mathematics used to investigate the cashflows is sometimes easier if we assume that the payments are made continuously rather that at regular intervals. This will become clearer when this mathematics is introduced to you later in the course.

Question 1.1
Describe two cashflows, one positive and one negative, that will occur in the next month where one of the parties involved in the cashflow is (i) you, (ii) your employer.
In some businesses, such as insurance companies, investment income will be received in relation to positive cashflows (premiums) received before the negative cashflows (claims and expenses).

For example, consider a premium received by an insurance company from a policyholder. Some of the premium might be used to cover the costs associated with setting up the insurance policy. The remainder of the premium could be put into a bank account. Investment income, in this case interest, will be earned on the money in the bank account until the money is needed for further expenses or payments back to the policyholder. Where there is uncertainty about the amount or timing of cashflows, an actuary can assign probabilities to both the amount and the existence of a cashflow. In this subject we will assume that the existence of the future cashflows is certain.

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The amount and timing of some cashflows will be known with great certainty. An employed person who gets paid on the last Friday of every month will be almost certain to receive a payment on the last Friday of this month. The amount of the payment is also likely to be known.
However, other cashflows are not so certain. If you buy a lottery ticket every week, you don’t know when, or if, you will win or how much you may win.
The probability that the payment will take place could be estimated by looking at past results. If there are no past data relating to the event being considered, then data from similar events would be used.

Question 1.2
The example above stated that the employed person will be almost certain to receive a payment on the last Friday of this month. Why is the person not completely certain?

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CT1-01: Cashflow models

Examples of cashflow scenarios
In this section some simple examples are given of practical situations which are readily described by cashflow models.

The first few examples are types of security or investment. A security is a tradeable financial instrument, ie a financial contract that can be bought and sold.

2.1

A zero-coupon bond
The term “zero-coupon bond” is used to describe a security that is simply a contract to provide a specified lump sum at some specified future date. For the investor there is a negative cashflow at the point of investment and a single known positive cashflow on the specified future date.

For example the investor may give the issuer of the zero-coupon bond £400,000, and in return the investor will receive £500,000 from the issuer in exactly 5 years’ time. The issuer may be a government or a large company.
The positive cashflow is paid on a set date and is of a set amount, but it is not certain that the payment will be made. There is a chance that the issuing organisation will not make the payment, ie that it will default. This risk is usually negligible for bonds issued by governments of developed countries, since the government can always raise taxes.
The risk of default is greater for issuing organisations that may go bust, eg companies.
You can think of a zero-coupon bond as a loan from the investor to the issuer. The loan is repaid by one single payment of a fixed amount at a fixed date in the future. It is a special case of a fixed-interest security with no interest payments before redemption.
We will study fixed-interest securities in the next section.
We can plot the cashflows of the investor on a timeline:
Cashflows

–£400,000

Time

0

+£500,000
5

Question 1.3
Describe the cashflows for an organisation that issues a zero-coupon bond.

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CT1-01: Cashflow models

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These cashflows could be shown on a timeline as:
Cashflows
Time

£400,000
0

–£500,000
5

The investor may also be referred to as the lender, and the issuer may be referred to as the borrower.

2.2

A fixed interest security
A body such as an industrial company, a local authority, or the government of a country may raise money by floating a loan on the stock exchange.

This means that the organisation borrows money by issuing a loan to investors. The loan is simultaneously listed on the stock exchange so that after issue the securities can be traded on the stock exchange. This means that investors can sell their right to receive the future cashflows.
In many instances such a loan takes the form of a fixed-interest security, which is issued in bonds of a stated nominal amount. The characteristic feature of such a security in its simplest form is that the holder of a bond will receive a lump sum of specified amount at some specified future time together with a series of regular level interest payments until the repayment (or “redemption”) of the lump sum.

The regular level interest payments are referred to as coupons. Thus a zero-coupon bond has no interest payments.
The investor has an initial negative cashflow, a single known positive cashflow on the specified future date, and a series of smaller known positive cashflows on a regular set of specified future dates.

An investor might buy a 20-year fixed-interest security of nominal amount £10,000.
This means that the face value of the loan is £10,000. The investor is unlikely to pay exactly £10,000 for this security but will pay a price that is acceptable to both parties.
This may be higher or lower than £10,000. The investor will then receive a lump sum payment in 20 years’ time. This lump sum is most commonly equal to the nominal amount, in this case £10,000, but could be a pre-specified amount higher or lower than this. The investor will also receive regular payments throughout the 20 years of, say,
£500 pa. These regular payments could be made at the end of each year or half-year or at different intervals.

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CT1-01: Cashflow models

We can again represent the cashflows of the investor on a timeline.
–£price +£500 +£500 +£500
0

1

2

3



+£500 +£10,500
19

20

In this case the payments are made at the end of each year. The last payment is made up of the final regular payment (£500) and the lump sum payment (£10,000).

2.3

An index-linked security
Inflation is a measure of the rate of change in the price of goods and services, including salaries. High inflation implies that prices are rising quickly and low inflation implies that prices are rising slowly.
If CDs cost £10 each then £50 could be used to buy 5 CDs. However, if inflation was high, then the cost of CDs in 1 year’s time might be £12.50. £50 would then only buy 4
CDs. This simple example shows how the “purchasing power” of a given sum of money, ie the quantity of goods that can be bought with the money, can diminish if inflation is high. In this case inflation was 25% over the year.
With a conventional fixed interest security, the interest payments are all of the same amount. If inflationary pressures in the economy are not kept under control, the purchasing power of a given sum of money diminishes with the passage of time, significantly so when the rate of inflation is high. For this reason some investors are attracted by a security for which the actual cash amount of interest payments and of the final capital repayment are linked to an
“index” which reflects the effects of inflation.
Here the initial negative cashflow is followed by a series of unknown positive cashflows and a single larger unknown positive cashflow, all on specified dates.
However, it is known that the amounts of the future cashflows relate to the inflation index. Hence these cashflows are said to be known in “real” terms.

Real terms means taking into account inflation, whereas nominal means ignoring inflation. For example if your wages are rising at 5% pa and inflation is 7% pa, your wages are falling in real terms (you will be able to buy less with your “higher wages”), even though your wages are rising in nominal terms.

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Question 1.4
Are the payments on fixed-interest securities known?
Both the regular payments and the final payments on an index linked security are linked to the inflation index. If inflation is high, then the regular payments will rise by larger amounts than if inflation is low.
If inflation is 10% per time period and the regular coupon after one time period is for
£500, then the payment after two time periods will be £550 ( 500 × 1.1 ), and the payment after three time periods will be £605 etc.
Inflation is often measured by reference to an index. For example an inflation index might take values as set out in the table below.

Date
Index

1.1.2001
100.00

1.1.2002
105.00

1.1.2003
108.00

1.1.2004
113.00

The rate of inflation during 2002 is 2.86% pa (ie 108 105 − 1 ).

Question 1.5
An investor purchased a three-year index-linked security on 1.1.2001. In return the investor received payments at the end of each year plus a final redemption amount, all of which were increased in line with the index given in the table above. The payments would have been £600 each year and £11,000 on redemption if there had been no inflation. Calculate the payments actually received by the investor.
Note that in practice the operation of an index-linked security will be such that the cashflows do not relate to the inflation index at the time of payment, due to delays in calculating the index. It is also possible that the need of the borrower
(or perhaps the investors) to know the amounts of the payments in advance may lead to the use of an index from an earlier period.

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CT1-01: Cashflow models

Question 1.6
Repeat Question 1.5 for a two-year index-linked security issued on 1.1.2002. The payments increase in line with the index with a one-year indexation lag, ie the index value one year before each payment is used.

2.4

Cash on deposit
If cash is placed on deposit, the investor can choose when to disinvest and will receive interest additions during the period of investment. The interest additions will be subject to regular change as determined by the investment provider.
These additions may only be known on a day-to-day basis. The amounts and timing of cashflows will therefore be unknown.

The Core Reading is describing a bank account that pays interest and allows instant access. Consider your own bank account. You can choose when to invest money, ie pay money in, and when to disinvest money, ie withdraw money. The interest you receive on your money will depend on the current interest rate and this may change with little or no notice.
This type of deposit is called a call deposit. Another type of deposit is a term deposit.
Term deposits are when the money is deposited for a fixed term usually for between one week and one year. The interest rate can be fixed for the term, or vary at specified intervals. Term deposits are not negotiable, ie you can’t sell a deposit to a third party.
The investor is committed until the end of the specified term, although if the investor had to have the funds back, the bank might agree (at a price!).

2.5

An equity
Equity shares (also known as “shares” or “equities” in the UK and as “common stock” in the USA) are securities that are held by the owners of an organisation.
Equity shareholders own the company that issued the shares. For example if a company issues 4,000 shares and an investor buys 1,000, the investor owns
25 per cent of the company. In a small company all the equity shares may be held by a few individuals or institutions. In a large organisation there may be many thousands of shareholders.
Equity shares do not earn a fixed rate of interest as fixed-interest securities do.
Instead the shareholders are entitled to a share in the company’s profits, in proportion to the number of shares owned.

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The distribution of profits to shareholders takes the form of regular payments of dividends. Since they are related to the company profits that are not known in advance, dividend rates are variable. It is expected that company profits will increase over time. It is therefore expected also that dividends per share will increase – though there are likely to be fluctuations. This means that in order to construct a cashflow schedule for an equity it is necessary first to make an assumption about the growth of future dividends. It also means that the entries in the cashflow schedule are uncertain – they are estimates rather than known quantities. In practice the relationship between dividends and profits is not a simple one.
Companies will, from time to time, need to hold back some profits to provide funds for new projects or expansion. Companies may also hold back profits in good years to subsidise dividends in years with poorer profits. Additionally, companies may be able to distribute profits in a manner other than dividends, such as by buying back the shares issued to some investors.

Share buy-backs will result in some investors having to sell their shares back to the company. The remaining shareholders will subsequently own a greater percentage of the company and should expect greater future profits.
The following table shows the projected future cashflows for a shareholder who has just purchased a block of shares for £6,000 and expects the dividends in each year to be 5% higher than the corresponding amounts in the previous year. Dividends are paid twice yearly. This shareholder expects the two dividends in the first year to be £100 each and intends to sell all the shares after 2 years.
In the table the time is measured from the date of purchase.
Time (years)
0
½
1

2

Purchase price (£)
–6,000

Dividends (£)

Sale proceeds (£)

+100
+100
+105
+105

+6,615

In this example we have assumed that the price also grows at 5% pa.

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CT1-01: Cashflow models

Since equities do not have a fixed redemption date, but can be held in perpetuity, we may assume that dividends continue indefinitely (unless the investor sells the shares or the company buys them back), but it is important to bear in mind the risk that the company will fail, in which case the dividend income will cease and the shareholders would only be entitled to any assets which remain after creditors are paid. The future positive cashflows for the investor are therefore uncertain in amount and may even be lower, in total, than the initial negative cashflow. Perpetuity means that payments continue forever.

Question 1.7
Complete the table below using the symbols:

Contract

(= yes), × (= no) or ? (= sometimes).

Absolute amount of payments known in advance? Timing of payments known in advance?

Zero-coupon bond
Fixed-interest security
Index-linked security
Call deposit
Equity

2.6

An annuity certain
An annuity certain provides a series of regular payments in return for a single premium (ie a lump sum) paid at the outset. The precise conditions under which the annuity payments will be made will be clearly specified. In particular, the number of years for which the annuity is payable, and the frequency of payment, will be specified. Also, the payment amounts may be level or might be specified to vary – for example in line with an inflation index, or at a constant rate.
The cashflows for the investor will be an initial negative cashflow followed by a series of smaller regular positive cashflows throughout the specified term of payment. In the case of level annuity payments, the cashflows are similar to those for a fixed-interest security.

However there will not be a redemption payment as there normally is for a fixedinterest security.

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From the perspective of the annuity provider, there is an initial positive cashflow followed by a known number of regular negative cashflows.

Annuity policyholders are usually called annuitants.
In the Subject CT5, Contingencies, the theory of this subject will be extended to deal with annuities where the payment term is uncertain, that is, for which payments are made only so long as the annuity policyholder survives.

An example of this is a whole life annuity where a policyholder receives a fixed amount of money per month until they die. In more common language this type of policy is a pension! 2.7

An “interest-only” loan
An “interest-only” loan is a loan that is repayable by a series of interest payments followed by a return of the initial loan amount.

So you still owe the capital amount that you borrowed throughout the term of the loan.
In the simplest of cases, the cashflows are the reverse of those for a fixed interest security. The provider of the loan effectively buys a fixed interest security from the borrower.
In practice, however, the interest rate need not be fixed in advance. The regular cashflows may therefore be of unknown amounts.
It may also be possible for the loan to be repaid early. The number of cashflows and the timing of the final cashflows may therefore be uncertain.

2.8

A repayment loan (or mortgage)
A repayment loan is a loan that is repayable by a series of payments that include partial repayment of the loan capital in addition to the interest payments.
In its simplest form, the interest rate will be fixed and the payments will be of fixed equal amounts, paid at regular known times.
The cashflows are similar to those for an annuity certain.
As for the “interest-only” loan, complications may be added by allowing the interest rate to vary or the loan to be repaid early. Additionally, it is possible that the regular repayments could be specified to increase (or decrease) with time.
Such changes could be smooth or discrete.

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CT1-01: Cashflow models

Each payment can be considered to consist of an interest payment and a capital repayment. The interest payment covers the interest that will be charged over the period since the previous payment. The interest payment will be calculated by reference to the amount of the loan outstanding just after the last payment. The remainder of each payment will be used to reduce the amount of the loan outstanding.
It is important to appreciate that with a repayment loan the breakdown of each payment into “interest” and “capital” changes significantly over the period of the loan. The first repayment will consist almost entirely of interest and will provide only a very small capital repayment. In contrast, the final repayment will consist almost entirely of capital and will have a small interest content.

This is because the amount of the loan outstanding will reduce throughout the term of the loan. At the start of the contract the entire loan will be outstanding and so the interest portion of the payment will be large. The remainder, the capital portion, will therefore be relatively small. At the end of the contract the amount of the loan outstanding will be small and so the interest due will also be small. The capital repayment will then be much larger.
The diagram below shows how the capital outstanding reduces for a repayment loan.
The graph is based on a loan of £50,000 being repaid by monthly instalments.

Capital outstanding
50,000
40,000
30,000
20,000
10,000
0

We’ll look at numerical examples of repayment loans in Chapter 9.

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Specimen exam questions
At the end of many of the chapters we have included a specimen exam question for you to attempt. The first one of these is in Chapter 5; thereafter they appear in more and more chapters as you cover an increasing proportion of the course material.
We suggest two different ways that you might wish to use these questions to help you progress through the course:
(1)

You could attempt the questions as soon as you reach them in your studies. You may find them quite difficult on the first attempt, especially the early ones, and we would expect you to refer back to the course notes in order to answer them.
By tackling them as you go through the course, however, you will get to know more quickly the level you need to be aiming for in order to pass the Subject
CT1 exam. But you should not be worried if your answers appear far from perfect on these first attempts.

(2)

Alternatively you could miss them out until you get to the end of Part 3. At this point you should be aiming to tackle a good sample of questions from the
Question and Answer Bank prior to attempting the relevant assignment, as usual.
Immediately before the assignment you could go back and attempt all the specimen exam questions from these first three parts, which should help your preparation for tackling the assignment.
Repeat this at the end of Part 4, using all the specimen questions in that part.

Whichever of these you follow, you are likely to benefit from a fresh second attempt at these questions as part of your revision. On these second attempts you should be looking to do the questions under exam conditions, and strictly within the time available according to the number of marks for the question (remember that 1 mark = 1.8 minutes of exam time!). We suggest you don’t try all the questions in one sitting but tackle them one at a time once you feel you have fully revised the topics involved.
Of course, you may wish to use the questions in other ways: the above are just suggestions! The Actuarial Education Company

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Chapter 1 Summary
Cashflows are simply sums of money that are paid or received at particular times.
Where there is uncertainty about the amount or timing of cashflows, an actuary can assign probabilities to both the amount and the existence of a cashflow.
A zero-coupon bond is a security that provides a specified lump sum at a specified future date and no other positive cashflows.
The holder of a fixed-interest security will receive a lump sum, of specified amount at a specified future time, together with a series of regular level interest payments until the repayment of the lump sum.
With a call deposit the amounts and timing of cashflows will usually be unknown.

Equity shares are securities that are held by the owners of an organisation.
Shareholders are entitled to a share in the company’s profits in proportion to the number of shares owned.
An annuity certain provides a series of regular payments in return for a single premium.
An interest-only loan is a loan that is repayable by a series of interest payments followed by a return of the initial loan amount.
A repayment loan is a loan that is repayable by a series of payments, each including a partial repayment of the loan capital in addition to the interest payments.

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Chapter 1 Solutions
Solution 1.1
(i)

Cashflows involving you

Positive: receiving your salary, borrowing some money from a friend.
Negative: repaying some borrowed money, buying something from a shop using cash.
(ii)

Cashflows involving your employer

Positive: receiving payments for supplying products or services, eg receiving premiums.
Negative: paying salaries, paying property expenses, eg electricity bills.

Solution 1.2
The payment might not be made if:


the employer makes an error or goes bankrupt



the bank makes an error and doesn’t make the payment on the due date



the person leaves the company before the end of the month



the last Friday is a public holiday and other arrangements are in place.

Solution 1.3
For the issuing organisation there is a positive cashflow at the point of investment and a single known negative cashflow on a specified future date.
Cashflows

+£price

Time

0

–£redemption n Solution 1.4
The payments are known in nominal terms but are unknown in real terms.

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CT1-01: Cashflow models

Solution 1.5
Date

Amount

Calculation
105
600 ×
100
108
600 ×
100

1.1.2002

£630

1.1.2003

£648

1.1.2004

£13,108

(11,000 + 600) ×

Date

Amount

1.1.2003

£630

Calculation
105
600 ×
100

1.1.2004

£12,528

113
100

Solution 1.6

(11,000 + 600) ×

108
100

Solution 1.7

Contract
Zero-coupon bond
Fixed-interest security
Index-linked security
Call deposit
Equity

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Absolute amount of payments known in advance? ×
×
×

Timing of payments known in advance?

×
×

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Chapter 2
The time value of money
Syllabus objective
(ii)

Describe how to take into account the time value of money using the concepts of compound interest and discounting.
1.

Accumulate a single investment at a constant rate of interest under the operation of:
– simple interest
– compound interest.

2.
3.

Discount a single investment under the operation of simple (commercial) discount at a constant rate of discount.

4.

0

Define the present value of a future payment.

Describe how a compound interest model can be used to represent the effect of investing a sum of money over a period.

Introduction
Interest may be regarded as a reward paid by one person or organisation (the borrower) for the use of an asset, referred to as capital, belonging to another person or organisation (the lender).

In return for the use of the investor’s capital, the borrower will be expected to pay interest to the lender. For example:


A bank will be expected to pay interest to its customers on money held in their savings accounts.



A company will be expected to pay interest to a bank on money lent to them for a business project.

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In this chapter we will look at the basic ideas underlying the theory of interest rates.
Interest rates are a fundamental part of actuarial work. We will need to make assumptions about interest rates in almost every calculation in this course.

Question 2.1
In what situations does a bank act as (i) a lender and (ii) a borrower?
Capital and interest need to be measured in terms of the same commodity, but when expressed in monetary terms, capital is also referred to as principal.
If there is some risk of default (ie loss of capital or non-payment of interest) a lender would expect to be paid a higher rate of interest than would otherwise be the case.

If you lent money to the US government (by purchasing a fixed interest security) and a property developer then you will probably demand a higher rate of interest from the property developer. This is because the property developer is more likely not to repay the loan or not pay interest due on the loan.
Another factor which may influence the rate of interest on any transaction is an allowance for the possible depreciation or appreciation in the value of the currency in which the transaction is carried out. This factor is obviously very important in times of high inflation.

If the current Euro / Yen exchange rate is 1 Euro = 150 Yen, but a few weeks later
1 Euro = 175 Yen, then because the value of the Euro has risen we say that the Euro has appreciated against the Yen. The discussion of the link between exchange rates, inflation and interest rates is covered in Subject CT7.
The most elementary concept is that of simple interest. This leads naturally to the idea of compound interest, which is much more commonly found in practice
– at least in relation to all but short-term investments. Both concepts are easily described within the framework of a savings account.

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Simple interest
If an amount C is deposited in an account which pays simple interest at the rate of i × 100% per annum and the account is closed after n years – there being no intervening payments to or from the account – then the amount paid to the investor when the account is closed will be

C (1 + ni )

(1.1)

This payment consists of a return of the initial deposit C, together with interest of amount

niC

(1.2)

So if an investor deposits £4,000 in a bank account that pays simple interest at a rate of
6% pa, then after 8 years there will be £5,920 ( 4, 000 × (1 + 0.06 × 8) ) in the account.
In our discussion so far we have implicitly assumed that, in each of these last two expressions, n is an integer. However, the normal commercial practice in relation to fractional periods of a year is to pay interest on a pro rata basis, so that expressions (1.1) and (1.2) may be considered as applying for all nonnegative values of n.
The essential feature of simple interest, as expressed algebraically by expression (1.1), is that interest, once credited to an account, does not itself earn further interest. This leads to inconsistencies which are avoided by the application of compound interest theory.

The following question demonstrates these inconsistencies.

Question 2.2
An investor puts £5,000 in a savings account that pays 10% simple interest at the end of each year. Compare how much the investor would have after 6 years if the money was:
(i)

invested for 6 years

(ii)

invested for 3 years, then immediately reinvested for a further 3 years.

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Compound interest
The essential feature of compound interest is that interest itself earns interest.
The operation of compound interest may be described as follows. Consider a savings account, which pays compound interest at rate i per annum, into which is placed an initial deposit C. (We assume that there are no further payments to or from the account.) If the account is closed after one year, the investor will receive C (1 + i ) . More generally, let An be the amount which will be received by the investor if the account is closed after n years.

Thus A1 = C (1 + i ) .

By

definition, the amount received by the investor on closing the account at the end of any year is equal to the amount which would have been received, if the account had been closed one year previously, plus further interest of i times this amount. Thus the interest credited to the account up to the start of the final year itself earns interest (at rate i per annum) over the final year.
Expressed algebraically, the above definition becomes:

An + 1 = An + iAn or: An + 1 = (1 + i )An

n≥1

(2.1)

Since (by definition):

A1 = C (1 + i ) equation (2.1) implies that, for n = 1,2,… :

An = C (1 + i )n

(2.2)

Thus, if the investor closes the account after n years, the amount received will be: C (1 + i )n
This payment consists of a return of the initial deposit, C, together with accumulated interest (ie interest which, if n > 1 , has itself earned further interest) of amount:

C [(1 + i )n − 1]

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So if an investor deposits £4,000 in a bank account that pays compound interest at a rate of 6% pa, then after 8 years there will be £6,375 ( 4,000 × 1068 ) in the account.
.

Question 2.3
Repeat Question 2.2 but with the savings account paying 10% compound interest at the end of each year.
The following graph shows how £1 invested in a savings account would grow over the next few years if the account paid either 15% pa simple interest (solid line) or 15% pa compound interest (broken line).

Amount

6
4
2
0
0

1

2

3

4

5

6

7

8

9

10

Y ear

Note that the simple interest account produces a straight-line graph whereas the graph of the compound interest account is exponentially shaped.

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Present values
In the previous section we saw how to answer the question: How much will a single payment accumulate to at a later time? In actuarial work, we are usually aiming to make payments at certain future dates, eg making pension payments to a worker after retirement or making a life assurance payment when an individual dies. So actuaries are usually more interested in answering the question: How much do we need now to provide payments at a later time? This amount is called the present value (PV) or discounted value of the payments.
Example
An investor must make a payment of £5,000 in 5 years’ time. The investor wishes to make provision for this payment by investing a single sum now in a deposit account that pays 10% per annum compound interest. How much should the initial investment be?
By the end of 5 years an initial payment of £X will have accumulated to: £ X (1 + 01)5
.
So:

X × 115 = 5,000
.

ie X = 5,000 / 115 = £3,105
.

We will now consider the more general case:
Let t1 ≤ t 2 .

It follows by formula (2.2) that an investment of C ( At 2 − t1 C ) , ie

C / (1 + i )t 2 − t1 at time t1 will produce a return of C at time t 2 . We therefore say that the discounted value at time t1 of C due at time t 2 is:
C / (1 + i )t 2 − t1
This is the sum of money which, if invested at time t1 , will give C at time t 2 . In particular, the discounted value at time 0 (the present time) of C due at time t ≥ 0 is called its discounted present value (or, more briefly, its present value); it is equal to:

C / (1 + i )t

(3.1)

We now define the function:

v = 1 / (1 + i )

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(3.2)

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It follows by formulae 3.1 and 3.2 that the discounted present value of C due at a non-negative time t is:

Cv t

Using this notation, we could have found the answer to the last example by calculating:

v=

1
1
=
= 0.90909
1 + i 11
.

So:

PV = 5,000v 5 = 5,000 × 0.90909 5 = 5,000 × 0.62092 = £3,105

Formulae and tables for actuarial examinations
Values of v n at various interest rates are tabulated in “Formulae and Tables for
Examinations”. You will need a copy if you haven’t yet got one. They are available from The Institute of Actuaries or The Faculty of Actuaries. From now on we will refer to this book as simply the Tables.

Question 2.4
Calculate v, assuming an effective annual rate of interest of 4%.

Note on rounding
How to round your answers is an important factor that you need to consider in your calculations. The rounding used should be appropriate for the level of accuracy used in your calculations. For example, if you are using numbers from the Tables then you shouldn’t quote an answer to more significant figures than given in the Tables. The number of significant figures that you quote is usually more important than the number of decimal places. When using your calculator you can keep the accurate values of intermediate calculations in your memories. Your final answer will then be more accurate. In an exam you need not write down the fully accurate intermediate figures.
Remember that 104.27 is rounded to 5 significant figures, but 2 decimal places!

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Simple discount
As has been seen with simple interest, the rate of simple interest is not itself subject to further interest.
Similarly with simple discount, which is defined as follows:
Suppose an amount C is due after n years and a rate of simple discount applies of d per annum.
Then the sum of money required to be invested now (to amount to C after n years) is C (1 − nd ) .
In normal commercial practice, d is usually encountered only for periods of less than a year. If a lender bases his short-term transactions on a simple rate of discount d then, in return for a repayment of X after a period t ( t < 1 ) he will lend
X (1 − td ) at the start of the period. In this situation, d is also known as a rate of
“commercial discount”.

There are no new concepts involved in rates of commercial discount. They are just a convenient way of expressing the amount of the repayment required.

Example
An 8-month loan, repayable by a single repayment, is issued at a rate of commercial discount of 15% per annum. If the amount of the repayment is £100,000, how much was initially lent to the borrower?
The amount lent was:

100,000 (1 −

8
× 015) = £90,000
.
12

Note that since the 15% was quoted per annum, we use

8
12

as the length of time, ie 8

months written in years.

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Question 2.5
A company is due to receive a payment of £500,000 from a customer in 6 months’ time.
To smooth its cashflows, the company would prefer to receive the payment immediately, and has agreed to transfer its entitlement to this payment to a third party (a discount house) in return for an immediate payment calculated using a rate of commercial discount of 16% per annum.
How much will the immediate payment made by the discount house be?

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CT1-02: The time value of money

Investing over a period
We begin by considering investments where capital and interest are paid at the end of a fixed term, there being no intermediate interest or capital payments.
It is essential in any compound interest problem to define the unit of time. This may be, for example, a month or a year, the latter period being frequently used in practice. In certain situations, however, it is more appropriate to choose a different period (eg six months) as the basic time unit.

For example, if £200 is invested in a savings account that pays compound interest at a rate of 2% per half-year then the amount in the account at the end of 18 months is
200 × 1023 = £212.24 .
.
Consider an investment of 1 for a period of 1 time unit, commencing at time t, and suppose that 1 + i (t ) is returned at time t + 1 . One sometimes refers to i(t) as the “effective rate of interest” for the period, to distinguish it from “nominal” and
“flat” rates of interest which will be discussed later.

The effective rate of interest over a given time period is the amount of interest a single initial investment will earn at the end of the time period, expressed as a proportion of the initial amount. It is denoted by the symbol i(t) or more commonly just i when i(t) is constant for all t.

Example
An investor makes an initial investment of £5,000 and is credited with £500 interest at the end of the year. What is the effective rate of interest and the value of i ?
Here the effective rate of interest is 10% per annum and i = 0.1 (per annum). This is often quoted as 10% pa effective.
If it is assumed that the rate of interest does not depend on the amount invested, the cash returned at time t + 1 from an investment of C at time t is C [1 + i (t )] .
It may easily be seen that the accumulation of C from time 0 to time n (where n is some positive integer) is:

C [1 + i (0)][1 + i (1)] … [1 + i ( n − 1)] since the proceeds C [1 + i (0)] at time 1 may be invested at this time to produce

C [1 + i (0)][1 + i (1)] at time 2, and so on.

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If the rate of interest per period does not depend on the time t at which the investment is made, we write i (t ) = i for all t. In this case the accumulation of an investment of C for any period of length n time units is:

C (1 + i )n
This formula is referred to as the “accumulation” of C for n time units under compound interest at rate i per time unit.

Question 2.6
Find the accumulation of a payment of £400 for 6½ years under compound interest at rate 12½% per annum.

Converting between different effective rates
The accumulated value of £100 in one year at 10% pa effective is 100 × 11 = £110 .
.
We could express the interest rate in terms of an effective half-yearly rate, j, such that
100 × (1 + j ) 2 = 110 . Solving for j gives j = 4.88% .
So 10% per annum effective is equivalent to 4.88% per half-year effective.
You will not need to use this logic every time you convert between different effective rates. If the effective annual rate is 10% then the effective half-yearly rate is

110.5 − 1 = 4.88% .
.
We can also move to a longer time period. If the effective half-yearly rate is 5%, then the effective annual rate is 1.052 - 1 = 10.25% .

Question 2.7
Calculate the equivalent effective monthly rate and biennial rate for 10% pa effective.

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Question 2.8
Consider two non-overlapping time periods. Period 1 has length l time units and Period
2 has length m time units. If the effective Period 1 interest rate is i, express the equivalent effective Period 2 interest rate in terms of i, l, and m.

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Chapter 2 Summary
Many financial arrangements involve a borrower and a lender. Borrowers reward lenders by paying interest to lenders.
Two factors that might influence the level of interest rates are the likelihood of default on payments and the possible appreciation or depreciation of currency.
The calculation of the amount of interest payable under a financial arrangement can be expressed in terms of compound interest or simple interest.
The essential feature of simple interest is that interest, once credited to an account, does not itself earn further interest.
The essential feature of compound interest is that interest itself earns interest.
The present value of a series of payments is a key concept used throughout actuarial work.
The formula for present value is PV = Xv n .
The effective rate of interest over a given time period is the amount of interest a single initial investment will earn at the end of the time period, expressed as a proportion of the initial amount. It is usually denoted by the symbol i.
The rate of commercial discount is a way of expressing the amount of the repayment required for a loan. In return for a payment of X after a period t ( t < 1 ) a sum of
X (1 - td ) will be lent at the start of the period, where d is the commercial rate of discount. The formula for accumulated values is X (1 + ni ) for simple interest and X (1 + i ) n for
1
compound interest. The formula for the discount factor is v =
.
1+ i

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This page has been left blank so that you can keep the chapter summaries together for revision purposes.

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Chapter 2 Solutions
Solution 2.1
(i)

A bank acts as a lender when:




it makes a business loan to a company



(ii)

it offers a mortgage to a person wanting to buy a house

it buys fixed interest securities.

A bank acts as a borrower when:


it accepts money from savers



it issues shares (ie the bank’s own shares) to investors



it sells fixed interest securities.

(These lists cover the most common situations. They are not exhaustive.)

Solution 2.2
(i)

At the end of 6 years the investor will have 5, 000 + 6 ¥ 500 = £8, 000 .

(ii)

At the end of 3 years the investor will have 5, 000 + 3 ¥ 500 = £6,500 .
At the end of 6 years the investor will have 6,500 + 3 ¥ 650 = £8, 450 .

Solution 2.3
(i)

At the end of 6 years the investor will have 5, 000 ¥ 1.16 = £8,858 .

(ii)

At the end of 3 years the investor has 5, 000 ¥ 1.13 = £6, 655 .
At the end of 6 years the investor will have 6, 655 ¥ 1.13 = £8,858 .

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Solution 2.4
Using the definition of v :

v=

1
1
=
= 0.96154
1 + i 104
.

Alternatively, you could have found this on page 56 of the Tables, either at the top of the v n column or among the “constants” listed on the left-hand side.

Solution 2.5
The amount of the immediate payment will be:
6
500,000 (1 − 12 × 016) = £460,000
.

Solution 2.6
The accumulation (or accumulated value) is 400 × 11256.5 = £860.10 .
.

Solution 2.7
.
The effective monthly rate is 111/12 − 1 = 0.797% .
.
The effective biennial rate is 112 − 1 = 21% .

Solution 2.8
Consider investing 1 unit for 1 time period.
The accumulated value equals:
(1 + i )1/ l = (1 + j )1/ m where j is the effective Period 2 interest rate and so: j = (1 + i ) m/ l − 1

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Chapter 3
Interest rates
Syllabus objective
(iii)

Show how interest rates or discount rates may be expressed in terms of different time periods.
1.

2.

Derive the relationships between the rate of interest payable once per effective period and the rate of interest payable p times per time period and the force of interest.

3.

Explain the difference between nominal and effective rates of interest and derive effective rates from nominal rates.

4.

0

Derive the relationship between the rates of interest and discount over one effective period arithmetically and by general reasoning.

Calculate the equivalent annual rate of interest implied by the accumulation of a sum of money over a specified period where the force of interest is a function of time.

Introduction
The last chapter considered the accumulation and present value of a single payment and introduced the idea of interest. This chapter describes the alternative way of expressing interest rates and shows the relationships between them.

Question 3.1
Define the effective rate of interest over a given time period.

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Nominal rates of interest
Definition
Consider transactions for a term of length h time units, where h > 0 and h need not be an integer. We define ih (t ) , the “nominal rate of interest” per unit time on transactions of term h beginning at time t, to be such that the effective rate of interest for the period of length h beginning at time t is hih (t ) . Thus, if the sum of
C is invested at time t for a term h, the sum to be received at time t + h is, by definition: C [1 + hi h (t )]

(1.1)

Example
Working in time units of years, we will let the nominal rate of interest per year on monthly transactions over the next year be 12%. If £100 is invested at time 0, how much is it worth at the end of the first month and at the end of the third month?
Solution
We have defined i1/12 (0) = 12% and so the effective monthly interest rate is
1
i1/12 (0) = 1% .
12
At the end of the first month we have 100(1 + 0.01) , ie £101.
At the end of the third month we have 100 × 1013 , ie £103.03
.

1.2

Simplifications
If h = 1 , the nominal rate of interest coincides with the effective rate of interest for the period t to t + 1 , so:

i1(t ) = i (t )
In many practical applications ih (t ) does not depend on t, in which case we may write i h (t ) = i h for all t
(1.2)

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If, in this case, we also have h = 1 p , where p is a positive integer (ie h is a simple fraction of a time unit), it is more usual to write i ( p ) rather than i1/ p .
Thus, as a definition, we have:

i ( p ) = i1/ p

(1.3)

It follows that, in this case, an investment of 1 for any period of length 1 p will produce a return of:

1+

i ( p) p This result comes directly from equations (1.1), (1.2) and (1.3).
1
In the previous example, h = 12 , and so p = 12 and i1/12 = i (12 ) = 12% . An investment

of 1 for a month will produce a return of 1 +

i (12)
= 101 .
.
12

Note that i ( p ) is often referred to as a nominal rate of interest per unit time payable pthly, or convertible pthly, or with pthly rests.

Therefore, working in years, i (12 ) is referred to as a nominal rate convertible monthly and i ( 4 ) as a nominal rate convertible quarterly etc.

Question 3.2
£250 is invested in a savings account. The nominal rate of interest convertible monthly for the first 3 months is 18% and the nominal rate of interest convertible quarterly for the next 9 months is 20%. How much is in the account at the end of the year?
We will consider nominal rates again at the end of this chapter.

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Accumulation factors
Let time be measured in suitable units (eg years). For t1 ≤ t 2 we define A(t1, t 2 ) to be the accumulation at time t 2 of an investment of 1 at time t1 for a term of

(t 2 − t1 ) . Thus A(t1, t 2 ) is the amount which will be repaid at time t 2 in return for an investment of 1 at time t1 .

Remember that we have defined hih (t ) to be the effective rate for the period of length h beginning at time t and so:
It follows by the definition of i h (t ) that, for all t and for all h > 0 :

A(t , t + h) = 1 + hih (t ) and hence that:

i h (t ) =

A(t , t + h ) − 1 h h>0

(2.1)

We also define A(t , t ) = 1 for all t.

The number A(t1, t 2 ) is often called an accumulation factor, since the accumulation at time t 2 of an investment of C at time t1 is, by proportion:

CA(t1, t 2 )

(2.2)

Example
£80 is invested at time 5 and the accumulated amount at time 8 is £100.

A(5,8 ) = 100 = 125 and i3 (5) =
.
80

125 − 1
.
= 8.33% .
3

Question 3.3
Calculate the equivalent effective annual rate of interest for the investment in the above example, assuming that time is in years.

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2.1

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Principle of consistency
Interest rates for time periods that have already elapsed are known but future interest rates are unlikely to be known unless the timing and amount of all future cashflows are known. In relation to the past, ie when the present moment is taken as time 0 and t and t + h are both less than or equal to 0, the factors A(t , t + h ) and the nominal rates of interest i h (t ) are a matter of recorded fact in respect of any given transaction.
As for their values in the future, estimates must be made (unless one invests in fixed interest securities with guaranteed rates of interest applying both now and in the future).
Now let t 0 ≤ t1 ≤ t 2 and consider an investment of 1 at time t 0 . The proceeds at time t 2 will be A(t 0 , t 2 ) if one invests at time t 0 for term t 2 − t 0 , or

A(t 0 , t1 )A(t1, t 2 ) if one invests at time t 0 for term t1 − t 0 and then, at time t1 , reinvests the proceeds for term t 2 − t1 . In a consistent market these proceeds should not depend on the course of action taken by the investor. Accordingly, we say that under the principle of consistency:

A(t 0 , t n ) = A(t 0 , t1)A(t1, t 2 ) … A(t n − 1, t n )

Question 3.4
£4,600 is invested at time 0 and the proceeds at time 10 are £8,200.
Calculate A(7,10) if A(0,9) = 18, A(2,4) = 11, A(2,7) = 132, A(4,9) = 145.
.
.
.
.

Question 3.5
Define a nominal rate of interest.

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The force of interest
An effective rate of interest is the amount of interest a single initial investment will earn at the end of the time period. So you can think of the interest being paid at the end of the time period. We now move on to the case where the interest is paid continuously throughout the time period.
If we consider a nominal interest rate convertible very frequently (eg every second), we are no longer thinking of a fund that suddenly acquires an interest payment at the end of each interval, but of a fund that steadily accumulates over the period as interest is earned and added. In the limiting case, the amount of the fund can be considered to be subject to a constant “force” causing it to grow. This leads us to the concept of a force of interest, which is the easiest way to model interest rates mathematically.
The force of interest at time t, denoted by δ (t ) , is the fraction that the interest earned during a very short time period represents of the current accumulated amount, expressed as an annualised rate.
We assume that for each value of t there is a number δ (t ) such that: lim ih (t ) = δ (t )

h→0+

(3.1)

It is usual to call δ (t ) the “force of interest per unit time” at time t. In view of formula (3.1), δ (t ) is sometimes called the nominal rate of interest per unit time at time t convertible momently.
By combining equations (2.1) and (3.1) we may define δ (t ) directly in terms of the accumulation factor as:

⎡ A(t , t + h ) − 1⎤

h



δ (t ) = lim ⎢ h→0+ Let us now define:

F (t ) = A(t 0 , t ) where t 0 is fixed and t 0 ≤ t . Thus F (t ) is the accumulation at time t of an investment of 1 at time t 0 .

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We can write δ (t ) in terms of F(t) by noticing that if the principle of consistency holds then: A(t , t + h) =

So,

A(t0 , t + h) F (t + h)
=
A(t0 , t )
F (t )

F ( t + h) − F ( t ) 1 F ′ ( t ) d
× =
= log e F (t ) . h→0 F (t ) h F (t ) dt

δ (t ) = lim

F (t + h) − F (t ) is the increase in the accumulated fund during the time period
( t , t + h) .
The factor of

1 h converts the ratio to an equivalent amount per unit time.

The middle equality follows from the mathematical definition of a derivative and the last equality follows from the “function of a function” rule for differentiation.

Question 3.6
If a large pension fund with a value of £1,000m is assumed to grow steadily subject to a constant force of interest of 10% per annum, how much interest is earned every second?
(Assume that there are 365¼ days in a year.)
For the remainder of the course we will usually write log as opposed to log e . However we may use log e , log and ln interchangeably.

3.1

Formulae for the accumulation factor
Question 3.7
⎡n

Show that F (n) = exp ⎢ ∫ δ (t ) dt ⎥ .
⎢t

⎣0


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The last question shows that the accumulated amount at time t = n of an initial investment of 1 unit at time t = t0 is:

⎡n

⎢ δ (t ) dt ⎥
F (n) = exp ∫
⎢t

⎣0

So in order to find this accumulated amount you should “integrate the force of interest

over the period, then apply e x ”.
The force of interest function δ (t ) is defined in terms of the accumulation function A(t1, t 2 ) , but when the principle of consistency holds it is possible, under very general conditions, to express the accumulation factor in terms of the force of interest. This result is contained in the following theorem.
If δ (t ) and A(t 0 , t ) are continuous functions of t for t ≥ t0 , and the principle of consistency holds, then, for t 0 ≤ t1 ≤ t 2 :

A(t1, t 2 ) = exp

z

LM t
Nt

2

δ (t )dt

1

OP
Q

(3.2)

This formula appears on page 31 of the Tables. For mathematical precision, if one of the limits in the integral is t then we would change the variable.

Explanation
By definition, A(t1 , t 2 ) is the ratio of the present value at times t1 and t 2 , ie:

A(t1 , t 2 ) =

F (t 2 )
F ( t1 )

Using the result of Question 3.7:

È t2
˘
A(t1 , t2 ) = exp Í Ú d (t ) dt ˙
Í0
˙
Î
˚

È t1
˘
exp Í Ú d (t ) dt ˙
Í0
˙
Î
˚

t1
È t2
˘
È t2
˘
Í Ú d (t ) dt - Ú d (t ) dt ˙ = exp Í Ú d (t ) dt ˙
= exp
Ít
˙
Í0
˙
0
Î
˚
Î1
˚

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Page 9

Also, from the definition of A(t 2 , t1 ) , the corresponding discount factor is:

A(t 2 , t1 ) =

F ( t1 )
F (t 2 )
1
=1
=
F (t 2 )
F (t1 ) A(t1 , t 2 )

As soon as δ (t ) , the force of interest per unit time, is specified, the accumulation factors A(t1, t 2 ) can be determined by formula (3.2). We may also find ih (t ) by formulae (3.2) and (2.1). Thus:

LMt + h OP exp MN t δ (s) ds PQ − 1 i h (t ) =

z

(3.3)

h

Example

0.02
,(t ≥ 0) , find an expression for the t +1
( t1 < t 2 ) .

If the force of interest is δ (t ) = 0.08 + accumulation factor from time t1 to t 2

From the formula for the accumulation factor:

È t2
˘
È t2
0.02 ˆ ˘
Ê
A(t1 , t2 ) = exp Í Ú d (t ) dt ˙ = exp Í Ú Á 0.08 +
˜ dt ˙ t + 1¯ ˙
Ít
˙
Ít Ë
Î1
˚
Î1
˚

(

t

= exp [0.08t + 0.02 log(t + 1)]t2
1

)

È
Ê t + 1ˆ ˘
0.08(t2 -t1 ) Ê t2 + 1ˆ
= exp Í 0.08(t2 - t1 ) + 0.02 log Á 2
¥Á
˜˙ = e
Ë t1 + 1 ¯ ˚
Ë t1 + 1 ˜
¯
Î

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We can also have a force of interest that changes over time.

Example
If the force of interest is δ (t ) = 0.08 for 0 ≤ t < 5 and δ (t ) = 0.13 − 0.01t for 5 ≤ t < 10 , find an expression for the accumulation factor from time 0 to t .
We need separate expressions here for the accumulation factor depending on whether t is less than 5 or greater than 5. From the formula for the accumulation factor:

⎡t

t
A(0, t ) = exp ⎢ ∫ 0.08 ds ⎥ = exp [ 0.08s ]0 = exp(0.08t )
⎢0




)

(

where 0 ≤ t < 5

However, where 5 ≤ t < 10 , we need to find the product of two accumulation factors:
È5
˘
Èt
˘
A(0, t ) = A(0,5) A(5, t ) = exp Í Ú 0.08 ds ˙ exp Í Ú 0.13 - 0.01s ds ˙
Í0
˙
Í5
˙
Î
˚
Î
˚

(

)


5
Ê
= exp [0.08s ]0 exp Á È 0.13s - 0.005s 2 ˘ ˜
˚5 ¯
ËÎ

(

)

= exp(0.4) exp 0.13t - 0.005t 2 - 0.65 + 0.125

(

= exp 0.13t - 0.005t 2 - 0.125

)

Question 3.8
The force of interest is given by:

Ï0.04 + 0.002t 0 £ t < 10
Ô
d (t ) = Ì0.015t - 0.08 10 £ t < 12
Ô
0.07 t ≥ 12
Ó
find an expression for the accumulation factor from time 0 to t .

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We have already shown that the next two results are true but you can also see that by formula (3.2):

log F (t ) =

z

t

t0

δ ( s )ds

and hence, for t > t0 :

δ (t ) =

F ′ (t ) d log F (t ) =
F (t ) dt The case when δ (t ) = δ for all t is of very great practical importance. It is clear that in this case:

A(t 0 , t 0 + n) = e δn

(3.4)

for all t 0 and n ≥ 0 . By formula (3.3), with h = 1 , the effective rate of interest per time unit is:

i = eδ − 1 and hence:

eδ = 1 + i
The accumulation factor A(t 0 , t 0 + n ) may thus be expressed in the alternative form: A(t 0 , t 0 + n ) = (1 + i )n

Question 3.9
If the force of interest is δ (t ) = 0.08 +

0.02
, (t ≥ 0) , calculate: t +1

(i)

the accumulated value at time t = 5 of an investment of £1,000 at time t = 0

(ii)

the present value at time t = 2 of a sum of £1,000 payable at time t = 8

In each case, apply a reasonableness check to support your answer.

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We can also calculate accumulated values when the force of interest changes over time.

Example
If the force of interest is δ (t ) = 0.04 for 0 ≤ t < 6 and δ (t ) = 0.2 − 0.02t for 6 ≤ t < 9 , find the accumulated value at time 8 of a payment of $400 at time 3.
The accumulated value will be:
È6
˘
È8
˘
400 A(3,8) = 400 A(3, 6) A(6,8) = 400 exp Í Ú 0.04 ds ˙ exp Í Ú 0.2 - 0.02 s ds ˙
Í3
˙
Í6
˙
Î
˚
Î
˚

(

)


6
Ê
= 400 exp [0.04s ]3 exp Á È 0.2 s - 0.01s 2 ˘ ˜
˚6 ¯
ËÎ

= 400 exp(0.12) exp (1.6 - 0.64 - 1.2 + 0.36)
= 400 exp (0.24) = $508.50

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Page 13

Present values
Let t1 ≤ t2 . ie C exp[ −

z

It follows by formula (2.2) that an investment of C / A(t1 , t2 ) ,

t2 δ (t )dt ] , t1 at time t1 will produce a return of C at time t2 . We therefore say

that the discounted value at time t1 of C due at time t2 is:

LM
N

C exp −

z

t2

t1

δ (t )dt

OP
Q

(4.1)

This is the sum of money which, if invested at time t1 , will give C at time t2 . In particular, the discounted value at time 0 (the present time) of C due at time t ≥ 0 is called its discounted present value (or, more briefly, its present value); it is equal to:

LM
N

z

t

C exp − δ ( s)ds
0

OP
Q

(4.2)

We now define the function:

LM
N

z

t

v (t ) = exp − δ ( s)ds o OP
Q

(4.3)

When t ≥ 0 , v ( t ) is the (discounted) present value of 1 due at time t. When t < 0 , the convention z δ (s)ds = − z δ (s)ds shows that v(t ) is the accumulation of 1 from time t to t 0

0 t time 0.
It follows that v (t ) = 1 A(0, t ) and by formulae (4.2) and (4.3) that the discounted present value of C due at a non-negative time t is:
Cv (t )
In the important practical case in which δ (t ) = δ for all t, we may write: v (t ) = v t

for all t

where v = v (1) = e −δ

Question 3.10
Calculate the discounted present value of £780 due in six years if the force of interest per annum is 5%.

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We can now calculate present values when the force of interest changes over time. We apply the same principles as when we were accumulating, but this time we remember to put a negative in the power of the exponential.

Question 3.11
The force of interest is given by:

⎧ 0.08 − 0.001t 0 ≤ t < 3

δ (t ) = ⎨0.025t − 0.04 3 ≤ t < 5

0.03 t ≥5

Calculate the present value at time 2 of a payment of £1,000 at time 10.

Question 3.12
Calculate the annual effective rate of interest from time 2 to 10 equivalent to the force of interest in Question 3.11.

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Page 15

The basic compound interest functions
The last chapter introduced the interest functions i and v, and in this chapter you have met δ . This section shows the relationships between these three rates and a fourth, d, the rate of discount.
The particular case in which δ (t ) , the force of interest per unit time at time t, does not depend on t is of special importance. In this situation we assume that, for all values of t:

δ (t ) = δ where δ is some constant.
The value at time s of 1 due at time s + t is: s +t
È
˘
Ê s +t
ˆ
exp Í d (r ) dr ˙ = exp Á d dr ˜
Ë s
¯
s
Î
˚

Ú

Ú

= exp( -d t ) which does not depend on s. Thus the value at any given time of 1 due after a further period t is:

v (t ) = e −δ t
= vt
= (1 − d )t where v and d are defined in terms of δ by the equations:

v = e −δ and: (1 − d ) = e − δ

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Thus, in return for a repayment of 1 at time 1, an investor will lend an amount
(1 − d ) at time 0. The sum of (1 − d ) may be considered as a loan of 1 (to be repaid after 1 unit of time) on which interest of amount d is payable in advance.
For this reason d is called the “rate of discount” per unit time. Sometimes, in order to avoid confusion with nominal rates of discount, d is called the effective rate of discount per unit time.

Consider the situation where an individual borrows a sum of £5,000 and agrees to pay this back at the end of 1 year with interest calculated at an effective rate of 10% per annum.
The total amount repayable will therefore be £5,500.
An alternative way of looking at this arrangement would be to say that the individual has borrowed £5,500 (the amount to be repaid) but the lender has deducted an interest payment of £500 at the time the money was lent. Presented in this way, it would seem logical to express the interest rate as 9.09% (ie 500 / 5,500 ) of the amount borrowed, where the interest is payable at the beginning of the year.
Therefore the effective rate of discount over a given time period is the amount of interest a single initial investment will require to be paid at the beginning of the time period, expressed as a proportion of the final amount.
If 1 unit is invested at the beginning of the year at an effective annual interest rate i , it will accumulate to 1 + i by the end of the year. By definition, the discount rate is the amount of the interest payment (which is deducted at the beginning of the year) divided i . by the amount at the end of the year ie
1+ i
Similarly, it follows immediately from equation (3.4) that the accumulated amount at time s + t of 1 invested at time s does not depend on s and is given by:

F (t ) = e δt
= (1 + i )t where i is defined by the equation:

1 + i = eδ
Thus an investor will lend an amount 1 at time 0 in return for a repayment of
(1 + i ) at time 1. Accordingly i is called the “rate of interest” (or the “effective rate of interest”) per unit time.

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This definition is equivalent to the one you met in the last chapter. It has just been reached from a different starting point.

Question 3.13
Calculate the discount rate d corresponding to an effective annual interest rate of 15%.

Question 3.14
Calculate the PV of a payment of £5,000 payable in 3 years’ time, assuming an effective annual rate of discount of 7.5%.
The relationships between δ , i, v, and d can therefore be seen in the following table: In terms of

Value of

δ δ d

v

i δ e −1

e

−δ

1 − e −δ

(1 + i )-1

i

log(1 + i )

v

- log v

v -1 - 1

d

- log(1 - d )

(1 - d )-1 - 1

i (1 + i )-1
1- v

1- d

Question 3.15
You have seen above that d = i (1 + i ) which is equivalent to d = iv . Explain why this relationship is true.

5.1

Approximations of d and δ
When i is small, approximate formulae for d and δ in terms of i may be obtained from well-known series by neglecting the remainder after a small number of terms. For example, since:

d = log(1 + i )
= i - 12 i 2 + 13 i 3 - 14 i 4 + ...

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CT1-03: Interest rates

it follows that, for small values of i:

δ ≈i−

1

2i

2

and for small values of δ :

i ≈ δ + ½δ 2
Similarly:

d = i (1 + i )-1
= i (1 - i + i 2 - i 3 +

)

(if i < 1)

= i - i2 + i3 - i4 + so, if i is small:

d ≈ i − i2 and for small values of δ :

d ≈ δ − ½δ 2

Question 3.16
If i = 6.5% , calculate d and δ accurately and using the approximations above.

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Page 19

Interest payable pthly
In Section 1 we introduced nominal rates of interest. This section gives an alternative way of considering nominal rates plus a few new important results.
Suppose that a borrower, who is lent 1 at time 0 for repayment at time 1, wishes to pay the interest on the loan in p equal instalments over the interval. We define

i ( p ) to be that total amount of interest, payable in equal instalments at the end of each pth subinterval (ie at times 1 p ,2 p ,3 p , … ,1).
Likewise we define d ( p ) to be that total amount of interest, payable in equal instalments at the start of each pth subinterval (ie at times
0,1 p ,2 p , … ,( p - 1) p .

Instead of paying one instalment of interest at time 1 we are paying p instalments throughout the time period. Therefore, the accumulated value of the p interest payments, each of amount i(p)/p, is equal to i.
We may easily express i ( p ) in terms of i. Since i ( p ) is the total interest paid, each interest payment is of amount i ( p ) p and our definition implies that:

i ( p)
(1 + i )( p − t )/ p = i t =1 p p Σ

Written out this series is: i( p)
(1 + i )1-1 p p

+

i( p)
(1 + i )1- 2 p p

+

+

i( p) p i( p )
1− 1
−1
(1 + i ) p , common ratio (1 + i ) p p and p terms. The sum of this progression is (providing that i ≠ 0 ):
This is a geometric progression, with first term

i( p )
×
p

1− 1 p (1 + i )

R1 − (1 + i) U
S
V
T
W

1 − (1 + i )

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CT1-03: Interest rates

This can be simplified to:

LM
NM

OP
QP

i ( p)
(1 + i ) − 1
=i
p (1 + i )1/ p − 1
Hence (rearranging this expression algebraically):

i ( p ) = p[(1 + i )1/ p − 1] and (again rearranging):

LM1 + i ( p) OP p = 1 + i
NM p QP

(6.1)

Likewise it is a consequence of our definition of d ( p ) that: d ( p)
(1 − d ) (t − 1) / p = d t =1 p p Σ

or, if d π 0 :

LM
MN

OP
PQ

d ( p)
1 − (1 − d )
=d
p 1 − (1 − d )1/ p
Hence:

d ( p ) = p[1 − (1 − d )1/ p ] and: LM1 − d ( p) OP p = 1 − d
MN p PQ

(6.2)

1

Since (1 − d ) = v , then we also have the expression d ( p ) = p[1 - v p ] = p[1 - (1 + i )

-1 p ].

The results for d ( p ) have been derived in a very similar way to those for i( p ) .

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Page 21

Explanation
Follow through this example to see how these interest rates work in practice:
A nominal rate of interest of 8% per annum convertible quarterly (ie i ( 4 ) = 0.08 ) is equivalent to an effective rate of 2% per quarter. So, an initial investment of 1 unit will
4

⎛ i (4) ⎞
.
amount to 102 ie ⎜1 +
⎟ by the end of the year. If the equivalent effective annual

4 ⎟

⎠ rate of interest is i , an initial investment of 1 unit would amount to 1 + i . So we must
4

⎛ i (4)
.
have 102 = 1 + i ie ⎜1 +

4

4

4


⎟ = 1 + i . So, in this example, the equivalent effective rate



of interest is i = 102 4 − 1 = 0.0824 ie 8.24%.
.
The formula for nominal discount rates is shown similarly by noting that an amount of
1 unit at the end of the year at an effective discount rate of d corresponds to an amount
1 − d (ie the final amount minus the interest d ) at the beginning of the year.

Example
Find the PV as at 1 January 1999 of a payment of £500 payable on 1 July 1999, assuming an interest rate of 9% pa convertible monthly.

Solution
9% pa convertible monthly is equivalent to an effective monthly interest rate of 0.75%.
So, working in terms of months with i = 0.0075 , the PV is:
500v 6 = 500 × 0.9562 = £478.08

Question 3.17
If the effective rate of interest is 10% per annum, calculate (i) i ( 4 ) and (ii) d (12 ) .

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It is essential to appreciate that, at force of interest δ per unit time, the five series of payments illustrated in Figure 3.1.1 all have the same value.

1 p d ( p) p ...

p−1 p d ( p) p d ( p) p ...

d ( p) p i ( p) p i ( p) p ...

i ( p) p 1

time

i ( p) p i

equivalent payments d

(2)

3 p i ( p) p (1)

2 p d ( p) p 0

(3)
(4)

δ

(5)

Figure 3.1.1 Equivalent payments
If we choose to regard i ( p ) or d ( p ) as the basic quantity, equation (6.1) or (6.2) may be used to define i in terms of i ( p ) or d in terms of d ( p ) . It is customary to refer to i ( p ) and d ( p ) as “nominal” rates of interest and discount “convertible” pthly. For example, if we speak of a rate of interest of 12% per annum convertible quarterly, we have i (4) = 0.12 (with one year as the unit of time).
4
Ê i (4) ˆ
Since (1 + i ) = Á 1 +
˜ , this means that i = 0.125509 .
4 ¯
Ë

Thus the equivalent

annual rate of interest is 12.5509%. Thus, if the nominal rate of interest convertible quarterly is 12%, the effective rate per annum is 12.5509%.
The treatment of problems involving nominal rates of interest (or discount) is almost always considerably simplified by an appropriate choice of the time unit.
For example, on the basis of a nominal rate of interest of 12% per annum convertible quarterly, the present value of 1 due after t years is:

(1 + i )

−t

LM1 + i (4) OP −4t
=
NM 4 QP
FG
H

= 1+

0.12
4

IJ −4t
K

(by equation (6.1))

(since i (4) = 0.12 )

= 103 −4t
.
Thus if we adopt a quarter-year as our basic time unit and use 3% as the effective rate of interest, we correctly value future payments.

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Page 23

The general rule to be used in conjunction with nominal rates is very simple.
Choose as the basic time unit the period corresponding to the frequency with which the nominal rate of interest is convertible and use

i ( p) as the effective rate p of interest per unit time. For example, if we have a nominal rate of interest of 18% per annum convertible monthly, we should take one month as the unit of time and
1½% as the rate of interest per unit time.

Question 3.18
The constant nominal rate of interest convertible four-monthly is 15% pa. Calculate the accumulated value after 7 years of a payment of £300.
Note that i ( p ) and d ( p ) are given directly in terms of the force of interest δ by the equations:

i ( p ) = p(e δ / p − 1)

(6.3)

and:

d ( p ) = p(1 − e −δ / p )
Since:

lim x (e δ / x − 1) = lim x (1 − e − δ / x ) = δ

x→∞

x→∞

it follows immediately from the equations (6.3) that:

lim i ( p ) = lim d ( p ) = δ

p→ ∞

p→ ∞

This is intuitively obvious from our original definitions, since a continuous payment stream may be regarded as the limit as p tends to infinity of a corresponding series of payments at intervals of time 1 p .

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CT1-03: Interest rates

It is easy to establish that:

i > i ( 2) > i ( 3) > and: d < d ( 2) < d ( 3) < so that the sequences { i ( p ) } and { d ( p ) } tend monotonically to the common limit δ from above and below respectively.

Question 3.19
Given δ = 8% , calculate i, i(4) and d(12).

Question 3.20
Arrange the following quantities in increasing order of numerical value (assuming that they all correspond to the same effective interest rate):

i , i ( 4 ) , i ( 365) , d , d (12 ) , δ

Question 3.21
Given i = 7% , calculate v , d , d (4) and i (2) .

Question 3.22
Given d = 9% , calculate i , d (2) and i (4) .

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Page 25

Chapter 3 Summary
Interest rates can be expressed in terms of an effective rate of interest or an effective rate of discount over a time period such as one year. Alternatively, a nominal rate of interest or a force of interest can be used.
The effective rate of interest refers to the amount of interest that would be paid at the end of the year. The effective rate of discount refers to the amount of interest that would be paid at the start of the year.
The force of interest is the amount of interest that would be paid continuously over a time period. The nominal rate of interest convertible pthly is the total amount of interest that would be paid in p equal instalments at the end of each pth subinterval.
The principle of consistency says that the accumulated proceeds of an investment in a consistent market will not depend on the action of the investor.

L
O
F (n) = exp Mz δ (t ) dt P
MN
PQ n The formulae you will need for accumulations are

and

0

O
L
A(t , t ) = exp M z δ (t ) dt P .
MN
PQ t2 1

2

t1

The discount rate is given by d =

i
= iv = 1 − v .
1+ i

The formulae connecting effective and nominal interest rates are:

F1 + i I
GH p JK
( p)

ib

p

p

= 1+ i

g = p[(1 + i )1/ p − 1]

F1 − d I
GH p JK
( p)

p

= 1− d

d ( p ) = p[1 − (1 − d ) 1/ p ] = p[1 − v 1/ p ]

The force of interest is defined as lim i ( p) = lim d ( p) = δ . The other formulae you need p→∞ p→∞

for the force of interest are δ = log e (1 + i ) , 1 + i = e δ and v = e −δ .

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This page has been left blank so that you can keep the chapter summaries together for revision purposes

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Page 27

Chapter 3 Solutions
Solution 3.1
The effective rate of interest over a given time period is the amount of interest a single initial investment will earn at the end of the time period, expressed as a proportion of the initial amount.

Solution 3.2
The effective monthly interest rate for the first three months is

18%
12

= 1.5% per month.

The effective quarterly interest rate for the next three quarters (nine months) is
20%
4

= 5% per quarter.

.
.
So at the end of the year we have 250 × 10153 × 1053 = £302.63 .

Solution 3.3
Effective annual rate of interest is

( 100 )
80

1/ 3

− 1 = 7.72% pa .

Solution 3.4
A(0,4) =

A ( 0 ,9 )
A ( 4 ,9 )

.8
= 1145 = 12414
.
.

A(4,7) =

A ( 2 ,7 )
A ( 2 ,4 )

.32
= 111 = 12
.
.

A(0,10) =

8,200
4 ,600

A(7,10) =

A( 0,10)
A ( 0, 4 ) A ( 4 , 7 )

= 17826
.
1.7826
= 1.2414 ×1.2 = 1197
.

Other approaches could have been taken to calculate A(7,10) .

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Solution 3.5 ih(t), the nominal rate of interest per unit time on transactions of term h beginning at

time t, is defined to be such that the effective rate of interest for the period of length h beginning at time t is hih(t).

Solution 3.6
During a short time interval h the fund will grow by h × δ (t ) × Fund value (very nearly).
So the interest earned in 1 second will be:

1
× 010 × 1,000m = £3.17
.
365¼ × 24 × 60 × 60

Solution 3.7

δ (t ) =

d log e F (t ) dt Integrating from t0 to n gives: n Ú d (t )

dt = log e F (n) - log e F (t0 )

t0

But F (t0 ) = A(t0 , t0 ) = 1 , so taking antilogs and swapping sides gives:
⎡n

F (n) = exp ⎢ ∫ δ (t ) dt ⎥
⎢t

⎣0


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Page 29

Solution 3.8
We need separate expressions here for the accumulation factor depending on whether t is less than 10, between 10 and 12 or greater than 12. From the formula for the accumulation factor:
When 0 ≤ t < 10 :
⎡t
⎤ t ⎞

A(0, t ) = exp ⎢ ∫ 0.04 + 0.002s ds ⎥ = exp ⎜ ⎡ 0.04s + 0.001s 2 ⎤ ⎟ = exp(0.04t + 0.001t 2 )
⎦0 ⎠
⎝⎣
⎢0




However, when 10 ≤ t < 12 , we need to find the product of two accumulation factors:

È10
˘
Èt
˘
A(0, t ) = A(0,10) A(10, t ) = exp Í Ú 0.04 + 0.002 s ds ˙ exp Í Ú 0.015s - 0.08 ds ˙
Í0
˙
Í10
˙
Î
˚
Î
˚ t ÊÈ
˘ ˆ
0.015s 2
ÊÈ
2 ˘10 ˆ
= exp Á Î 0.04 s + 0.001s ˚ ˜ exp Á Í
- 0.08s ˙ ˜
Ë
0 ¯
ÁÎ 2
Í
˙10 ¯
˚ ˜
Ë

(

= exp(0.4 + 0.1) exp 0.0075t 2 - 0.08t - 0.75 + 0.8

(

= exp 0.0075t 2 - 0.08t + 0.55

)

)

Also, when t ≥ 12 , we need to find the product of three accumulation factors:

A(0, t ) = A(0,10) A(10,12) A(12, t )
È10
˘
È12
˘
Èt
˘
= exp Í Ú 0.04 + 0.002s ds ˙ exp Í Ú 0.015s - 0.08 ds ˙ exp Í Ú 0.07 ds ˙
Í0
˙
Í10
˙
Í12
˙
Î
˚
Î
˚
Î
˚
12
ÊÈ
˘ ˆ
0.015s 2 t ÊÈ
2 ˘10 ˆ
= exp Á Î 0.04 s + 0.001s ˚ ˜ exp Á Í
- 0.08s ˙ ˜ exp [0.07 s ]12
Ë
0 ¯
ÁÍ 2
˙10 ˜
˚ ¯
ËÎ

(

)

= exp(0.4 + 0.1) exp (1.08 - 0.96 - 0.75 + 0.8) exp (0.07t - 0.84)
= exp (0.07t - 0.17 )
Notice that three versions of the force of interest leads to three versions of the accumulated value.

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CT1-03: Interest rates

Solution 3.9
(i)

Using the formula from a previous example, the accumulated value will be
1, 000 A(0,5) = 1, 000e

0.08(5- 0)

Ê 5 + 1ˆ
¥Á
Ë 0 + 1˜
¯

0.02

= 1, 000e0.4 ¥ 60.02 = £1,546.25
Reasonableness check:
The force of interest over the period 0 < t < 5 averages around 8½% (calculate δ when t = 2½ ). So the effective annual rate of interest would be a bit more (since i = eδ − 1 = δ + ½δ 2 + approximately: > δ ), say 8.9%. So we would expect the answer to be

1,000 × 10895 = £1,532
.
So our answer seems reasonable.
(ii)

The present value will be
1, 000 A(8, 2) = 1, 000e

0.08(2 -8)

Ê 2 + 1ˆ
¥Á
Ë 8 + 1˜
¯

0.02

= 1, 000e -0.48 ¥ 3-0.02 = £605.34
Reasonableness check:
The force of interest over the period 2 < t < 8 averages just over 8%. So the effective annual rate of interest would be a bit more, say 8.3%. So we would expect the answer to be approximately:

1,000 × 1083−6 = £620
.
So our answer seems reasonable.

Solution 3.10
PV = 780v 6 = 780e −6×0.05 = £577.84

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Solution 3.11
The present value is:
È 3
˘
È 5
˘
È 10
˘
1, 000 exp Í - Ú 0.08 - 0.001s ds ˙ exp Í - Ú 0.025s - 0.04 ds ˙ exp Í - Ú 0.03 ds ˙
Í 2
˙
Í 3
˙
Í 5
˙
Î
˚
Î
˚
Î
˚

(



10
Ê
Ê
= 1, 000 exp Á - È 0.08s - 0.0005s 2 ˘ ˜ exp Á - È 0.0125s 2 - 0.04s ˘ ˜ exp - [0.03s ]5
˚2 ¯
˚3 ¯
Ë Î
Ë Î

)

= 1, 000 exp( -[0.2355 - 0.158]) exp ( -[0.1125 + 0.0075]) exp ( -[0.3 - 0.15])
= 1, 000 exp ( -0.3475) = £706.45

Solution 3.12
If the annual effective rate of interest is i , we have:
706.45(1 + i )8 = 1, 000 ⇒ i = 4.44%

Solution 3.13
If i = 015 , then d =
.

015
.
= 013043 ie 13.043%.
.
115
.

(Alternatively, you can look this up on page 64 of the Tables.)

Solution 3.14
We can use the factor 1 − d (which is the same as v ) to discount the value of a future payment. So, the PV is:
5,000(1 − d ) 3 = 5,000(1 − 0.075) 3 = 5,000 × 0.79145 = £3,957

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CT1-03: Interest rates

Solution 3.15
With 1 unit of initial capital, d is the interest payable at the beginning of the year that corresponds to an interest payment of i payable at the end of the year. So d must equal the present value at the beginning of the year of the payment i payable at the end of the year ie d = iv .

Solution 3.16
Accurately, d = i (1 + i ) = 0 .065 1065 = 610% .
.
.
Approximately, d ≈ i − i 2 = 0.065 − 0.0652 = 6.08% .
Accurately, δ = log(1 + i ) = log(1065) = 6.30% .
.
Approximately, δ ≈ i − ½i 2 = 0.065 − ½ × 0.0652 = 6.29% .

Solution 3.17
(i)

The annual effective rate of interest is i = 01 .
.

F1 + i I
GH 4 JK
( 4)

4

= 1 + i = 11
.

.
So i ( 4 ) = 4(111/ 4 − 1) = 0.096455 ie 9.6455% pa convertible quarterly.
(ii)

Again the annual effective rate of interest is i = 0.1 :

F1 − d I
GH 12 JK

(12 ) 12

= 1− d = v =

1
.
11

So d (12 ) = 12(1 − 11−1/12 ) = 0.094933 ie 9.4933% pa convertible monthly.
.

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Solution 3.18
The effective four-monthly interest rate is 5% and so the accumulated value is:
300 × 10521 = £835.79
.

Solution 3.19 i = eδ − 1 = e 0.08 − 1 = 8.33% i ( p ) = p(eδ / p − 1) = 4(e 0.02 − 1) = 8.08% d ( p ) = p(1 − e −δ / p ) = 12(1 − e −0.08 12 ) = 7.97%

Solution 3.20
Assuming a positive rate of interest: d < d (12 ) < δ < i ( 365) < i ( 4 ) < i
The order reflects how late interest is paid.
For example, d corresponds to interest paid immediately, which requires a smaller payment amount.
If you are not convinced by this argument, pick an interest rate (say i = 8% ), go to the
Tables page 60 and look up the relevant figures.

Solution 3.21
We have:

v=

1
= 0.93458 ,
1.07

1
Ê
ˆ d (4) = 4 Á1 - (1 - d ) 4 ˜ = 0.06709
Ë
¯

d=

0.07
= 0.06542
1.07

1
Ê
ˆ i (2) = 2 Á (1 + i ) 2 - 1˜ = 0.06882
Ë
¯

Alternatively, we can look these values up in the Tables.

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CT1-03: Interest rates

Solution 3.22
We have:

v = 1 - d = 0.91 fi i =

1
- 1 = 0.09890
0.91

1
Ê
ˆ d (2) = 2 Á1 - (1 - d ) 2 ˜ = 0.09212
Ë
¯

1
Ê
ˆ i (4) = 4 Á (1 + i ) 4 - 1˜ = 0.09543
Ë
¯

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CT1-04: Real and money interest rates

Page 1

Chapter 4
Real and money interest rates
Syllabus objectives
(iv)

0

Demonstrate a knowledge and understanding of real and money interest rates.

Introduction
This chapter looks at two types of interest rate, namely real and money rates. Real rates of interest are needed when inflation needs to be taken into account. Money rates of interest are used when inflation does not need to be taken into account.
An actuary will use either the real or money rate of interest depending on whether inflation has already been allowed for or not.
You will meet real rates of interest again in Chapter 12.

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CT1-04: Real and money interest rates

Definition of real and money interest rates
Accumulating an investment of 1 for a period of time t from time 0 produces a new total accumulated value A(0, t ) , say.
Typically the investment of 1 will be a sum of money, say £1 or $1 or 1 Euro.

Question 4.1
Find the accumulated value if $1 is invested for 7 years at an interest rate of 6.5% pa effective. In this case, if we are given the information on the initial investment of 1 in the specified currency, the period of the investment, and the cash amount of money accumulated, then the underlying interest rate is termed a “money rate of interest”. In the above question, we have a money rate of interest of 6.5%.
More generally, given any series of monetary payments accumulated over a period, a money rate of interest is that rate which will have been earned so as to produce the total amount of cash in hand at the end of the period of accumulation. In practice, most such accumulations will take place in economies subject to inflation, where a given sum of money in the future will have less purchasing power than at the present day. It is often useful, therefore, to reconsider what the accumulated value is worth allowing for the eroding effects of inflation.

Inflation is a measure of the increase in costs, for example the price of a loaf of bread or a litre of petrol.
Purchasing power is the amount of goods that your money can buy. When inflation occurs you can buy less goods with the same amount of money.

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Example
If a CD cost £13 on 1/1/06 and £14 on 31/12/06, calculate the rate of inflation on CDs during 2006.
Solution
The rate of inflation, j , is given by:

1+ j =

14
13

Therefore j = 7.69% .

Example
If you had £182 on 1/1/06 and the same amount on 31/12/06, how many CDs could you purchase on both dates? Comment on your answer.
Solution
On 1/1/06, you could buy:

182
= 14 CDs
13
On 31/12/06, you could buy:

182
= 13 CDs
14
Since there has been inflation during 2006, the same amount of money buys fewer CDs at the end of the year. You have less purchasing power with your money at the end of the year than you had at the start of the year.
Since you can only buy 13 CDs at the end of the year, the initial £182 is now only worth
182
13 ¥ £13 = £169 . Notice that
= 169 , ie you divide by (1 + the rate of inflation)
1.0769
to calculate how much your money is worth at the end of the year.

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CT1-04: Real and money interest rates

Returning to the initial Core Reading example above, suppose the accumulation took place in an economy subject to inflation so that the cash A(0, t ) is effectively worth only A* (0, t ) after allowing for inflation, where A* (0, t ) < A(0, t ) .
In this case, the rate of interest at which the original sum of 1 would have to be accumulated to produce the sum A* is lower than the money rate of interest.
The sum A* (0, t ) is referred to as the real amount accumulated, and the underlying interest rate, reduced for the effects of inflation, is termed a “real rate of interest”.

Example
£1 accumulates to £1.05 after one year if it is invested at an effective rate of interest of
5% pa.
However, the effects of inflation means that £1.05 is only worth £1.04.
The original £1 would only have to be accumulated at an effective rate of interest of 4% pa to accumulate to £1.04.
The money rate of interest here is 5% pa, but the real rate of interest is 4% pa.

Question 4.2
A bank offers an effective annual rate of interest on one of its accounts of 4.2%.
Inflation is 3% pa effective. Calculate the real rate of interest.
You will meet these calculations in more detail later in this course.
More generally, given any series of monetary payments accumulated over a period, a real rate of interest is that rate which will have been earned so as to produce the total amount of cash in hand at the end of the period of accumulation reduced for the effects of inflation.
Chapter 12 of this subject will describe ways of calculating real rates of interest given the money rates of interest (and vice versa).

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2

Page 5

Deflationary conditions
The above descriptions assume that the inflation rate is positive. Where the inflation rate is negative, termed “deflation”, the above theory still applies and

A* (0, t ) > A(0, t ) , giving rise to the conclusion that the real rate of interest in such circumstances would be higher than the money rate of interest.

Some countries do have a negative inflation rate, eg Japan had a negative inflation rate during early parts of this decade.

Example
£1 accumulates to £1.05 after one year if it is invested at an effective rate of interest of
5% pa.
However, the effects of inflation means that £1.05 is now worth £1.08 at the end of the year. The original £1 would have to be accumulated at an effective rate of interest of 8% to accumulate to £1.08.
The money rate of interest here is 5% pa, but the real rate of interest is 8% pa.

Question 4.3
A bank offers an annual effective rate of interest on one of its accounts of 4.2%.
Inflation is -2% pa effective. Calculate the real rate of interest.
As might be expected, where there is no inflation A* (0, t ) = A(0, t ) , and the real and money rates of interest are the same.

This is because any amount is worth the same at the end of the year as it was at the start of the year.

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CT1-04: Real and money interest rates

Usefulness of real and money interest rates
We assume here that we have a positive inflation rate.
Which of the two rates of interest, real or money, is the more useful will depend on two main factors:


the purpose to which the rate will be put



whether the underlying data have or have not already been adjusted for inflation. The purpose to which the rate will be put
Generally, where the actuary is performing calculations to determine how much should be invested to provide for future outgo, the first step will be to determine whether the future outgo is real or monetary in nature. The type of interest rate to be assumed would then be, respectively, a real or a monetary rate.
For example, first suppose that an actuary was asked to calculate the sum to be invested by a person aged 40 to provide for a round-the-world cruise when the person reaches 60, and where the person says the cruise costs £25,000.
Unless the person has, for some reason, already made an allowance for inflation in suggesting a figure of £25,000 then that amount is probably today’s cost of the cruise. In this case, the actuary would be wise to assume (checking his understanding with the person) an inflation rate and this could be achieved by assuming a real rate of interest.

Here the future outgo of £25,000 is real in nature in that it is likely to be more than
£25,000 in 20 years’ time.
As an alternative example, suppose that a person has a mortgage of £50,000 to be paid off in twenty years’ time. Here, the party that granted the mortgage would contractually be entitled to only £50,000 in twenty years’ time.
Accordingly, in working out how much should be invested to repay the outgo in this case, a money rate of interest would be assumed.

Here the future outgo of £50,000 is monetary in nature in that it is going to be exactly
£50,000 in 20 years’ time.

Whether the underlying data has or has not already been adjusted for inflation
In the first example above, we see that the data may already have been adjusted for inflation and in that case it would not be appropriate to allow for inflation again. A money rate would then be assumed.

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Page 7

More generally in actuarial work, the nature of the data provided must be understood before choosing the type and amount of assumptions to be made.

Question 4.4
In each of the following circumstances state whether the calculations should use a money or real rate of interest.
(i)

An actuarial student wants to invest an amount of money now to buy a BMW Z3 in one year’s time. Today’s list price of the car is available.

(ii)

A woman wants to invest a lump sum today in order to provide her with a fixed income of £25,000 pa for the rest of her life.

(iii)

A man buys a zero-coupon bond that will provide him with £100,000 in 10 years’ time. He is trying to calculate an appropriate purchase price.

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Chapter 4 Summary
Real rates of interest allow for future inflation. Money rates of interest ignore the effects of inflation.
In periods of positive inflation, the real rate of interest will be lower than the money rate of interest.
In periods of negative inflation, the real rate of interest will be higher than the money rate of interest.
In periods of zero inflation, the real rate of interest will be equal to the money rate of interest. Which of the two rates of interest, real or money, is the more useful will depend on two main factors:




the purpose to which the rate will be put whether the underlying data has or has not been adjusted for inflation

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Chapter 4 Solutions
Solution 4.1
The accumulated value is:
1.0657 = 1.554
So $1 accumulates to $1.55.

Solution 4.2
£1 accumulates in the bank account over one year to £1.042.
At the end of the year, the £1.042 is only worth:

1.042
= 1.0117
1.03
Therefore the real rate of interest is 1.17%.

Solution 4.3
£1 accumulates in the bank account over one year to £1.042.
At the end of the year, the £1.042 is worth:

1.042
= 1.0633
0.98
Notice here that (1 + the rate of inflation) < 1 since inflation is negative.
Therefore the real rate of interest is 6.33% pa.

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CT1-04: Real and money interest rates

Solution 4.4
(i)

You would use a real rate since the cost of the car will be quoted as today’s value. (ii)

You would use a money rate since the income of £25,000 is fixed.

(iii)

You would use a money rate since the value of £100,000 (the redemption value) is fixed.

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CT1-05: Discounting and accumulating

Page 1

Chapter 5
Discounting and accumulating
Syllabus objective
(v)

Calculate the present value and the accumulated value of a stream of equal or unequal payments using specified rates of interest and the net present value at a real rate of interest, assuming a constant rate of inflation.
1.

Discount and accumulate a sum of money or a series (possibly infinite) of cashflows to any point in time where:



the rate of interest or discount varies with time but is not a continuous function of time



2.

the rate of interest or discount is constant

either or both the rate of cashflow and the force of interest are continuous functions of time

Calculate the present value and accumulated value of a series of equal or unequal payments made at regular intervals under the operation of specified rates of interest where the first payment is:


deferred for a period of time



not deferred

Real rates of interest are dealt with in Chapter 12.

0

Introduction
So far we have calculated present values and accumulated values of single payments.
This chapter starts to look at present values and accumulations of a series of payments and continuous payments.

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CT1-05: Discounting and accumulating

Present values of cashflows
In many compound interest problems one must find the discounted present value of cashflows due in the future. It is important to distinguish between
(a) discrete and (b) continuous payments.

In Section 1.1 we will consider discrete payments before looking at continuous payments in Section 1.2.

1.1

Discrete cashflows
We have already seen that the present value of a cashflow, C, due at time t is Cvt where v = 1/(1 + i ) and i is the effective rate of interest per annum. Here we are assuming that we are working in years and that the effective rate of interest is constant over the period.
What if we have two payments, C1 due at time t1 , and C2 at time t2 ? The present value of the these payments is the amount we would have to invest, say in a bank account, to be able to pay each of the payments at the times they are required. Rather than investing a single sum into a single bank account to provide for the payments, we could have set up a separate bank account to cater for each payment and invested the present value of each payment in the corresponding account.
This alternative arrangement would have exactly the same result. So, we see that the present value of the two payments is just the sum of the individual present values.
More generally, the present value of a series of payments of ct1 , ct2 , ..., ctn due at times t1 , t2 , ..., tn is given by: n  ct v j =1

tj

j

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Example
Under its current rent agreement, a company is obliged to make annual payments of
£7,500 for the building it occupies. Payments are due on 1 January 2006, 1 January 2007 and 1 January 2008. If the company wishes to cover these payments by investing a single sum in its bank account that pays 7.5% pa compound, what sum must be invested on
1 January 2005?

1/1/2005

£7,500

1/1/2006

£7,500

1/1/2007

£7,500

1/1/2008

Here: v = 1 / (1 + i ) = 1 / 1075 = 0.93023
.
So:

PV payment due on 1 January 2006 = 7,500v = 7,500 × 0.93023 = 6,976.74
PV payment due on 1 January 2007 = 7,500v 2 = 7,500 × 0.930232 = 6, 489.99
PV payment due on 1 January 2008 = 7,500v3 = 7,500 × 0.930233 = 6, 037.20

So:

PV all payments = 6,976.74 + 6, 489.99 + 6, 037.20 = £19,504

Question 5.1
Calculate the present value on 1 September 2002 of payments of £280 due on
1 September 2004 and £360 due on 1 March 2005. Interest is 15% pa effective.
If the effective interest rate is not constant then we could write the present value in terms of the function v (t ) , where v(t ) is the (discounted) present value of 1 due at time
t.
The present value of the sums ct1 , ct 2 , ..., ct n due at times t1, t2 , ..., t n (where

0 £ t1 < t2 < … < t n ) is: ct1v (t1) + ct2 v (t2 ) +

+ ct n v (t n ) =

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 ct v (t j ) j =1

j

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CT1-05: Discounting and accumulating

If the number of payments is infinite, the present value is defined to be:


 ct v (t j ) j =1

j

provided that this series converges. It usually will in practical problems.

Question 5.2
Express the present value of the sums ct1 , ct2 , ..., ctn due at times t1 , t2 , ..., tn in terms of the force of interest at time t, denoted by δ (t ) .

Example
Find the value at time t = 0 of $250 due at time t = 6 and $600 due at time t = 8 if d (t ) = 3% pa for all t.

Solution
If the force of interest, d , is constant then the present value of a payment of C due at time t is:
Ce -d t
Therefore the present value of the two payments in this example is:
250e -6¥0.03 + 600e -8¥0.03 = $680.79

1.2

Continuously payable cashflows (payment streams)
Suppose that T > 0 and that between times 0 and T an investor will be paid money continuously, the rate of payment at time t being £ ρ (t ) per unit time.
What is the present value of this cashflow?

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In order to answer this question it is essential to understand what is meant by the “rate of payment” of the cashflow at time t. If M(t) denotes the total payment made between time 0 and time t, then by definition:

r (t ) = M ¢(t )

for all t

Then, if 0 £ a £ b £ T , the total payment received between time α and time β is: b M ( b ) - M (a ) =

Ú M ¢(t )dt

a

b

=

(1.1)

Ú r (t )dt

a

Thus the rate of payment at any time is simply the derivative of the total amount paid up to that time, and the total amount paid between any two times is the integral of the rate of payments over the appropriate time interval.

You can think of Formula (1.1) as the sum of lots of small payments, each of amount r (t )dt . It may help to consider this simple example:
If the rate of payment is a constant £24 pa then in any one year the total amount paid is
£24, but this payment is spread evenly over the year.
In half a year, the total paid is 24 ¥ ½ , ie £12.
In one month, the total paid is 24 ¥ 112 , ie £2.
So, in a small time period dt, the total paid is £24dt .

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CT1-05: Discounting and accumulating

Example
A life office starts issuing a new type of 10-year savings policy to young investors who pay weekly premiums of £10. Assuming that the life office sells 10,000 policies evenly over each year and that no policyholders stop paying premiums, what will the rate of premium income be for the life office during the first few years?

Solution
After t years the office will have sold 10,000t policies.
So the weekly premium income will be: 10,000t × £10 = £100,000t
Since there are 52.18 ( 365.25 / 7 ) weeks in a year, this corresponds to an annual rate of income of:
52.18 × £100,000t = £5,218,000t

Question 5.3
Calculate the total premium income that would have been received during the first 3 years. Between times t and t + dt the total payment received is M (t + dt ) - M (t ) . If dt is very small this is approximately M ¢(t )dt or r (t )dt . Theoretically, therefore, we may consider the present value of the money received between times t and t + dt as v (t ) r (t )dt . The present value of the entire cashflow is obtained by integration as: T

Ú v (t ) r (t )dt
0

where T is the time of the last cashflow.
For those who don’t like seeing integrals in formulae, don’t be put off. View this as summing, between times t = 0 and t = T , the present values of an infinite number of payments. At each time point, t, the payment made (in the time interval of length dt) is r (t )dt . We use v(t ) to discount the payment from time t to time 0, and the integral sums the infinite number of payments.

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If T is infinite we obtain, by a similar argument, the present value:


Ú v (t ) r (t )dt
0

By combining the results for discrete and continuous cashflows, we obtain the formula: •

 ct v (t ) + Ú v (t ) r (t )dt

(1.2)

0

for the present value of a general cashflow (the summation being over those values of t for which ct , the discrete cashflow at time t, is non-zero).

Assuming a constant interest rate this simplifies slightly to the important result for the present value of a series of discrete cashflows and a continuous cashflow:


Present value = Â ct vt + Ú vt r (t )dt
0

Example
A company expects to receive for the next five years a continuous cashflow with a rate of payment of 100 ¥ 0.8t at time t (years). Calculate the present value of this cashflow assuming a constant force of interest of 8% pa.

Solution
5

5

PV = Ú v(t ) r (t )dt = Ú e
0

-0.08t

0

5

¥ 100 ¥ 0.8 dt = 100 Ú (e -0.08 ¥ 0.8)t dt

È 100(e -0.08 ¥ 0.8)t

-0.08
¥ 0.8)
Í log(e
Î

t

0

5

˘
˙
˙0
˚

È 100[(e -0.08 ¥ 0.8)5 - 1] ˘

˙ = £257.42
Í -0.08 + log(0.8) ˙
Î
˚

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So far we have assumed that all payments, whether discrete or continuous, are positive. If one has a series of income payments (which may be regarded as positive) and a series of outgoings (which may be regarded as negative) their net present value is defined as the difference between the value of the positive cashflow and the value of the negative cashflow.

The net present value will often be abbreviated to NPV. We will study net present values in Chapter 10.

Question 5.4
A company expects to receive for the next five years a continuous cashflow of £350 pa.
It also expects to have to pay out £600 at the end of the first year and £400 at the end of the third year. Calculate the net present value of these cashflows if v (t ) = 1 − t 100 for
0 ≤ t ≤ 5.

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Valuing cashflows
Consider times t1 and t2 , where t2 is not necessarily greater than t1 . The value at time t1 of the sum C due at time t2 is defined as:
(a)

If t1 ≥ t2 , the accumulation of C from time t2 until time t1 ; or

(b)

If t1 < t2 , the discounted value at time t1 of C due at time t2 .

In both cases the value at time t1 of C due at time t2 is:
È t2
˘
C exp Í - d (t )dt ˙
Î t1
˚

Ú

(2.1)

This result was derived in the last chapter.

Question 5.5
Write down an expression for the value at time t1 of C due at time t2 if δ (t ) = δ for all
t.
(Note the convention that, if t1 > t2 ,

t2

Út

1

t1

d (t )dt = - Ú d (t )dt ) t2 Since: t2 Út

1

t2

t1

0

0

d (t )dt = Ú d (t )dt - Ú d (t )dt

it follows immediately from Equation (2.1) that the value at time t1 of C due at time t2 is:
C

v (t2 ) v (t1)

This could also be written in the form C

(2.2)

1
= Cv (t 2 ) A(0, t1 ) , remembering that
A(t1 , t 2 )

v (t ) = 1 A(0, t ) .

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Finding the value at time t1 of C due at time t2 involves discounting the payment C from time t2 to time t1 . This could be performed directly using the expression
C A(t1 , t2 ) .
Alternatively, we could first discount back to time 0 by multiplying by v (t 2 ) , and then accumulate to time t1 by multiplying by A(0, t1 ) . The last step is equivalent to dividing by v (t1 ) and so this alternative uses Formula (2.2).
This is represented diagrammatically below:
0

t1

t2
1 A(t1 , t2 )

v (t2 )
1 v (t1 )

Question 5.6

(easy)

Find the value at time 4 of a payment of 860 at time 10 if v(10) = 0.76 and v(4) = 0.91 .
The value at a general time t1 of a discrete cashflow of ct at time t (for various values of t) and a continuous payment stream at rate r (t ) per time unit may now be found, by the methods given in Section 1, as: v (t )



v (t )

 ct v (t1) + Ú-• r (t ) v (t1) dt

(2.3)

where the summation is over those values of t for which ct π 0 .

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We note that in the special case when t1 = 0 (the present time), the value of the cashflow is:


 ct v (t ) + Ú-• r (t )v (t )dt where the summation is over those values of t for which ct π 0 .

This is a

generalisation of Formula (1.2) to cover the past as well as present or future payments. If there are incoming and outgoing payments, the corresponding net value may be defined, as in Section 1, as the difference between the value of the positive and the negative cashflows. If all the payments are due at or after time t1 , their value at time t1 may also be called their “discounted value”, and if they are due at or before time t1 , their value may be referred to as their
“accumulation”.
It follows that any value may be expressed as the sum of a discounted value and an accumulation. This fact is helpful in certain problems. Also, if t1 = 0 and all the payments are due at or after the present time, their value may also be described as their “(discounted) present value”, as defined by Formula (1.2).

Question 5.7
Consider the following four payments:
£100 on 1 January 2005, £130 on 1 January 2006, £150 on 1 January 2008 and £160 on
1 January 2009.

( t − 2) 3
, calculate the value of these
100
payments on 1 January 2007 and express the value as a sum of a discounted value and an accumulation.
If t = 0 on 1 January 2004 and v (t ) = 0.92 −

It follows from Formula (2.3) that the value at any time t1 of a cashflow may be obtained from its value at another time t2 by applying the factor v (t2 ) / v (t1) , ie:

È Value at time t1 ˘ È Value at time t2 ˘ È v (t2 ) ˘
˙
Í
˙= Í
˙ Í
Î of cash flow ˚ Î of cash flow ˚ Î v (t1) ˚ or: È Value at time t1 ˘
È Value at time t2 ˘
Í
˙ [v (t1)] = Í
˙ [v (t2 )]
Î of cash flow ˚
Î of cash flow ˚

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Each side of Equation (2.4) is the value of the cashflow at the present time (time
0).
In particular, by choosing time t2 as the present time and letting t1 = t , we obtain the result:

È Value at time t ˘ È Value at the present ˘ È 1 ˘
Í of cash flow ˙ = Í time of cash flow ˙ Í
˙
Î
˚ Î
˚ Î v (t ) ˚
These results are useful in many practical examples. The time 0 and the unit of time may be chosen so as to simplify the calculations.

2.1

Constant interest rate
The special case when we assume that interest rates remain constant is of particular importance. Using this assumption v (t ) = v t for all t. Remember also that v = 1 (1 + i ) and so 1 v t = (1 + i ) t .
In the diagram below, each of the three payments P1 , P2 and P3 has the same present value. Payments
Time

P = Xv n
1
–n

P3 = X (1 + i ) n

P2 = X
0

n

This shows that we can think of the factors (1 + i ) n and v n as a way of adjusting payments to a different point on the time line.
If the present value of a series of definite payments at a particular date is X, then:


the accumulated value at a date n years later is:

X (1 + i ) n



the present value at a date n years earlier is:

Xv n

Note that n does not have to be a whole number in these formulae.

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Example
Under its current rent agreement, a company is obliged to make annual payments of
£7,500 for the building it occupies. Payments are due on 1 January 2004, 1 January 2005 and 1 January 2006. The nominal rate of interest is 8% per annum, convertible quarterly.
(Remember this means that the effective quarterly rate of interest is 2%.)
Working in quarters, the present value of these payments on 1 January 2003 is:
7,500(v 4 + v8 + v12 ) = £19, 243.72

where v = 1 1.02 = 0.980392

Alternatively, you might prefer to first calculate the effective annual rate, i, as:
1.024 - 1 = 8.243216%
Working in years, the present value of these payments on 1 January 2003 is:
7,500(v + v 2 + v3 ) = £19, 243.72

where v = 1 1.08243216 = 0.923845

The accumulated value of the payments on 1 January 2007 is:
PV ¥ (1 + i ) 4 = 19, 244 ¥ 1.082432164 = 19, 244 ¥ 1.0216 = £26, 418

Question 5.8
What would the present value of the rent payments in the above example be as at
(i) 1 January 2002, (ii) 1 January 2005 and (iii) 1 July 2017?

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Example
The force of interest takes the following values:

d (t ) = 0.04

0 < t ≤ 10

d (t ) = 0.001(t - 10) 2 + 0.04

10 < t

Calculate the accumulation of £150 from time t = 0 to time t = 20.

Solution
The accumulation is:

150 A(0, 20) = 150 A(0,10) ¥ A(10, 20)
Calculating the two accumulation factors:
A(0,10) = e10¥0.04 = 1.4918
Ê 20
ˆ
A(10, 20) = exp Á Ú 0.001(t - 10) 2 + 0.04 dt ˜
Ë 10
¯
20

È 0.001
˘
= exp Í
(t - 10)3 + 0.04t ˙
Î 3
˚10
= exp(0.7333) = 2.0820

Thus the accumulation is equal to:

150 A(0, 20) = 150 ¥ 1.4918 ¥ 2.0820 = £466

Question 5.9
In the above example, a continuous payment stream is paid at rate e -0.03t from time t = 0 to time t = 10 . Calculate the present value of this payment stream at time t = 0 .

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2.2

Page 15

Payment streams
In the last question, we saw how to evaluate the present value of a particular payment stream. We will now consider the general case, finding formulae to evaluate the present and accumulated values of payment streams, and then look at some examples and questions to see how the formulae are used in practice.
A payment stream ρ (t ) is received from time a to time b, during which time the force of interest is δ (t ) .
The present value at time a of this payment stream is: b ⎛

t



a




a




∫ ρ (t ) exp ⎜ − ∫ δ (s) ds ⎟ dt
Question 5.10
Explain where this formula has come from.
The accumulated value at time b of this payment stream is:
⎛b
⎞ ρ (t ) exp ⎜ ∫ δ ( s ) ds ⎟ dt



a
⎝t
⎠ b Question 5.11
Explain where this formula has come from.
Very often in questions involving payment streams, you will be using the integral:

∫ f ′( x) exp [ f ( x)] dx = exp [ f ( x)] + constant
You should have covered this in A-Level Maths, Higher Level Maths or the equivalent.
Notice that in the formulae for the present and accumulated values, the integral in the exponential term has t as one of the limits. Students very often make a mistake here and substitute in the wrong limits.

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Example
The force of interest is:

d (t ) = 0.01t + 0.04

0≤t≤5

Find the present value at time 0 of the payment stream 0.5t + 2 , which is received between time 0 and 5.

Solution
The present value is: b Ê

t

ˆ

5

Ê

t

ˆ

a

Ë

a

¯

0

Ë

0

¯

Ú r (t ) exp Á - Ú d (s) ds˜ dt = Ú (0.5t + 2) exp Á - Ú 0.01s + 0.04 ds˜ dt
5

t
Ê È
˘ ˆ
= Ú (0.5t + 2) exp Á - Î 0.005s 2 + 0.04s ˚ ˜ dt
Ë

0

(

5

)

= Ú (0.5t + 2) exp - È 0.005t 2 + 0.04t ˘ dt
Î
˚
0

5

)

(

)

(

5

But ∫ −(0.01t + 0.04) exp − ⎡0.005t 2 + 0.04t ⎤ dt = ⎡exp − ⎡0.005t 2 + 0.04t ⎤ ⎤ , so:




⎦ ⎦0


0

5
Ê t
ˆ
2
Ú r (t ) exp Á - Ú d (s) ds˜ dt = 50Ú (0.01t + 0.04) exp - È0.005t + 0.04t ˘ dt
Î
˚
Ë a
¯
0 a (

b

(

)

)

5

= -50 È exp - È 0.005t 2 + 0.04t ˘ ˘
Í
Î
˚ ˙0
Î
˚
= -50 (exp( -0.125 - 0.2) - exp(0) )
= 13.87

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Question 5.12
Find the accumulated value of a payment stream of 0.3 + 1.5t that is received continuously from time 4 to time 8 during which time the force of interest is
0.01 + 0.05t .

2.3

Sudden changes in interest rates
Where the force of interest is not a continuous function of time, it is necessary to break up the calculations at the points where the interest rate changes. Where payments do not start immediately, the appropriate discount factor must be included.

Example
Calculate the present values as at 1 January 2005 of the following payments:
(i)

a single payment of £2,000 payable on 1 July 2009

(ii)

a single payment of £5,000 payable on 31 December 2016.

Assume effective rates of interest of 8% per annum until 31 December 2011 and 6% per annum thereafter.

Solution
(i)

Here, the interest rate is constant throughout the relevant period, so the present value is just:
2, 000 v 4½ @8% = 2, 000 ¥ 0.70728 = £1, 415

(ii)

Here, we need to break the calculation up at 31 December 2011 when the interest rate changes:
5, 000 v 7@8% ¥ v5@ 6% = 5, 000 ¥ 0.58349 ¥ 0.74726 = £2,180

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Question 5.13
An investment of £1,000 made at time 0 is accumulated at the following rates: 8% per annum simple for two years, followed by a rate of discount of 6% per annum convertible monthly for two years. Calculate the accumulated amount of the investment after 4 years.

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Interest income
Consider now an investor who wishes not to accumulate money but to receive an income while keeping his capital fixed at C. If the rate of interest is fixed at i per time unit, and if the investor wishes to receive income at the end of each time unit, it is clear that the income will be iC per time unit, payable in arrear, until such time as the capital is withdrawn.

This is because the effective rate of interest, i, is defined to be the amount of interest a single initial investment will earn at the end of the time period.

Simple example
An investor who wishes to receive income deposits £1,000 in a bank account that pays an effective rate of interest of 8% per annum. The interest income is paid to the investor at the end of year. The amount of each payment is £80, (8% of 1,000).
More generally, suppose that t > t0 and that the investor wishes to deposit C at time t0 for withdrawal at time t. Suppose further that n > 1 and that the investor wishes to receive interest on the deposit at the n equally spaced times t0 + h, t0 + 2h, … , t0 + nh , where h = (t - t0 ) n . The interest payable at time

t0 + ( j + 1)h , for the period t0 + jh to t0 + ( j + 1)h , will be:
Chih (t0 + jh )

Again, this is true because of our definition of ih (t ) .

Question 5.14
What is this definition and what is ih (t ) called?

Question 5.15
Look again at the previous simple example. If the investor wishes to receive 12 payments of interest each year so that interest is paid at the end of each month, how much is each interest income payment?

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CT1-05: Discounting and accumulating

The total interest income payable between times t0 and t will be: n -1

C

 hih (t0 + jh) j =0

Since, by assumption, ih (t ) tends to d (t ) as h tends to 0, it is fairly easily shown
(provided that d (t ) is continuous) that as n increases (so that h tends to 0) the total interest received between times t0 and t converges to:

I (t ) =

t

Út Cd (s )ds

(4.1)

0

Hence, in the limit, the rate of payment of interest income per unit time at time t,
I ¢(t ) , equals:

Cd (t )

If interest is paid continuously to the investor then we are just considering a continuous cashflow with a rate of payment of Cδ (t ) . The total amount of interest received can therefore be found by applying Formula (1.1), which gives Formula (4.1). The total amount of interest received is the sum, between t0 and t, of lots of small interest payments, each of amount Cd ( s )ds .
If the investor withdraws the capital at time T, the present values of the income and capital at time 0 are:

C

T

Ú0 d (t )v (t )dt

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Since:
T

È

T

t

˘

Ú0 d (t )v (t )dt = Ú0 d (t ) exp Í - Ú0 d (s )ds ˙ dt
Î
˚
T
È
Ê t
ˆ˘
= Í - exp Á - d ( s )ds ˜ ˙
Ë 0
¯˚
Î
0

Ú

= 1 - v (T ) we obtain:

C =C

T

Ú0 d (t )v (t )dt + Cv (T )

as one would expect by general reasoning.

If we invest an amount of capital C, then the present value of the proceeds we receive from this investment should equal our original amount of capital.

Example
An investor deposits £2,000 in a bank account and receives income at the end of each of the next three years. The rate of interest is 4% pa effective. The investor withdraws the capital after three years.
At the end of each year the investor receives 0.04 ¥ 2, 000 = £80 .
The present value of the interest received is:
80(v + v 2 + v3 ) = £222.01
The present value of the capital received after three years is:
2, 000v3 = £1, 777.99
The present value of the capital plus the present value of the interest equals the initial investment, ie:

1, 777.99 + 222.01 = £2, 000

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Question 5.16
A woman deposits £200 in a special bank account. Interest is paid to the woman every year on her birthday for five years. The capital is returned after exactly five years, along with any interest accrued since her last birthday. Interest is calculated at an effective rate of 6% pa. Calculate the present value of the interest received by the woman. So far we have described the difference between money returned at the end of the term and the cash originally invested as “interest”. In practice, however, this quantity may be divided into interest income and capital gains, the term capital loss being used for a negative capital gain.

If you invest some capital then not only might you receive income but the value of your capital may also increase (or decrease). Equities or shares are a good example of this.
These were introduced earlier in the course. If you buy some shares in a company then you should receive dividends or interest from the company. However the capital value that you receive back will depend upon the market price of the shares when you decide to sell. We will consider this in more detail later in the course.

Question 5.17
A rich woman pays £2m and in return expects to receive a continuous cashflow for the next six years with a constant rate of payment. Calculate the annual payment from this cashflow and the accumulated amount of the cashflow after six years if the interest rates is 9% pa effective.

Question 5.18
True or false:
A(0, t2 )
A(0, t1 )

(i)

A(t1 , t2 ) =

(ii)

A(0, t2 ) = A(0, t1 ) + A(t1, t2 )

(iii)

v(t2 ) = v(t1 )v(t2 − t1 )

(iv)

1
= A(0, t1 ) A(t1, t2 ) v(t2 )

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Page 23

Exam-style question
This is a typical exam-style question on this chapter. Have a go at it before turning over and having a look at the solution.
Question

The force of interest at any time t (measured in years) is given by:

0.04
0 < t ≤1 δ (t ) = 0.05t − 0.01 1 < t ≤ 5
0.24

t >5

(i)

What is the total accumulated value at any time t ( > 0 ) of investments of 1 at times 0, 4 and 6?

(ii)

What is the present value at time 0 of a payment stream paid at a rate of ρ (t ) = 5t − 1 received between t = 1 and t = 5 ?

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Solution

(i)

We have to break down the times for when there is a payment or when the force of interest changes. If we let A(t ) represent the accumulated value of the total investments made to date at time t, then:
0 < t ≤ 1:
A(t ) = e 0.04 t

1< t ≤ 4:
A(t ) = e

0.04

t

× exp

∫ 0.05s − 0.01ds
1

= e0.04 × exp 0.025s 2 − 0.01s
= e0.04 × e0.025t
= e0.025t

2

2

t
1

− 0.01t − 0.025+ 0.01

− 0.01t + 0.025

4 < t ≤ 5:
A(t ) = e

0.025×16− 0.01× 4+ 0.025

t

exp

∫ 0.05s − 0.01ds

t

+ exp

4

= (e0.385 + 1)e0.025t

2

∫ 0.05s − 0.01ds
4

− 0.01t −0.36

5< t ≤ 6:

A(t ) = (e

0.385

+ 1)e

0.625− 0.05− 0.36

t

exp

∫ 0.24 ds
5

= (e0.385 + 1)e0.215 e0.24t −1.2
= (e0.385 + 1)e0.24t −0.985

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t>6
A(t ) = (e0.385 + 1)e0.455 exp

t

∫ 0.24 ds + exp
6

t

∫ 0.24 ds
6

= (e0.84 + e0.455 + 1)e0.24t −1.44

(ii)

It is easiest to calculate the present value at time 1 and then discount back to time 0. The present value at time 1 is:
5

t

1

1

∫ (5t − 1) exp − ∫ 0.05s − 0.01 ds
5

dt t = ∫ (5t − 1) exp − 0.025s 2 − 0.01s dt
1

1

5

= ∫ (5t − 1) exp[ −0.025t 2 + 0.01t + 0.025 − 0.01] dt
1

5

= ∫ (5t − 1) exp[ −0.025t 2 + 0.01t + 0.015] dt
1

(

= −100 exp −0.025t 2 + 0.01t + 0.015

)

5
1

= −100 e −0.625+0.05+0.015 − e −0.025+0.01+0.015
= −100 e −0.56 − 1
= 42.879

We then need to discount this back to time 0:
1

PV = 42.879 × exp − ∫ 0.04 dt = 42.879e −0.04 = 41.20
0

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Chapter 5 Summary
In many compound interest problems you may need to find the discounted present value of cashflows due in the future. It is important to distinguish between discrete and continuous payments.
The present value of a series of discrete payments is the sum of the individual present values. The present value of continuous payments is found by integrating the rate of payment multiplied by a discount factor. So the formula for present value is:


∑ ct v(t ) + ∫ v(t ) ρ (t )dt
0

The net present value is defined as the difference between the value of the positive cashflow and the value of the negative cashflow.
The value of payments that are due after the time of valuation is called a discounted value. The value of payments that are due before the time of valuation is called an accumulated value.
The value of a cashflow at one particular time can easily be found from the value of the cashflow at a different time. The formula for moving along the timeline is:
Value at time t1 of cash flow

=

Value at time t2 of cash flow

v(t2 ) v(t1 )

An investor may wish to receive an income while keeping the amount of capital fixed.
The present value of the income plus the present value of the returned capital equals the initial capital invested ie:
T

C = C ∫ δ (t )v(t )dt + Cv(T )
0

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Chapter 5 Solutions
Solution 5.1
PV = 280v 2 + 360v 2.5 = 280 × 1.15−2 + 360 × 1.15−2.5 = 211.72 + 253.84 = £465.56

Solution 5.2 n tj

PV = ∑ ct j exp − ∫ δ ( s )ds
0

j =1

Solution 5.3
Total premium income:
3

5, 218, 000t 2
= ∫ 5, 218, 000tdt =
2
0

3

= 2, 609, 000 × 9 = £23, 481, 000
0

Solution 5.4
Present value of the income:
5

5

0

0

∫ 350v(t )dt = ∫ 350(1 − t 100)dt
= 350t − 1.75t 2

5
0

= £1, 706.25

Present value of the outgo:
600v(1) + 400v(3) = 600 × 0.99 + 400 × 0.97 = £982
So NPV = 1, 706.25 − 982 = £724.25 .

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CT1-05: Discounting and accumulating

Solution 5.5
Value = Ce − (t2 −t1 )δ
= Ce(t1 −t2 )δ

Solution 5.6
860 ×

v(10)
0.76
= 860 ×
= 718.24 v(4) 0.91

Solution 5.7
(t − 2)3
, we can calculate v(1) = 0.93 , v(2) = 0.92 ,
From the formula v(t ) = 0.92 −
100
v(3) = 0.91 , v(4) = 0.84 , v(5) = 0.65 .
The accumulation is the value on 1 January 2007 of the first two payments which is equal to:
1
(100v(1) + 130v(2)) = £233.63 v(3) The discounted value is the value on 1 January 2007 of the last two payments which is equal to:
1
(150v(4) + 160v(5)) = £252.75 v(3) So the net value = 233.63 + 252.75 = £486.38 .

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Solution 5.8
(i)

To find the PV as at 1 January 2002, we must move our reference point back in time 1 year (from 1 January 2003 to 1 January 2002) ie we need to multiply
19,243.72 by a factor of v 4 @ 2% = 1/1.024 = 0.923845 .
So PV as at 1 January 2002 = 19, 244v 4 = £17, 778 .
(The PV will be less, since the payments are now more distant from the reference point.) (ii)

PV as at 1 January 2005 = 19, 244 × 1.028 = £22,547 .

(iii)

PV as at 1 July 2017 = 19, 244 × 1.0258 = £60, 688 .

Solution 5.9 t 10 −0.03t − ∫ 0.04 ds e e 0 dt 0

PV = ∫

0

10

1 −0.07t e = −
0.07
=

10

= ∫ e −0.07t dt

0

1
(1 − e −0.7 ) = 7.192
0.07

Solution 5.10 b The formula is

t

∫ ρ (t ) exp −∫ δ (s) ds a dt . Consider the payment at time t, which is at a

a

rate of ρ (t ) . This payment needs to be discounted back to time a and the discount t factor will be exp − ∫ δ ( s ) ds . a Finally, we need to add together all the present values of the payments at the different times. Since we are receiving payments continuously, we integrate these present values between the limits a and b, ie the times between which the payment comes in.

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CT1-05: Discounting and accumulating

Solution 5.11 b The formula is

∫ ρ (t ) exp a b

∫ δ (s) ds

dt . Consider the payment at time t, which is at a

t

rate of ρ (t ) . This payment needs to be accumulated to time b and the accumulation b factor will be exp

∫ δ (s) ds

.

t

Finally, we need to add together all the accumulated values of the payments at the different times. Since we are receiving payments continuously, we integrate these present values between the limits a and b, ie the times between which the payment comes in.

Solution 5.12
The accumulated value is: b ∫ ρ (t ) exp a b

8

8

∫ δ (s) ds dt = ∫ (0.3 + 1.5t ) exp t ∫ 0.01 + 0.05s ds

4

t

8

= ∫ (0.3 + 1.5t ) exp

0.01s + 0.025s 2

4

8

dt
8
t

dt

(

= ∫ (0.3 + 1.5t ) exp 0.08 + 1.6 − 0.01t − 0.025t 2
4

8

(

= ∫ (0.3 + 1.5t ) exp 1.68 − 0.01t − 0.025t 2
4

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) dt

) dt

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Page 33

But:
8

(

)

(

2
2
∫ −(0.01 + 0.05t ) exp 1.68 − 0.01t − 0.025t dt = exp 1.68 − 0.01t − 0.025t
4

)

8
4

so: b b

a

t

∫ ρ (t ) exp ∫ δ (s) ds

8

(

)

dt = 30∫ (0.01 + 0.05t ) exp 1.68 − 0.01t − 0.025t 2 dt
4

(

= −30 exp 1.68 − 0.01t − 0.025t 2

)

8
4

= −30 ( exp(1.68 − 0.08 − 1.6) − exp(1.68 − 0.04 − 0.4) )

(

)

= −30 e0 − e1.24 = 73.67

Solution 5.13
The accumulated value of the investment is:

(

1, 000 (1 + 2 × 0.08) 1 − 0.06
12

)

−24

= £1,308.29

Solution 5.14 ih (t ) is called the nominal rate of interest per unit time on transactions of term h beginning at time t. It is defined to be such that the effective rate of interest for the period of length h beginning at time t is hih (t ) .

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CT1-05: Discounting and accumulating

Solution 5.15
The effective monthly interest rate is:
(1.081/12 − 1) = 0.643%

Therefore each interest payment is £6.43.
Relating this to the algebra given in the Core Reading: h = 1 12 and ih (t ) = ih = 12 × (1.081/12 − 1) = 7.721%

(This is the nominal interest rate convertible monthly, i (12) .)
The amount of each payment is therefore 1, 000 × 1 12 × 0.07721 = £6.43 .

Solution 5.16
The present value of the interest received plus the present value of the capital returned will equal the initial deposit. Therefore the present value of the interest received equals the initial deposit less the present value of the capital returned:
200 − 200v5 = 200 −

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200
1.065

= £50.55

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Page 35

Solution 5.17
If C is the constant rate of payment per annum then we need to solve the equation:
6

2, 000, 000 = ∫ Cvt dt
0

vt
=C
log v

6

0

1 − v6
=C
log(1 + i )
⇒C =

2, 000, 000
= £0.427m
4.68489

Because we know the present value of the cashflow is £2m we can easily work out the accumulated value after six years by multiplying this by (1 + i )6 :
Accumulated value = 2, 000, 000 × 1.096 = £3.354m

Solution 5.18
(i)

True

(ii)

False

(iii)

False

(iv)

True

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CT1-06: Level annuities

Page 1

Chapter 6
Level annuities
Syllabus objective
(vi)

Define and use the more important compound interest functions, including annuities certain.
1.

Derive formulae in terms of i, v, n, d, δ , i ( p ) and d ( p ) for an| ,

(
(
(
(
sn| , an|p ) , sn|p ) , an| , sn| , an|p ) , sn|p ) , an| and sn| .

0

Introduction
In the next two chapters we will learn how to find the present value of a series of payments. We will also meet some actuarial symbols representing compound interest functions, which we will use very frequently in this course.

Terminology
Here is a brief summary of the key terms that we will use:
An annuity is a regular series of payments. An annuity certain is an annuity payable for a definite period of time: the payments do not depend on some factor, such as whether a person is alive or not.
If payments are made at the end of each time period, they are paid in arrear. If they are made at the beginning of each time period, they are paid in advance. An annuity paid in advance is also known as an annuity-due.
Where the first payment is made during the first time period, this is an immediate annuity.
Where no payments are made during the first time period, this is a deferred annuity.
If each payment is for the same amount, this is a level annuity. If payments increase
(decrease) each time by the same amount, this is a simple increasing (decreasing) annuity. The Actuarial Education Company

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CT1-06: Level annuities

This chapter covers level immediate annuities. Chapter 7 then covers increasing and decreasing annuities as well as deferred annuities (those that start at some point in the future). Ultimately, Chapter 7 deals with variable payments that can be evaluated using similar techniques to the ones detailed here.

Question 6.1

(easy)

State whether each of the following annuities is paid in arrear or in advance, immediate or deferred, level, simple increasing or simple decreasing.
(i)

Payments of £2 paid at the start and halfway through each of the next five years.

(ii)

A payment of £5 in five years, £10 in ten years, … , £40 in forty years.

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Page 3

1

Present values

1.1

Payments made in arrear
Consider a series of n payments, each of amount 1, to be made at time intervals of one unit, the first payment being made at time t + 1 .
1
t

1

1

...

1

1

payment

t +1

t+2

t+3

...

t + n−1

t+n

time

Such a sequence of payments is illustrated in the diagram above, in which the rth payment is made at time t + r .
The value of this series of payments one unit of time before the first payment is made is denoted by an| .

So in the above example, a n| is the value at time t of the payments shown.
The symbol a n| (pronounced “A.N.”) represents the PV of an annuity consisting of n payments of 1 unit made at the end of each of the next n time periods. This is called an annuity paid in arrear.
Clearly, if i = 0 , then an| = n ; otherwise:

an|

= v + v2 + v3 +

=
=

=

+ vn

v (1 - v n )
1- v
1- vn v -1 - 1
1− vn i The Actuarial Education Company

(1.1)

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CT1-06: Level annuities

Lots of derivations in this course will use the formula for the summation of a geometric series, which you will have covered in A-Level Maths (or equivalent). This is covered in the ActEd course, Foundation ActEd Course (FAC), but in case you’ve forgotten it: a + ax +

+ ax

n -1

n -1

a (1 - x n )
= Â ax =
1- x k =0 k for x ≠ 1

An alternative way is prove result (1.1) is to multiply the first equation through by (1 + i ) , so that v 3 becomes v 2 etc. This gives a series very similar to the original series. We can then obtain a formula for a n| by subtracting the two equations: an| =

v + v 2 + v3 +

(1 + i )an| = 1 + v + v 2 + v3 +
\

ian| = 1

a n| =

+ v n -1 + v n
+ v n -1
- vn

1 − vn i So:

If n = 0 , an| is defined to be zero.
Thus an| is the value at the start of any period of length n of a series of n payments, each of amount 1, to be made in arrear at unit time intervals over the period. It is common to refer to such a series of payments, made in arrear, as an immediate annuity-certain and to call an| the present value of the immediate annuity-certain. When there is no possibility of confusion with a life annuity (ie a series of payments dependent on the survival of one or more human lives), the term annuity may be used as an alternative to annuity-certain.

an is called an immediate annuity even though the first payment is made at the end of the first time period, and hence not “immediately”. Immediate annuities are annuities where the first payment is made during the first time period, including at the end of the period. If no payments are made during the first time period then the annuity is deferred. Deferred annuities are covered in the next chapter.

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Page 5

1− vn is a very important formula. It is used time and time again in this course i and it’s worthwhile memorising it now. a n| =

In the Tables you will find values of an tabulated at various interest rates from ½% to
25%.
@
You may see an calculated at i% effective written as an| i % , an|@i % or an|, i % .

Question 6.2
Verify the value of a10| given in the Tables, calculated at 4% interest.

Example
Calculate the present value as at 1 March 2005 of a series of payments of £1,000 payable on the first day of each month from April 2005 to December 2005 inclusive, assuming a rate of interest of 6% pa convertible monthly.

Solution
An interest rate of 6% pa convertible monthly is equivalent to an effective monthly interest rate of ½%.
There are 9 payments of £1,000 each, starting in one month’s time.
So, working in terms of months, the PV of the payments is:
@½%
1, 000a9|
@½%
You can look up a9| in the Tables or alternatively apply the formula:

1,000a

@½%
9|

= 1000 ×

1 − 1005−9
.
= 1000 × 8.7791 = £8,779
0.005

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1.2

CT1-06: Level annuities

Payments made in advance
Now consider a series of n payments made at the start of each time period as represented on the time line below. The first payment is made at time zero and the last at time n − 1.
Payment

1

1

1

1

Time

0

1

2



3

1

n–1

n

The value of this series of payments at the time the first payment is made is denoted by an| .

The symbol an| is pronounced “A due n”.
If i = 0 , then an| = n ; otherwise:

an|

= 1+ v + v2 +

=

1− vn
1− v

=

+ v n −1

1− vn d (1.2)

Again, this series is a geometric progression and the formula given on page 4 has been used to sum it.
Thus an| is the value at the start of any given period of length n of a series of n payments, each of amount 1, to be made in advance at unit time intervals over the period. It is common to refer to such a series of payments, made in advance, as an annuity-due and to call an| the present value of the annuity-due.
It follows directly from the above definitions that: an| = (1 + i )an| and that, for n ≥ 2 : an| = 1 + an − 1|

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U
|
|
V
|
|
|
W

(1.3)

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Page 7

The payments for a n| correspond exactly with those for a n| , except that each payment is made one year earlier ie each payment has a value that is greater by a factor of (1 + i ) . So an| = (1 + i )an| .
The above definitions also lead directly to this result since d = i (1 + i ) . Alternatively, consider the two series:

an = 1 + v + v 2 +

+ v n -1

an = v + v 2 + v3 +

+ vn

It is easy to see that an| = (1 + i )an| .

Question 6.3
Prove algebraically and by general reasoning that a n| = a n −1| + 1.

Question 6.4
Calculate a25| and a15| at 13½% pa effective.

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2

CT1-06: Level annuities

Accumulations
The value of the series of payments at the time the last payment is made is denoted by sn| . The value one unit of time after the last payment is made is denoted by s n| .

In other words, sn| considers the same series of payments as an| but it is the accumulated value at time n, as opposed to the present value at time 0. Similarly, sn| is the accumulated value at time n of the annuity we looked at above when defining an| .
Value

a n|

Time

0

sn|
1

n −1

2

n

If i = 0 then sn| = sn| = n ; otherwise

sn|

= (1 + i )n − 1 + (1 + i )n − 2 + (1 + i )n − 3 +

+1

= (1 + i )n an|

=

(1 + i )n − 1 i (2.1)

and:

sn|

= (1 + i )n + (1 + i )n − 1 + (1 + i )n − 2 +

+ (1 + i )

= (1 + i )n an|

=

(1 + i )n − 1 d © IFE: 2009 Examinations

(2.2)

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Page 9

Thus s n| and s n| are the values at the end of any period of length n of a series of n payments, each of amount 1, made at unit time intervals over the period, where the payments are made in arrear and in advance respectively. Sometimes s n| and s n| are called the “accumulation” (or the “accumulated amount”) of an immediate annuity and an annuity-due respectively. When n = 0 , s n| and s n| are defined to be zero. It is an immediate consequence of the above definition that: s n| = (1 + i )s n| and that: s n + 1| = 1 + sn| or: s n| = sn + 1| − 1

Question 6.5
Calculate s10| and s13| at 3½% pa effective.

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3

CT1-06: Level annuities

Continuously payable annuities
Let n be a non-negative number. The value at time 0 of an annuity payable continuously between time 0 and time n, where the rate of payment per unit time is constant and equal to 1, is denoted by an| .

This is a mathematical idealisation, which makes calculations easier. With a continuously payable annuity, the payments are considered as a continuous cashflow that is payable at a given rate over a given period of time. The symbol an| is pronounced “A bar n”.
Clearly:

=

an|

=

=

z

n

0

1.v t dt

=

z

n − δt e dt
0

LM
N

1
= − e −δt

δ

OP
Q

n
0

1 − e − δn

δ
1− vn

δ

(if δ ≠ 0 )

(4.1)

Note that an| is defined even for non-integral values of n.

If δ = 0 (or,

equivalently, i = 0 ), an| is of course equal to n.
Since equation (4.1) may be written as:

an| =

F
GH

i 1− vn δ i

I
JK

it follows immediately that, if n is an integer:

an| =

i

δ

an|

(if δ ≠ 0 )

Note that this relationship can be very useful, especially if you like using the Tables.
Although values of an| are not tabulated, values of an| and i δ are.
The accumulated amount of such an annuity at the time the payments cease is denoted by s n| .

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Page 11

By definition, therefore:

sn| =

z

n δ (n − t ) e dt
0

Hence:

sn| = (1 + i )n an|
If the rate of interest is non-zero:

=

s n|

=

(1 + i )n − 1

δ i δ

sn|

Question 6.6
Find an approximate value for the present value of a series of payments of £1 each, payable for 1 year at the beginning of each week, assuming an effective rate of interest of
8% per annum.
(Assume that there are 52.18 weeks in a year.)
Notice the similarity between the formulae for the present and accumulated values of annuities: an =

1 - vn i an =

1 - vn d an =

sn =

(1 + i ) n - 1 i sn =

(1 + i ) n - 1 d sn =

1 - vn

d
(1 + i ) n - 1

d

The numerators are the always consistent, the only difference between the formulae are the denominators.

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CT1-06: Level annuities

4

Annuities payable pthly

4.1

Present Values
Where annuity payments are made p times a year (eg p = 12 for a monthly annuity), a superscript ( p) is added in the top right hand corner of the symbol. Note that the annuity is still payable for n years and still refers to a total annual amount of 1 unit ie the annuity consists of np payments, each of amount 1 / p units.
If p and n are positive integers, the notation a( p ) is used to denote the value at n| time 0 of a level annuity payable pthly in arrear at the rate of 1 per unit time over the time interval [0, n ] . For this annuity the payments are made at times

1 p , 2 p , 3 p,…, n and the amount of each payment is 1 p .
By definition, a series of p payments, each of amount i ( p ) p in arrear at pthly subintervals over any unit time interval, has the same value as a single payment of amount i at the end of the interval. By proportion, p payments, each of amount 1 / p in arrear at pthly subintervals over any unit time interval, have the same value as a single payment of amount i i ( p ) at the end of the interval.
Consider now that annuity for which the present value is a( p ) . The remarks in n| the preceding paragraph show that the p payments after time r − 1 and not later than time r have the same value as a single payment of amount i i ( p ) at time r.
This is true for r = 1, 2, … , n , so the annuity has the same value as a series of n payments, each of amount i i ( p ) , at times 1, 2, … , n . This means that:

a( p ) =
|
n

i i ( p)

an|

(5.1)

(
Note that i i ( p ) is tabulated in the Tables and so an|p ) can quickly be calculated for

some interest rates by looking up i i ( p ) and an| . Alternatively it can be calculated directly from the formula:
(
an|p ) =

1− vn i ( p)

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Page 13

An alternative approach, from first principles, is to write:

a( p )
|

np

n

1

∑ p vt /p

=

t =1

1 v 1/ p (1 − v n ) p 1 − v 1/ p

=

1− vn

=

p (1 + i )1/ p − 1
1− vn

=

(5.2)

i ( p)

which confirms equation (5.1).

Question 6.7
(
Calculate a64 ) at 1½% pa, first without using the Tables and then with the Tables.
|

Likewise we define a( p ) to be the present value of a level annuity-due payable n| pthly at the rate of 1 per unit time over the time interval [0, n ] . (The annuity payments, each of amount 1 p , are made at times 0, 1 p , 2 p ,… , n − (1 p ) .)
By definition, a series of p payments, each of amount d ( p ) p , in advance at pthly subintervals over any unit time interval has the same value as a single payment of amount i at end of the interval. Hence, by proportion, p payments, each of amount 1 p in advance at pthly subintervals, have the same value as a single payment of amount i d ( p ) at the end of the interval. This means (by an identical argument to that above) that:

a( p ) =
|
n

i d ( p)

an|

(5.3)

Alternatively:
(
an|p )

=

1− vn d ( p)

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CT1-06: Level annuities

Alternatively, from first principles, we may write: a( p )
|
n

=

np

1

∑ p v (t −1)/ p

t =1

=

1− vn

(5.4)

d ( p)

(on simplification), which confirms equation (5.3). Note that: a( p ) = v 1/ p a( p )
|
| n (5.5)

n

each expression being equal to

(1 − v n ) i ( p)

.

(
Equation (5.5) can be derived by general reasoning. The payments for an|p ) correspond
(
exactly with those for an|p ) , except that each payment is made a period of length 1 p

earlier, ie each payment has a value that is greater by a factor of (1 + i )1/ p .
(
(
So an|p ) = (1 + i )1/ p an|p ) which is equivalent to equation (5.5).

Note that, since:

lim i ( p ) = lim d ( p ) = δ

p→ ∞

p→ ∞

it follows immediately from equation (5.2) and (5.4) that:

lim a( p ) = lim a( p ) = an| p →∞ n| p → ∞ n|

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Page 15

Example
Find the present value as at 1 January 2004 of a series of payments of £100 payable on the first day of each month during 2005, 2006 and 2007, assuming an effective rate of interest of 8% per annum.

Solution
The present value of the payments as at 1 January 2005 is 1, 200 a (12) .
|
3

So the present value as at 1 January 2004 is:
1, 200 v a (12) = 1, 200 v
|
3

1 - v3 d (12)

= 1, 200 ¥ 0.92593 ¥

1 - 0.79383
= £2,986
0.076714

using the values given in the Tables.

Question 6.8
Find the present value as at 1 June 2004 of payments of £1,000 payable on the first day of each month from July 2004 to December 2004 inclusive, assuming a rate of interest of 8% per annum convertible quarterly.

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4.2

CT1-06: Level annuities

Accumulations
Similarly, we define s ( p ) and s ( p ) to be the accumulated amounts of the n| n| corresponding pthly immediate annuity and annuity-due respectively. Thus: s ( p)
|
n

= (1 + i )n a( p )
|
n

= (1 + i )n

=

=

i i ( p)

i i ( p)

an|

(by (5.1))

sn|

(1 + i ) n − 1 i ( p)

Also:

s ( p)
|
n

= (1 + i )n a( p )
|
n

= (1 + i )n

=

=

i d ( p)

i d ( p)

an|

(by (5.3))

sn|

(1 + i ) n − 1 d ( p)

As previously, notice the similarity between the formulae for the present and accumulated values of annuities: a( p) =

1 - vn

s( p) =

(1 + i ) n - 1

n

n

i( p)

i( p)

a( p) =

1 - vn

s ( p) =

(1 + i ) n - 1

n

n

d ( p)

d ( p)

The numerators are the always consistent, the only difference between the formulae are the denominators.

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Page 17

Question 6.9
You have just invested £1,000 in a fixed interest security. In return you will receive
£40 at the end of each half year plus your money back on redemption in 12 years. You intend to deposit all of the proceeds in a bank account that will pay an effective rate of interest of 8% pa.
How much money do you expect there to be in the bank account after 10 years?
The above proportional arguments may be applied to other varying series of payments. Consider, for example, an annuity payable annually in arrear for n years, the payment in the tth year being x t . The present value of this annuity is obviously: a=

n

∑ xt v t

(5.6)

t =1

Consider now a second annuity, also payable for n years with the payment in the tth year, again of amount x t , being made in p equal instalments in arrear over that year. If a( p ) denotes the present value of this second annuity by replacing the p payments for year t (each of amount x t p ) by a single equivalent payment at the end of the year of amount x t [i i ( p ) ] , we immediately obtain

a( p ) =

i i ( p)

a

where a is given by equation (5.6) above.

Question 6.10
The present value at 6% pa of the following series of payments is $245.32.
$1 at time 1, $4 at time 2, $9 at time 3, … ,$100 at time 10.
What is the present value of the series of payments if, instead of being paid at the end of each year, the payments are made in three equal instalments at the end of each third of a year? The Actuarial Education Company

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4.3

CT1-06: Level annuities

Annuities payable pthly where p < 1
If p < 1 then we are considering an annuity payable less frequently than annually. For example, if p = 0.5 , the annuity is payable biennially (every two years). The formulae
(
( for an|p ) and an|p ) are still valid for p < 1.

In Section 5.1 the symbol a( p ) was introduced. Intuitively, with this notation one n| considers p to be an integer greater than 1 and assumes that the product n. p is also an integer. (This, of course, will be true when n itself is an integer, but one might for example, have p = 4 and n = 5.75 so that np = 23 .) Then a( p ) denotes n| the value at time 0 of n. p

payments, each of amount 1 p , at times

1 p , 2 p , ... ,(np ) p .

Non-integer values of n will be looked at more closely in the next section.
From a theoretical viewpoint it is perhaps worth noting that when p is the reciprocal of an integer and n. p is also an integer (eg when p = 0.25 and

n = 28 ), a( p ) still gives the value at time 0 of n. p payments, each of amount 1 p , n| at times 1 p , 2 p , ... ,(np) p .
For example, the value at time 0 of a series of seven payments, each of amount
4, at times 4, 8, 12 ,..., 28 may be denoted by a( 0.|25 ) .
28
It follows that this value equals:

1 − v 28
(0.25) ⎡(1 + i )4 − 1⎤


This last expression may be written in the form:

LM
OP 28
MM 44 PP. 1 − iv = s4 . a28|
4|
MN (1 + ii) − 1 PQ

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5

Page 19

Non-integer values of n
Let p be a positive integer. Until now the symbol a( p ) has been defined only n| when n is a positive integer. For certain non-integral values of n the symbol a( p ) n| has an intuitively obvious interpretation.

For example, it is not clear what

meaning, if any, may be given to a23.5| , but the symbol a( 4 ) | ought to represent
23.5
the present value of an immediate annuity of 1 per annum payable quarterly in arrear for 23.5 years (ie a total of 94 quarterly payments, each of amount 0.25).
On the other hand, a( 2) | has no obvious meaning.
23.25
Suppose that n is an integer multiple of 1 p , say n = r p , where r is an integer.
In this case we define a( p ) to be the value at time 0 of a series of r payments, n each of amount 1 p , at times 1 p , 2 p , 3 p , ..., r p = n .

If i = 0 , then clearly

a( p ) = n . If i ≠ 0 , then:
|
n

1 1/ p
(v
+ v 2/ p + v 3/ p + p 1 1/ p 1 − v r / p v p
1 − v 1/ p

=

n

=

=

a( p )
|

1
1− vr /p p (1 + i )1/ p − 1

F
GH

LM
NM

+ vr /p )

I
JK

OP
QP

Thus:

R1 − v n
|
a( p ) = S i ( p )
|
n
|n
T

if i ≠ 0

(5.1)

if i = 0

(
The standard formula for an|p ) therefore applies for non-integer values of n when n is an

integer multiple of

1
.
p

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CT1-06: Level annuities

Note that, by working in terms of a new time unit equal to

1 times the original p time unit and with the equivalent effective interest rate of

i ( p) per new time unit, p we see that:

a( p ) (at rate i ) = n 1 i ( p)
)
anp (at rate p p

This formula is useful when

i ( p) is a tabulated rate of interest. p Example
(
Calculate a312 ) given that i = 19.5618% .
.5|

Solution
If i = 19.5618% then

i (12)
= 1.5% .
12

@
Therefore it is easier to find a421.5% since it is tabulated in the Tables.
|
(
1 @ a312 ) = 12 a421.5% = 2.583
|
.5|

Question 6.11
(4
Calculate a13.)25| at 10.3813% pa effective.

Note that the definition of a( p ) given by equation (5.1) is mathematically n meaningful for all non-negative values of n. For our present purpose, therefore, it is convenient to adopt Equation (5.1) as a definition of a( p ) for all n. n This is only a mathematical definition. It is not easily translated into the present value of a series of payments (although it can be done).

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CT1-06: Level annuities

Page 21

If n is not an integer multiple of

1
, there is no universally recognised definition p of a( p ) . For example, if n = n1 + f , where n1 is an integer multiple of 1 p and n 0 < f < 1 p , some writers define a( p ) as: n a( p|) + fv n n1 With this alternative definition:

a( 2) | = a( 2) | + 1 4 v 23.75
23.75
23.5 which is the present value of an annuity of 1 per annum, payable half-yearly in arrear for 23.5 years, together with a final payment of 0.25 after 23.75 years. Note that this is not equal to the value obtained from Definition (5.1).

Question 6.12
(2
Confirm this last statement by calculating a23.)75| using both definitions given that

i = 0.03 .
(
(
(
We can extend the above results to find formulae for an|p ) , sn|p ) and sn|p ) for all non-

negative n.
If i ≠ 0 , we define for all non-negative n:
( p)

( p)

a | = (1 + i )1/ p a | = n n
( p)

( p)

s | = (1 + i )n a | = n n
( p)

( p)

s | = (1 + i )n a | = n n

¸
Ô
Ô
Ô
Ô
˝
Ô
Ô
Ô
Ô
Ô
˛

(1 - v n ) d ( p)

(1 + i )n - 1 i ( p)
(1 + i )n - 1 d ( p)

(5.2)

If i = 0 , each of these last three functions is defined to equal n.

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CT1-06: Level annuities

Whenever n is an integer multiple of 1 p , say n = r p , then a( p ) , s ( p ) , s ( p ) , are n| n| n| values at different times of an annuity-certain of r payments, each of amount 1 p , at intervals of 1 p time unit.
As before, we use the simpler notations an| , an| , sn| and sn| to denote

a(1) , a(1) , s (1) and s (1) respectively, thus extending the definition of an| etc, to all n| n| n| n| non-negative values of n. It is a trivial consequence of our definitions that the formulae: i
( p) a | = a ( p) n | n i i ( p)

a | = n ( p) s | n =

d

i i ( p) i ( p)

s | = n ( p)

d

( p)

an | sn | sn |

¸
Ô
Ô
Ô
Ô
Ô
Ô
˝
Ô
Ô
Ô
Ô
Ô
Ô
˛

(5.3)

(valid when i ≠ 0 ) now hold for all values of n.

Question 6.13
(
(
True or false: sn|p ) = (1 + i ) sn|p )

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6

Page 23

Perpetuities
We can also consider an annuity that is payable forever. This is called a perpetuity. For example, consider an equity that pays a dividend of £10 at the end of each year. Equities are covered in more detail in Chapter 12, Section 2.
An investor who purchases the equity pays an amount equal to the present value of the dividends. The present value of the dividends is:

10v + 10v 2 + 10v 3 +
This can be summed using the formula for an infinite geometric progression:

10v + 10v 2 + 10v 3 +

=

10v 10
=
1− v i Recall the formula for the present value of an annuity of £10 pa that continues for n years:

10an =

10(1 − v n ) i We have let n → ∞ in this expression in order to arrive at the formula

10
.
i

Note that this formula only holds when i is positive.

Question 6.14
Calculate the present value of an annuity that pays £150 pa annually in arrears forever using an annual effective rate of interest of 8%.
In general:

Perpetuity
The present value of payments of 1 pa payable at the end of each year forever is
1
1
. This present value is written as a∞ , ie a∞ = . i i
The present value of payments of 1 pa payable at the start of each year forever is
1
1
. This present value is written as a∞ , ie a∞ = . d d

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CT1-06: Level annuities

Question 6.15
Calculate the present value of payments of £2,000 at times 0,1, 2,… using i = 7.6% pa effective. Perpetuities payable pthly
1
The present value of payments of 1 pa payable in instalments of p at the end of

each pthly time period forever is:

1 a( p ) = ( p )

i
1
The present value of payments of 1 pa payable in instalments of p at the start of

each pthly time period forever is:

a( p ) =


1

d

( p)

Question 6.16
Calculate the present value of an annuity that pays £300 pa monthly in arrears forever using an annual effective rate of interest of 6%.

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CT1-06: Level annuities

Page 25

Chapter 6 Summary
An annuity consists of a regular series of payments. Payments continue for a specified period. The amounts of the payments may be level, increasing or decreasing. Continuous annuities involve continuous payments paid at specified rates.
The symbol a n| ( an| ) represents the PV of an annuity consisting of n payments of 1 unit made at the end (start) of each of the next n time periods. This is called an annuity paid in arrear (advance). The formulae for the present values are:

1 - vn i an| =
= an| d d

1− vn a n| = i = an -1| + 1 sn| and sn| are the values at the end (accumulated values) of any period of length n of a series of n payments, each of amount 1, made at unit time intervals over the period, where the payments are made in arrear and in advance respectively. The formulae for the accumulated values are:

(1 + i ) n - 1 i sn| =
= sn| d d

(1 + i ) n − 1 sn| = i = sn +1| - 1
The value at time 0 of an annuity payable continuously between time 0 and time n, where the rate of payment per unit time is constant and equal to 1, is denoted by an| .
The formulae for continuous payments are:

a n| =

1− vn

δ

=

i

δ

a n|

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sn| =

(1 + i ) n − 1

δ

=

i

δ

z n sn|

PV = r (t )v t dt
0

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CT1-06: Level annuities

(
If p and n are positive integers, the notation an|p ) is used to denote the value at time 0 of

a level annuity payable pthly in arrear at the rate of 1 per unit time over the time interval
[0, n] . For this annuity the payments are made at times 1 p , 2 p , 3 p,… , n and the amount of each payment is 1 p . The required formulae are: a ( |p ) =

1− vn

a ( |p ) n 1− vn

n

=

i

d

( p)

( p)

=

=

i i ( p)

i d ( p)

a n|

a n|

s ( |p ) =

(1 + i ) n − 1

s ( |p ) n (1 + i ) n − 1

n

=

i

( p)

d

( p)

=

=

i i ( p)

i d ( p)

sn|

sn|

(
There is more than one definition of an|p ) when n is not an integer multiple of 1 p .

An annuity that is payable forever is called a perpetuity. The required formulae are:

a• =

1 i a• =

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1 d a( p) =


1 i ( p)

a( p) =


1 d ( p)

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CT1-06: Level annuities

Page 27

Chapter 6 Solutions
Solution 6.1
(i)

This is a level immediate annuity payable half-yearly in advance for five years.

(ii)

This is a simple increasing, immediate annuity payable in arrear (at the end of every five years) for forty years.

Solution 6.2 a10| =

1 − v 10 1 − 104 −10
.
=
= 81109
.
i
0.04

This answer agrees with the value given on page 56 of the Tables.

Solution 6.3
Algebraically:
RHS = an -1| + 1
=

1 - v n -1
(1 + i ) - v n -1
1 - vn
1 - vn
+1 =
=
= i i i /(1 + i ) d = an| = LHS
Alternatively, you could prove it without using the formula used above: a n| = 1 + v + v 2 +

+ v n −1 = 1 + (v + v 2 +

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+ v n −1 ) = 1 + an −1|

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CT1-06: Level annuities

By general reasoning: a n| is the PV at time t = 0 of a series of payments of 1, payable at times t = 0, 1, 2, … , n − 1 .
This is the same as:


an initial payment of 1 (which has a PV of 1) plus



a series of payments of 1, payable at times t = 1,2,… , n − 1 (which has a PV of an −1| ).

So: an| = an −1| + 1

Solution 6.4 a25| =

1 − v 25 1 − 1135−25
.
=
= 7.095 i 0135
.

a15| =

1 − v15 1 − 1135−15
.
=
= 7.149
0135 1135
.
. d Solution 6.5 s10| =

(1 + i )10 − 1 103510 − 1
.
=
= 11731
.
0.035 i (1 + i )13 − 1 103513 − 1
.
s13| =
=
= 16.677
0.035 1035
.
d

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Page 29

Solution 6.6
If we approximate the weekly payments by a continuously payable annuity, and assume that there are 52.18 weeks on average in a year, then the PV of the payments is:

52.18a1@8% = 52.18
|

1− v

δ

= 52.18 ×

1 − 0.92593
= 52.18 × 0.96244 = £50.22
0.076961

(Note that, using this approximation, we would have arrived at exactly the same answer if payments were payable at the end of each week.)

Solution 6.7
First without the Tables:
(
a64) =
|

1− v6 i ( 4)

=

1 − 1015−6
.
4(10150.25 − 1)
.

= 5.729

Now with the Tables:
(
a64 ) =
|

i i ( 4)

a6| = 1005608 × 5.6972 = 5.729
.

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CT1-06: Level annuities

Solution 6.8
An interest rate of 8% pa convertible quarterly is equivalent to an effective quarterly interest rate of 2%. There are 6 monthly payments of £1,000 each, starting in one month’s time. If we work in terms of quarters, the payments are £3,000 per quarter, payable 3 times per quarter and starting in one month’s time.
So, the PV of the payments is:
(
3,000a23)@2% = 3,000
|

1 − v2 i ( 3)

= 3,000 ×

1 − 0.96117
.
= 3,000 × 19542 = £5,863
0.01987

(We have to work out i ( 3) from the formula i ( 3) = 3[(1 + i )1/ 3 − 1] , since i ( 3) is not shown in the Tables.)
Alternatively, you could have worked in months using an effective monthly interest rate. The calculation is then:
@
1, 000a6| 0.66227% = 1, 000 ×

1 − 1.0066227 −6
= £5,863
0.0066227

Or, using an annuity payable in advance, the calculation is:
@
1, 000va6| 0.66227%

−1

= 1, 000 × 1.0066227 ×

1 − 1.0066227 −6
0.0066227
1.0066227

= £5,863

Solution 6.9
You will receive £80 pa payable half-yearly in arrear. After 10 years you should have:
(2
80s10|)@8% = 80

i i ( 2)

s10| = 80 × 1019615 × 14.4866 = £1,182
.

Solution 6.10
The value will just be a factor of i i ( 3) greater.
PV = 245.32 ×

i i ( 3)

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= 245.32 ×

0.06
= $250.16
0.058838

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Page 31

Solution 6.11
10.3813% pa effective is equivalent to 2.5% pq effective but a53| is not tabulated and so using the formula:
(4
a13.)25|

=

1 a @2.5%
4 53|

1 − v 53
=
= 7.298
4i

Solution 6.12
Using definition (6.1): a (2)

23.75|

=

1 - v 23.75 i (2)

= 16.9391
Using the alternative definition:

a (2)

23.75|

= a (2) | + 1 4 v 23.75
23.5

= 16.9395

Solution 6.13
(
(
False. The correct expression is sn|p ) = (1 + i )1/ p sn|p ) .

Solution 6.14
The present value is:

150a• =

150 150
=
= £1,875 i 0.08

Note that this could also have been calculated using a geometric progression as follows: 150v + 150v 2 + 150v3 +

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=

150v
= £1,875
1- v

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CT1-06: Level annuities

Solution 6.15
The present value is:
2, 000
2, 000
=
= £28,315.79 d 0.076 1.076

2, 000a• =

Note that this could also have been calculated using a geometric progression as follows: 2, 000 + 2, 000v + 2, 000v 2 + 2, 000v3 +

=

2, 000
= £28,315.79
1- v

Solution 6.16
The present value is:
300a (12) =


300 i (12)

=

300
= £5,136.05
0.0584106

Note that this could also have been calculated using a geometric progression as follows: 3
1
2
300 12 300 12 300 12 v + v + v +
12
12
12

1

300 v 12
=
¥
= £5,136.05
1
12 1 - v 12

Alternatively, you can adjust the interest rate to use a monthly interest rate:

300
300 2 300 3 v+ v + v +
12
12
12

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=

1
300
v
¥
@1.0612 - 1 = £5,136.05
12 1 - v

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CT1-07: Deferred and increasing annuities

Page 1

Chapter 7
Deferred and increasing annuities
Syllabus objective
(vi)

Define and use the more important compound interest functions.
2.

Derive formulae in terms of i, v, n, d, δ , i ( p ) and d ( p ) for
( p) m| an| , m| an|

3.

and

( p) m| an| , m| an| ,

m| an| .

Derive formulae in terms of i, v, n, δ , an| and an| for ( Ia ) n| , ( Ia ) n| , ,
( Ia ) n| , ( Ia ) n| and the respective deferred annuities.

0

Introduction
In this chapter we will derive formulae for calculating the present value and accumulated value of deferred annuities and simple increasing annuities. Deferred annuities are annuities where no payment is made during the first time period.
At the end of the chapter we give some special cases that use the techniques that you have developed so far in the course.

Question 7.1
Define a simple increasing annuity.

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CT1-07: Deferred and increasing annuities

1

Deferred annuities

1.1

Annual payments
Suppose that m and n are non-negative integers. The value at time 0 of a series of n payments, each of amount 1, due at times (m + 1), (m + 2), … , (m + n ) is denoted by m| an| (see the figure below).
1
1
0

1

…m

m +1

m +2

...

1

payment



m+n

time

Such a series of payments may be considered as an immediate annuity, deferred for m time units. When n > 0 : m| an|

= v m +1 + v m + 2 + v m + 3 +
= (v + v 2 + v 3 +

+ v m+n

+ v m + n ) − (v + v 2 + v 3 +

= v m (v + v 2 + v 3 +

+ vm)

+ vn)

The last two equations show that:

= am + n| − am|

(1.1)

= v m an|

m| an|

(1.2)

Either of these two equations may be used to determine the value of a deferred immediate annuity. Together they imply that:

am + n| = am| + v m an|

This formula could easily be deduced using general reasoning. The present value of a series of ( n + m ) payments of one unit payable at the end of each time period is equal to the sum of
(a)

present value of m payments of one unit payable at the end of each time period

(b)

present value of n payments of one unit payable at the end of each time period, deferred for m years.

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Question 7.2
Using both Equations (1.1) and (1.2), calculate 8| a12| at 6.2% pa convertible half yearly.
We may define the corresponding deferred annuity-due as: m| an|

= v m an|

Question 7.3
Express

1.2

m| an|

in terms of an| .

Continuously payable annuities
If m is a non-negative number, we use the symbol m| an| to denote the present value of a continuously payable annuity of 1 per unit for n time units, deferred for m time units. Thus: m | an|

=

m + n -d t e dt m Ú

= e -d m
=

n -d s

Ú0 e

m + n -d t e dt
0

Ú

ds

-

m -d t e dt
0

Ú

Hence: m| an|

= am + n| − a m|
= v m an|

Question 7.4
Write down the formula for an| .

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CT1-07: Deferred and increasing annuities

Question 7.5
Write down a formula for

m| an|

in terms of an| .

Question 7.6
A being on the planet Xi is currently aged exactly 48. When the being retires at exact age 60 (in ‘Earth’ years) it will receive an income of 35 Dafts (the local currency) per
‘Earth’ week. All beings on the planet Xi die at exact age 76. Calculate the present value of the retirement benefits at 10% pa effective.

1.3

Annuities payable pthly
The present values of an immediate annuity and an annuity-due, payable pthly at the rate of 1 per unit time for n time units and deferred for m time units, are denoted by:
( p) m | an|

and

= v m a( p )
|

( p) m | an|

= v m a( p) n| n

¸
Ô
Ô
˝
Ô
Ô
˛

(1.3)

respectively.

Question 7.7
(2
(2
Which is higher, 5| a10|) or 6| a10|) at 5% pa effective?

1.4

Non-integer values of n
We may also extend the definitions of m| a( p ) and m| a( p ) to all values of n by the n| n| formulae: ( p) m | an |

= v m a( p)
|

( p) m | an |

= v m a( p) n| n

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¸
Ô
Ô
˝
Ô
Ô
˛

(1.4)

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CT1-07: Deferred and increasing annuities

Page 5

and so:
( p) m | an |
( p) m | an |

¸
Ô
Ô
˝
Ô
Ô
˛

= a( p ) | - a ( p|) n +m m =

a( p ) | n +m

- a ( p|) m (1.5)

This is easily proved:
(
v man|p ) = v m

1 − vn i ( p)

=

(1 − v n +m ) − (1 − v m ) i ( p)

(p
(
= an +) | − amp )
|
m

( and similarly for v man|p ) .

Question 7.8
Why is there not a section in this chapter covering the functions:
( p) m| sn|

and

( p) m| sn| , m| sn| , m| sn| ,

m| sn| ?

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CT1-07: Deferred and increasing annuities

2

Varying annuities

2.1

Annual payments
For an annuity in which the payments are not all of an equal amount it is a simple matter to find the present (or accumulated) value from first principles. Thus, for example, the present value of such an annuity may always be evaluated as: n ∑ X iv t

i

i =1

where the ith payment, of amount X i , is made at time t i .

If there are also continuous payments then the present value may be calculated as: n ∑ Xivt i =1

i

+

z



−∞

ρ (t )v t dt

where ρ (t ) is the rate of payment per time unit at time t.
In the particular case when X i = t i = i the annuity is known as an “increasing annuity” and its present value is denoted by (Ia)n| .

( Ia ) n| , therefore, represents the present value of payments of 1 at the end of the first time period, 2 at the end of the second time period, …, n at the end of the nth time period. Thus:

(Ia )n| = v + 2v 2 + 3v 3 +

¸
Ô
˝
Ô
˛

+ nv n

Hence: (1 + i )(Ia )n| = 1 + 2v + 3v 2 +

+ nv n -1

(2.1)

By subtraction, we obtain:

i (Ia )n| = 1 + v + v 2 + v 3 +

+ v n -1 - nv n

= an| - nv n

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So:

(Ia)n| =

an| − nv n i This formula can be found in the Tables but in the exam you won’t have time to look up formulae like this so learn it!
The graph below shows the profiles of the payments of ( Ia ) 5| .

5
Amount

4
3
2
1
0
0

1

2

3

4

5

Time

The present value of any annuity payable in arrear for n time units for which the amounts of successive payments form an arithmetic progression can be expressed in terms of an| and (Ia)n| . If the first payment of such an annuity is P and the second payment is (P + Q ) , the tth payment is (P − Q ) + Qt , then the present value of the annuity is
(P − Q )an| + Q(Ia)n|

(2.2)

Alternatively, the present value of the annuity can be derived from first principles. On a time line we can show the payments as:
P

P+ Q

P + 2Q P + 3Q

P + (n – 1)Q Payments

Time
1

2

3

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4

n

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CT1-07: Deferred and increasing annuities

Alternatively, these payments can be thought of as the sum of the following two sets of payments: P –Q

P– Q

P –Q

P –Q

P– Q

1

2

3

4

n

Q

2Q

3Q

4Q

nQ

1

2

3

4

n

Payments

Time

Payments

Time

ie we have a level annuity with payments of P - Q and an increasing annuity that increases by Q each time.

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Page 9

Example
Find the present value as at 1 January 2005 of a series of 10 annual payments starting at
£500 on 1 January 2006 and increasing by £100 each year. Assume an effective rate of interest of 8% pa.

Solution
We can think of this series of payments as a combination of: a level annuity of £400 and an increasing annuity of £100
So the present value of the payments as at 1 January 2005 is:

400 a10| + 100( Ia )10|

Ê a | - 10v10 ˆ
10
= 400 a10| + 100 Á
˜
i
Á
˜
Ë
¯
= 400 ¥ 6.7101 + 100 ¥

7.2469 - 10(0.46319)
0.08

= £5,953 or this can be calculated using the Tables.
The notation (Ia)n| is used to denote the present value of an increasing annuity-due payable for n time units, the tth payment (of amount t) being made at time t − 1 . Thus:

(Ia)n| = 1 + 2v + 3v 2 +

+ nv n − 1

= (1 + i )(Ia)n|

= 1 + an − 1| + (Ia)n − 1|

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CT1-07: Deferred and increasing annuities

Question 7.9
Derive a formula for ( Ia ) n|
(i)

algebraically and

(ii)

by general reasoning, starting from the formula for ( Ia ) n| .

The graph below shows the profiles of the payments of ( Ia )5| .

5
Amount

4
3
2
1
0
0

1

2

3

4

5

Time

Question 7.10
Calculate the present value at time 0 of payments of £50 at time 0, £60 at time 1, £70 at time 2 and so on. The last payment is at time 10. Assume that the annual effective rate of interest is 4.2%.

2.2

Continuously payable annuities
For increasing annuities which are payable continuously it is important to distinguish between an annuity which has a constant rate of payment r (per unit time) throughout the rth period and an annuity which has a rate of payment t at time t. For the former the rate of payment is a step function taking the discrete values 1,2,… . For the latter the rate of payment itself increases continuously. If the annuities are payable for n time units, their present values are denoted by

(Ia )n| and (Ia)n| respectively.

A bar over the a indicates that the payments are made continuously and a bar over the I indicates that increases occur continuously, rather than at the end of the year.

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You will sometimes see ( Ia ) n| written as ( Ia ) n| . The two are equivalent.
The graphs below show the profiles of the payments.
( Ia )5|

( Ia )5|

Clearly:

(Ia )n| =

n

z

r

∑ ( r −rv t dt )
1

r =1

and:

(Ia)n| =

z

n

0

tv t dt

and it can be shown that:

(Ia )n| =

an| − nv n

δ

and:

(Ia)n| =

an| − nv n

δ

The present values of deferred increasing annuities are defined in the obvious manner, for example: m | (Ia )n

= v m (Ia )n

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Explanation
The formula for ( Ia ) n| is derived by integrating by parts, using u = t : n n

( Ia) n| = Ú tv dt = Ú te -d t dt t 0

0

n

n

È Ê 1
1
ˆ˘
= Ít Á - e -d t ˜ ˙ - Ú - e -d t dt
¯ ˚0 d Î Ë d
0
=-

nv n

d

+

1

d

n

Úv

t

dt =

an| - nv n

0

d

The formula for ( Ia ) n| can be derived by breaking it up into individual years: n ( Ia ) n| = Ú Èt ˘vt dt
Í ˙
0

1

n

2

= Ú v dt + Ú 2v dt + t 0

t

+

1

= a1| + 2v a1| +

Ú

nvt dt

n -1

+ nv n -1a1|

or, by general reasoning, by noting that, the payments made in each year are the same as for ( Ia ) n| , but they are paid continuously throughout the year, rather than at the end of the year. So, by proportioning:
( Ia ) n| = ( Ia ) n| × a1| / a1| which simplifies to the formula given.

Question 7.11
Rent on a property is payable continuously for 5 years. The rent in the first year is
£3,000, thereafter the annual rent increases by £500 pa. Calculate the present value of the rent at the start of the 5 years, using an annual effective rate of interest of 6%.

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Question 7.12
A man agrees to make investments continuously for the next 10 years. He decides that he can afford to invest £20t at time t , 0 £ t £ 10 . Calculate the:
(i)

present value of the investment at time 0

(ii)

accumulated value of the investment at time 10.

You may assume an annual effective rate of interest of 3.7% throughout the 10 years.

2.3

Decreasing payments
We can also use increasing annuities to find the present value of annuities where the payments decrease by a fixed amount each time.

Example
The following payments are received:
50 at time 1, 45 at time 2, 40 at time 3 etc
The last payment is received at time 6.
These payments are equivalent to:
55 at time 1, 55 at time 2, 55 at time 3, etc
LESS the following payments:
5 at time 1, 10 at time 2, 15 at time 3, etc
In symbols this is:
55a6 - 5( Ia )6

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CT1-07: Deferred and increasing annuities

Question 7.13
A man makes payments into an investment account of $200 at time 5, $190 at time 6,
$180 at time 7, and so on until a payment of $100 at time 15. Assuming an annual effective rate of interest of 3.5%, calculate:
(i)

the present value of the payments at time 4

(ii)

the present value of the payments at time 0

(iii)

the accumulated value of the payments at time 15

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3

Special cases

3.1

Page 15

Irregular payments
Where the interest rate is constant, we can use the trick illustrated in the following example. This involves converting the payments into a simpler series of payments with the same present value.

Example
Find an expression in terms of annuity functions for the present value as at 1 January
2004 of the following payments under the operation of a constant rate of interest:
£100 on 1 January, 1 April, 1 July and 1 October 2004
£200 on 1 January, 1 April, 1 July and 1 October 2005
£300 on 1 January, 1 April, 1 July and 1 October 2006
£400 on 1 January, 1 April, 1 July and 1 October 2007
£500 on 1 January, 1 April, 1 July and 1 October 2008

Solution
We can convert the payments for each calendar year to an equivalent single payment with the same present value payable on 1 January that year. For example, the payments in
2006 are equivalent to a single payment of 1200 a (|4 ) payable on 1 January 2006. So, the
1

payments are equivalent (in terms of present value) to the following five payments:

1 × 400 a (|4 ) on 1 January 2004
1

2 × 400 a (|4 ) on 1 January 2005
1

...
5 × 400 a (|4 ) on 1 January 2008
1

This is just a simple increasing annuity (payable annually in advance) where the payment amounts increase by 400 a (|4 ) each year. So the present value is ( Ia ) 5| × 400 a (|4 ) .
1

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3.2

CT1-07: Deferred and increasing annuities

Sudden changes in interest rates
Sudden changes in interest rates were considered for single payments earlier in the course.
We now consider a series of payments.

Example
Calculate the present value as at 1 January 2004 of the following payments:
£100 on the first day of each quarter during calendar years 2006 to 2015 inclusive.
Assume effective rates of interest of 8% per annum until 31 December 2010 and 6% per annum thereafter.

Solution
Here we must value a ten-year annuity payable quarterly in advance, deferred for two years. We have to split the ten-year annuity into two five-year annuities to allow for the change in interest rates.
The present value here is:
(4)@8%
(4)@
400v 2@8% (a5|
+ v5@8% a5| 6% )

1 - 0.74726 ˆ
Ê 1 - 0.68058
= 400 ¥ 0.85734 Á
+ 0.68058 ¥
˜
Ë 0.076225
0.057847 ¯
= 400 ¥ 6.1420 = £2, 457

Question 7.14
The following payments are made:
£1,000 pa payable quarterly in arrears from 1/1/05 to 31/12/10
The annual effective rate of interest is 3.4% for calendar years 2005-2008 and 4.2% thereafter. Calculate:
(i)

the present value of the payments at 1/1/05

(ii)

the accumulated value of the payments at 1/1/12.

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3.3

Page 17

Compound increasing annuities
We have already looked at simple increasing annuities, such as ( Ia ) n| , where the payments increase by a constant amount each time. We will also need to be able to value compound increasing annuities where the payments increase by a constant factor each time. Example
Calculate the present value as at 1 January 2003 of an annuity payable annually in arrear for 15 years. The first payment is 500 and subsequent payments increase by 3% per annum compound.
Assume effective rates of interest of 10% per annum.

Solution – Method 1
A payment of 500 is made on 1 January 2004. A payment of 500 ¥ 1.03 is made on 1
January 2005. A payment of 500 ¥ 1.032 is made on 1 January 2006, and so on.
The final payment of 500 ¥ 1.0314 is made on 1 January 2018.
From first principles, we can write down an expression for the present value of this annuity as follows:
PV = 500v + 500 ¥ 1.03v 2 + 500 ¥ 1.032 v3 +

+ 500 ¥ 1.0314 v15

(*)

This can be rearranged as follows:

(

PV = 500v 1 + 1.03v + 1.032 v 2 +

+ 1.0314 v14

)

The expression in brackets is simply a geometric progression of the form:
2

1+ x + x +

where x = 1.03v =

+x

n -1

1 - xn
=
1- x

1.03 1.03
=
= 0.936364 and n = 15 .
1 + i 1.1

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CT1-07: Deferred and increasing annuities

Using simple substitution we get:

PV = 500v ¥

1-

(1.03 )
1.1

15

1 - 1.03
1.1

=

500
¥ 9.853407 = 4, 479
1.1

Method 2
We’ll now consider a slightly different way of solving this problem. The equation (*) could be rearranged and written as:

PV =
=

(

500
1.03v + 1.032 v 2 +
1.03

(

500 v ¢ + v ¢2 +
1.03

v ¢15

1.0315 v15

)

)

where v ¢ = 1.03v .
Remember that:

an = v + v 2 +

+ v n @ i % where v =

1
1+ i

If we introduce a new interest rate i ¢ , we can define:

an ¢ = v ¢ + v ¢ 2 +

+ v ¢n

@ i ¢% where v ¢ =

1
= 1.03v
1 + i¢

(**)

This is just the value, an ¢ , of an annuity certain, calculated at interest rate i¢ .
From (**) we get:

1
1.03
1.1
=
fi i¢ =
- 1 = 6.7961%
1 + i ¢ 1.1
1.03

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Page 19

So we can now write the present value of the annuity as:

PV =

500 a ¢
1.03 15

@ 6.7961%

(

)

15 ˆ
Ê
1
500 Á 1 - 1.067961 ˜ 500
=
=
¥ 9.2264 = 4, 479
1.03 Á 0.067961 ˜ 1.03
Ë
¯

The value of a compound increasing annuity can be valued using the present of a level annuity at a different interest rate.

Method 3
Finally, we’ll consider a slight adaptation of Method 2, using an annuity due instead of an annuity in arrear. In Method 1, we rearranged the equation (*) as:

(

PV = 500v 1 + 1.03v +

)

(

1.0314 v14 = 500v 1 + v ¢ + v ¢ 2 +

v ¢14

)

where v ¢ = 1.03v .
This time the expression in brackets is an annuity-due and so we can write:

PV = 500va15 ¢
We must be slightly careful when evaluating this expression. The v in the expression must be calculated at 10%, whereas the annuity must be calculated at our new interest rate i ¢ = 6.7961% as before. So:

(

)

15 ˆ
Ê
1
Ê 1 - v ¢15 ˆ 500 1 - 1.067961
Á
˜ = 500 ¥ 9.8534 = 4, 479
PV = 500v Á
˜ = 1.1 ¥ Á 0.067961
˜ 1.1
Ë d¢ ¯
1.067961
Ë
¯

We have considered three slightly different methods here. There is no “best method” to use. You can choose which ever you feel most comfortable with. Methods 2 and 3 may appear to be a little more complicated at first but it’s worth trying them out. They’re not quite as complex as they might first appear.
Before you have a go at an exercise yourself, here’s another look at Method 2 in a more general context.

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CT1-07: Deferred and increasing annuities

Present value of a compound increasing annuity
If payments increase in such a way that the payment at time t is (1 + e) t (where e is any constant), then: the PV of a single payment paid at time t = n is v ′ n the PV of a series of payments paid at times t = 1,2,… , n is an|′ where the functions with dashes are calculated at the rate of interest i ′ =

i−e
.
1+ e

“To value compound increases, subtract the rate of increase from the interest rate and divide by 1-plus-the-rate-of-increase”

Proof
The present value of a single escalated payment at time t is

(1 + e) t
(1 + i ) t

If we introduce a new interest rate i ′ defined by the equation the PV of a payment at time t = n

F 1 + e IJ .
=G
H 1+ i K t 1
1+ e
=
, we find that
1+ i′ 1+ i

F 1 IJ can be expressed as G
H 1 + i′K

n

, ie it is just v ′ n ,

calculated using the interest rate i ′ .
Similarly, the present value of a compound increasing annuity is: v′ + v′2 +

+ v′n

which is just the value a n|′ of an annuity certain, calculated at interest rate i ′ .
We can rearrange the equation for i ′ to find an expression for i ′ in terms of i and e :

1
1+ e
=
1+ i′ 1+ i

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⇒ i′ =

1+ i
(1 + i ) − (1 + e) i − e
−1=
=
1+ e
1+ e
1+ e

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Page 21

Question 7.15
Calculate the present value (at time t = 0 ) of payments of £20,000 × 10381t −1 payable at
.
times t = 1,2,3,… ,10 , assuming a constant annual effective rate of interest of 9%.

Question 7.16
Calculate numerical values for the following actuarial functions assuming the interest rates shown in brackets:
(i) a 7| (7½% pa)

(ii) ( Ia ) 5| (10% pa)

(iii) s ( 4|) (10% pa convertible quarterly)

(iv) s (½) (1% pa)
|

(v) a ( 4 ) | (4% pa)

(vi) 5| a (3) (6% pa)

10

14½

15

8

Question 7.17
Calculate the present value as at 1 January 2005 of a 5-year annuity consisting of payments of £100, starting on 1 January 2005 and payable on the first day of each month with an “R” in its name. Assume a constant annual effective rate of interest of
8%.

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4

CT1-07: Deferred and increasing annuities

Exam-style question
Here’s an example of an exam-style question involving compound increasing annuities.
Have a go at it yourself before turning the page and working through the solution.

Question
An investor wishes to find the present value of a stream of property income payments.
She proposes to make the following assumptions.


The level of current payments is £20,000 per annum, paid quarterly in advance.



Payments will remain fixed for 5-year periods. At the end of each 5-year period the payments will rise in line with total inflationary growth over the previous five years.



Inflation is assumed to be constant at 3% per annum.



The interest rate for the calculation is 12% per annum effective.

Find the present value of the income stream assuming that the payments continue for 50 years. [6]

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Page 23

Solution
(i)

The present value of the first five years’ worth of payments (working in
(
thousands of pounds) is 20 a5|4 ) (calculated at i = 12% ).

The present value (still at time 0) of the next 5 years worth of payments is:
(
20 × 1035 × a5|4 ) × v 5
.

Note that we must increase the annual payment by 1.035 because we are given an annual rate of inflation and “the payments will rise in line with total inflationary growth over the previous five years”.

The next 5 years’ worth of payments will be worth:
(
20 × 10310 × a5|4 ) × v10 and so on
.

So the total present value will be:
(4)
20 a5|

@i%

È1 + 1.035 v5 + 1.0310 v10 +
Î

(4)
+ 1.0345 v 45 ˘ = 20 a5| a10|
˚

@ j%

where the last annuity term above is calculated at the rate j for which
5

vj =

1
Ê 1.03 ˆ

˜ . This gives a rate of interest for j of 52.021%.
1 + j Ë 1.12 ¯

So the total present value is:

20

(4) a5| @i%

a10|

@ j%

= 20 ¥

1 - v5 d (4)

¥

1 - v10 j dj

= 20 ¥ 3.87131 ¥ 2.87797 = 222.830
So the present value of the income stream is £222,830.

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5

CT1-07: Deferred and increasing annuities

End of Part 1
You have now completed Part 1 of the Subject CT1 Notes.

Review
Before looking at the Question and Answer Bank we recommend that you briefly review the key areas of Part 1, or maybe re-read the summaries at the end of Chapters 1 to 7.

Question and Answer Bank
You should now be able to answer the questions in Part 1 of the Question and Answer
Bank. We recommend that you work through several of these questions now and save the remainder for use as part of your revision.

Assignments
On completing this part, you should be able to attempt the questions in Assignment X1.

Reminder
If you have not booked a tutorial, then maybe now is the time to do so.

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Page 25

Chapter 7 Summary
Deferred annuities are annuities where no payment is made during the first time period.
The value at time 0 of a series of n payments, each of amount 1, due at times
(m + 1), (m + 2), … , (m + n) is denoted by m| an| : m| an|

= am+n| − am| = v man|

Other functions exist for annuities payable in advance, continuously and pthly: m| an|

= am+n| − am| = v man|

( p) m| an|

m| an|

(
= v man|p )

= am+ n| − am| = v man|

( p) m| an|

(
= v man|p )

( Ia ) n| represents the present value of payments of 1 at time 1, 2 at time 2, …, n at time n: n

( Ia ) n| = ∑ tv = t a n| − nv n i t =1

An increasing annuity but with payments in advance is given by: n −1

( Ia ) n| = ∑ (t + 1)v = t a n| − nv n d t =0

For the continuous annuity ( Ia ) n| , the rate of payment is a step function taking the discrete values 1, 2, .... For ( Ia ) n| , the rate of payment itself increases continuously.
The rate of payment at time t is t. The formulae are:

z n ( Ia ) n| =

t v dt = t a n| − nv n

0

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δ

z n ( Ia ) n| = tv dt =
0

t

an| − nv n

δ

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CT1-07: Deferred and increasing annuities

This page has been left blank so that you can keep the chapter summaries together for revision purposes.

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Page 27

Chapter 7 Solutions
Solution 7.1
A simple increasing annuity is an annuity where the payments increase each time by a fixed amount. For example £3 at time 1, £6 at time 2, £9 at time 3 etc.

Solution 7.2
6.2% pa convertible half yearly is equivalent to an effective rate of 6.2961% pa.
Using Equation (1.1)
8| a12| = a20| − a8| =

1 − v 20 1 − v 8

= 1119923 − 613767 = 5.06156
.
. i i

Using Equation (1.2)
8| a12|

= v 8a12| = 0.613566 × 8.249406 = 5.06156

Solution 7.3 m| an|

= v m−1an|

Solution 7.4

an| =

1 − vn

δ

Solution 7.5 m| an|

= v m a n| = v m

i

δ

a n|

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CT1-07: Deferred and increasing annuities

Solution 7.6
Approximating the weekly payments by a continuous annuity:

PV = 52.18 × 35×12|a16| = 1826.3v12

1 − v16

δ

= 4,777 Dafts

Solution 7.7
Both annuity functions value a ten year annuity payable half yearly. The first payment
(2
(2 of 5| a10|) is at time 5½ whereas the first payment of 6| a10|) is at time 6. Therefore since
(2
the payments are made sooner, 5| a10|) will be greater.
(2
(2
In fact 5| a10|) = (1 + i ) 0.5 6| a10|) .

Alternatively, you could calculate each function:
( 2)
5| a10|

(2
= 6125 and 6| a10|) = 5.977
.

Solution 7.8
The functions represent the value of the series of payments at time m + n . They are unlikely ever to be used since

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m| sn|

= sn| and

( p) m| sn|

(
= sn|p ) etc.

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Page 29

Solution 7.9
Algebraically ( Ia ) n| is equal to 1 + 2v + 3v 2 + … + nv n −1 .

(i)

In this case, it is easiest to derive the formula by multiplying by v and subtracting (rather than multiplying through by (1+ i ) ). So we get:

+ ( n - 1)v n - 2 + nv n -1

( Ia) n| = 1 + 2v + 3v 2 + v( Ia ) n| =

v + 2v 2 +

\ (1 - v)( Ia ) n| = 1 + v + v 2 + ie d ( Ia ) n| = an| - nv
(ii)

n

+ (n - 2)v n - 2 + (n - 1)v n -1 + nv n
+

\ ( Ia ) n| =

vn-2 +

v n -1 - nv n

an| - nv n d The formula for ( Ia ) n| follows by general reasoning by noting that the payments for ( Ia ) n| are the same as for ( Ia ) n| , but advanced by 1 year.
So:

( Ia ) n| = (1 + i )( Ia ) n| = (1 + i )

a n| − nv n i =

a n| − nv n d Solution 7.10
On a time line we can show the payments as:
50

60

70

80

150

0

1

2

3

10

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Payments

Time

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CT1-07: Deferred and increasing annuities

Notice that there are 11 payments. Alternatively, these payments can be thought of as the sum of the following two sets of payments:

10

20

30

40

110

0

1

2

3

10

40

40

40

40

0

1

2

3

Payments

Time

40

Payments

Time
10

ie we have a level annuity with payments of 40 and an increasing annuity which increases by 10 each time.
The present value of these payments is:

40a11 + 10( Ia )11
We have: a11 1 - 1.042 -11
=
= 9.03074
0.042 1.042

( Ia )11 =

a11 - 11v11 d =

9.03074 - 11 ¥ 1.042-11
= 50.48174
0.042 1.042

So the present value is:
40 ¥ 9.03074 + 10 ¥ 50.48174 = £866.05

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Solution 7.11
On a time line the rent is as follows:

3,000
0

3,500
1

4,000
2

4,500
3

Payments

5,000
5

4

Time

The present value of these payments is:
2,500a5 + 500( I a )5
Evaluating these annuities:

a5 =

1 - 1.06-5
= 4.3375 ln1.06 a5 =

1 - 1.06 -5
= 4.4651
0.06 1.06

( I a )5 =

4.4651 - 5 ¥ 1.06-5
= 12.5078 ln1.06 So the present value is:

2,500 ¥ 4.3375 + 500 ¥ 12.5078 = £17, 097.66

Solution 7.12
(i)

The present value of these payments is:

a10 - 10v

10

20( I a )10 = 20
(ii)

d

1 - 1.037 -10
- 10 ¥ 1.037 -10
= 20 ln1.037
= 787.82 ln1.037 The accumulated value is the present value accumulated for 10 years:
787.82 ¥ 1.03710 = 1,132.96

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CT1-07: Deferred and increasing annuities

Solution 7.13
The payments can be thought of as:
210 at time 5, 210 at time 6, 210 at time 7, …, 210 at time 15
LESS the following payments:
10 at time 5, 20 at time 6, 30 at time 7, …, 110 at time 15
(i)

The present value of these at time 4 is:

210a11 - 10( Ia)11
Evaluating these, we get:

a11

1 - 1.035-11
=
= 9.0016
0.035

a11 =

1 - 1.035-11
= 9.3166
0.035 1.035

( Ia)11 =

9.3166 - 11 ¥ 1.035-11
= 50.9201
0.035

So the present value is:

210 ¥ 9.0016 - 10 ¥ 50.9201 = 1,381.13
(ii)

To find the present value at time 0, we need to discount the answer to part (i) by
4 years:
1,381.13 ¥ 1.035-4 = 1, 203.57

(iii)

To find the accumulated value at time 15, we need to accumulate the answer to
(i) by 11 years:
1,381.13 ¥ 1.03511 = 2, 016.40

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Solution 7.14
(i)

The present value is:
1, 000a (4)

4 @3.4%

4
+ 1, 000v@3.4% a (4)

2 @ 4.2%

Evaluating these:
1
Ê
ˆ
(4) i@3.4% = 4 Á1.034 4 - 1˜ = 0.03357
Ë
¯
1
Ê
ˆ
(4) i@ 4.2% = 4 Á1.042 4 - 1˜ = 0.04135
Ë
¯

The present value is then:
-2
1 - 1.034 -4
-4 1 - 1.042
1, 000
+ 1, 000 ¥ 1.034
0.03357
0.04135

= 3, 728.432 + 1, 670.965 = 5,399.40
(ii)

We need to accumulate the answer to part (i) by 7 years. Four of these years have an interest rate of 3.4% pa and the remainder have an interest rate of
4.2% pa:
5,399.40 ¥ 1.0344 ¥ 1.0423 = 6,982.81

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CT1-07: Deferred and increasing annuities

Solution 7.15

(

PV = 20, 000 v + 1.0381v 2 +

=

1.03819 v10

)

20, 000 a ¢
1.0381 10

Note that in this case we have had to divide by 1.0381 to ensure that the power we raise
1.0381 to matches the power we raise v to in each term of the original expression. This won’t always be the case. It’s usually best to work form principles and use simple algebraic manipulation to obtain an expression of the form:
(1 + e)v + (1 + e) 2 v 2 +

(1 + e)n v n

which can be valued as an annuity in arrear (or as a sum of a geometric progression). or of the form:
1 + (1 + e)v +

(1 + e) n -1 v n -1

which can be valued as an annuity-due.
However, once you have practised quite a few of these you may be able to spot a few
“short-cuts” but be careful not to slip up.
The interest rate to use for the annuity is:

i′ =

i − e 0.09 − 0.0381
=
= 0.05000
1+ e
1 + 0.0381

which is almost exactly 5%.
So the present value of the payments is:

20,000
20,000
a10|@5% =
× 7.7217 = £148,800
10381
.
10381
.

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Solution 7.16
(i)

a7| =

1 - v7

d

1 - 1.075-7
=
= 5.4928 log1.075 a5| − 5v 5

( Ia ) 5| =

(iii)

10% pa convertible quarterly is equivalent to an effective rate of 2½% per quarter. So, converting to quarterly time periods gives:

δ

1
1
1 s60| = (1 + i ) s60| = ¥ 1.025 ¥ 135.9916 = 34.848
4 @2½% 4
4

s (4) =
|
15

(iv)

s (½) =
|
10

i i (½)

=

4.1699 − 5 × 0.62092
= 11177
.
0.095310

(ii)

s10| =

0.01
½[1012 − 1]
.

× 10.4622 = 10.410

or:
(½)@1%
10|

s

@ 2.01%
5|

= 2s

=2

LM (1 + i) − 1OP = 2F 10201 − 1I = 10.410
GH . 0.0201 JK
NM i PQ
5

5

or:
(½)
s10| = 2[1 + (1 + i ) 2 + (1 + i ) 4 + (1 + i )6 + (1 + i )8 ]

= 2[1 + 1.02010 + 1.04060 + 1.06152 + 1.08286] = 10.410
1 − v 14½

1 − 104 −14½
.
= 11005
.
0.039414

(v)

a ( 4) | =

(vi)

(3)
= v5 a (3) = 1.06 -5 ¥
5| a

14½

8

i ( 4)

=

8

1 - 1.06 -8 d (3)

1
-1 ˆ
Ê
ˆ
Ê
where d (3) = 3 Á1 - (1 - d ) 3 ˜ = 3 Á1 - 1.06 3 ˜ = 0.057707 . So:
Ë
¯
Ë
¯

(3)
5| a8

= 4.8247

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CT1-07: Deferred and increasing annuities

Solution 7.17
Payments are made at the beginning of:
JanuaRy, FebRuaRy, MaRch, ApRil and
SeptembeR, OctobeR, NovembeR, DecembeR ie the first 4 months and the last 4 months of each calendar year.
First, consider the first 4 payments in 2005. We have payments of 100 made at the start of the next four months.
Working in years, the present value of these payments at 1 January 2005 is:
1, 200a (12)

@8% pa

4 /12

(Note that the normal annuity formulae work for fractions of a year too, provided that the fraction of a year is compatible with the payment interval, ie working in months, the fraction should be a multiple of 112 .)
Looking at each element of this expression:
The length of the annuity is 4 months and so working in years the term must be

4

12 .

The annuity is paid monthly and so this is represented by (12). The payments are made at the start of each month, which gives a . Finally, because we are working in years, we must multiply the annuity by the amount that we would have paid in a whole year had the payments continued, ie 12 payments of 100, which of course equals 1,200.
Similarly, the present value as at 1 January 2005 of the payments made in September to
December is:
1, 200v8 /12 a (12)

4 /12

@8% pa

because these payments commence in 8 months’ time.

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So the payments in each calendar year have the same value as a single payment on
1 January of:
1,200(a (12 ) | + v 8/12 a (12 ) | ) = 1,200(1 + v 8/12 )a (12 ) |
4 /12

4 /12

4 /12

(*)

So the value of the 5-year annuity is the value of these payments made at the beginning of each year is:

PV = 1, 200(1 + v8 /12 )a (12) | ¥ a5|
4 /12

= 1, 200(1 + 1.08-8/12 ) ¥ 1.042824 ¥

1 - 1.08-4 /12 1 - 1.08-5
¥
0.08
0.08 /1.08

= £3,331.32
(*) You could have solved this by working in monthly time periods instead.
The present value as at 1 January of the payments made in the coming year is:

(

PV = 100 a4 + v8 a4

) @ 0.643403% (the effective monthly rate).

= 100a4 (1 + v8 )
= 100 ¥ 3.96181 ¥ 1.94999
= 772.547
So, reverting back to working in years and using an effective interest rate of 8% pa, the present value of the annuity is:

772.547 a5|
= 772.547 ¥

1 - 1.08-5
0.08 /1.08

= £3,331.32

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CT1-08: Equations of value

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Chapter 8
Equations of value
Syllabus objective
(vii)

Define an equation of value.
1.
2.

Describe how an equation of value can be adjusted to allow for uncertain receipts or payments.

3.

0

Define an equation of value, where payment or receipt is certain.

Understand the two conditions required for there to be an exact solution to an equation of value.

Introduction
An equation of value equates the present value of money received to the present value of money paid out:
“PV income = PV outgo” or equivalently:
“PV income – PV outgo = 0”
Equations of value are used throughout actuarial work. For example:
The “fair price” to pay for an investment such as a fixed interest security or an equity (ie,
PV outgo) equals the present value of the proceeds from the investment, discounted at the rate of interest required by the investor.
The premium for an insurance policy is calculated by equating the present value of the expected amounts received in premiums to the present value of the expected benefits and other outgo.

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CT1-08: Equations of value

1

The equation of value and the yield on a transaction

1.1

The theory
Consider a transaction that provides that, in return for outlays of amount at1 , at 2 , … , at n at time t1, t 2 , … , t n , an investor will receive payments of

bt1 , bt 2 , … , bt n at these times respectively. (In most situations only one of at r and bt r will be non-zero.) At what force or rate of interest does the series of outlays have the same value as the series of receipts? At force of interest δ the two series are of equal value if and only if: n ∑

at r e − δt r =

r =1

n

∑ bt e −δt

r =1

r

r

This equation may be written as: n ∑ ct e − δt

r =1

r

r

=0

(1.1)

where ct r = bt r − at r is the amount of the net cashflow at time t r . (We adopt the convention that a negative cashflow corresponds to a payment by the investor and a positive cashflow represents a payment to the investor.)
Equation (1.1), which expresses algebraically the condition that, at force of interest δ , the total value of the net cashflows is 0, is called the equation of value for the force of interest implied by the transaction. If we let e δ = 1 + i , the equation may be written as: n ∑ ct (1 + i )−t

r =1

r

r

=0

(1.2)

The latter form is known as the equation of value for the rate of interest or the
“yield equation”. Alternatively, the equation may be written as: n ∑ ct v t

r =1

r

r

=0

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In relation to continuous payment streams, if we let ρ 1(t ) and ρ 2 (t ) be the rates of paying and receiving money at time t respectively, we call ρ (t ) = ρ 2 (t ) − ρ 1(t ) the net rate of cashflow at time t. The equation of value (corresponding to
Equation (1.1)) for the force of interest is:


∫0

ρ (t )e −δ t dt = 0

When both discrete and continuous cashflows are present, the equation of value is: n



∑ ct e −δ t + ∫0 r r =1

r

ρ (t )e −δ t dt = 0

(1.3)

and the equivalent yield equation is: n ∞

∑ ct (1 + i )−t + ∫0 r r =1

r

ρ (t )(1 + i )−t dt = 0

(1.4)

For any given transaction, Equation (1.3) may have no roots, a unique root, or several roots. If there is a unique root, δ 0 say, it is known as the force of interest implied by the transaction, and the corresponding rate of interest

i 0 = e δ 0 − 1 is called the “yield” per unit time. (Alternative terms for the yield are the “internal rate of return” and the “money-weighted rate of return” for the transaction.) Thus the yield is defined if and only if Equation (1.4) has precisely one root greater than −1 and, when such a root exists, it is the yield.

Question 8.1
Why must the yield be greater than –1?

Question 8.2
An investor pays £100 now in order to get £60 back in 5 years’ time and £60 back in 10 years’ time. What is the annual effective rate of interest earned on this investment?
We will look at solving equations of value shortly.

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CT1-08: Equations of value

The analysis of the equation of value for a given transaction may be somewhat complex depending on the shape of the function f (i ) denoting the left hand side of Equation (1.4). However, when the equation f (i ) = 0 is such that f is a monotonic function, its analysis is particularly simple. The equation has a root if and only if we can find i1 and i 2 with f (i1 ) and f (i 2 ) of opposite sign. In this case, the root is unique and lies between i1 and i 2 . By choosing i1 and i 2 to be
“tabulated” rates sufficiently close to each other, we may determine the yield to any desired degree of accuracy.

Linear interpolation will usually be used to obtain i0 from i1 and i2 . Examples of this method are given a little later in this chapter.
It should be noted that, after multiplication by (1 + i )t 0 , Equation (1.2) takes the equivalent form: n ∑ ct (1 + i )t − t
0

r =1

r

r

=0

This slightly more general form may be called the equation of value at time t 0 . It is of course directly equivalent to the original equation (which is now seen to be the equation of value at time 0). In certain problems a particular choice of t 0 may simplify the solution.

1.2

Solving for an unknown quantity
Many problems in actuarial work can be reduced to solving an equation of value for an unknown quantity. We will look at how to do this next.
Our examples are based on a hypothetical financial security which operates as follows:
Security S
A price P is paid (by the investor) in return for a series of interest payments of I payable at the end of each of the next n years and a final redemption payment of R payable at the end of the n years.
R
–P
Time

I



I

I

0

1



n–1

n

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The equation of value for this investment is:
P = Ia n| + Rv n

calculated at interest rate i

In the remainder of this section we will consider how to solve this equation when each of the quantities P, I or R, n and i in turn is unknown.

Solving for the present value (P)
The present value (which, in this case, represents the price) can be found using the formulae we derived earlier for compound interest functions:
Example
Find P , if I = 5 , R = 125 , i = 10% and n = 10 .
Solution
The price P can be calculated directly (using 10% interest):
P = 5a10| + 125v 10 = 5 × 61446 + 125 × 0.38554 = £78.92
.
Note from this example that the equation of value “works” when P = 78.92 , if I = 5 ,
R = 125 , i = 10% and n = 10 . We will treat these values as our reference values.

Question 8.3
Find P , if I = 5 , R = 125 , i = 10% and n = 20 .
The following result is sometimes useful:
Value of Security S when I = iR
If I = iR for Security S , then P = R .

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CT1-08: Equations of value

Proof (algebraic)
The price P is:
P = iRa n| + Rv n (calculated at interest rate i )
Using the formula for the annuity and simplifying gives:

Ê 1 - vn ˆ n n n P = iR Á
˜ + Rv = R(1 - v ) + Rv = R
Ë i ¯
Proof (general reasoning)
Suppose I have a sum of money R and I pay this into a bank account that pays an effective annual interest rate i . If I ask for my interest to be paid to me by cheque at the end of each year and I close my account at the end of n years, I will receive interest payments of iR at the end of each year and my initial capital of R will be repaid at the end of n years.
Under this arrangement the cashflows I will receive exactly match the cashflows I would have received if I’d invested an amount P in Security S . Also, the rate of return I will obtain from the bank account will be i (by definition), which is the same as the interest rate i that I require from Security S (ie, the rate I’m valuing it at).
So, investing R in the bank account or P in Security S leads to the same cashflows and gives the same rate of return. So P and R must be equal.

Question 8.4
Calculate P , if I = 10 , R = 125 , i = 8% and n = 10 . Comment on your answer.

Solving for the amount of a payment (I or R)
Solving the equation of value for I or R is straightforward.

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Example
Find R , if P = 78.92 , I = 5 , i = 10% and n = 10 .
Solution
The equation here is:
78.92 = 5a10| + Rv 10 ie 78.92 = 5 × 61446 + 0.38554 R
.

This can be rearranged to find R :

R=

78.92 − 5 × 61446
.
= 125.01
0.38554

Question 8.5
Find I , if P = 127.12 , R = 125 , i = 7.75% and n = 10 .

Question 8.6
An investor is to pay £800 for a property. The investor will then be entitled to receive rent payments for 99 years payable at the end of each year at a constant rate for the first
33 years, increasing to double that rate for the next 33 years and three times that rate for the remaining 33 years. The value of the property at the end of the 99 years is expected to be £250,000. Find the amount of the rent payable in the first year, if the investor expects to obtain a rate of return of 8% on the purchase.

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CT1-08: Equations of value

Solving for the timing of a payment (n)
We can solve the equation of value for n by expressing the annuity function in terms of v .
Example
Find n , if P = 78.92 , I = 5 , R = 125 and i = 010 .
.
Solution
The equation here is:
78.92 = 5a n| + 125v n
Substituting the formula for a n| gives:

78.92 = 5 ×

ie

1− vn
+ 125v n
010
.

78.92 = 50(1 − v n ) + 125v n = 50 + 75v n

This can be rearranged to find v n :

vn =

ie

78.92 − 50
= 0.38560
75

110 − n = 0.38560
.

So:

−n log 110 = log 0.38560
.

ie

n=−

log 0.38560
= 10.00 log 110
.

Question 8.7
If I = 5 , R = 125 and i = 010 , what can you say about the range of possible values of
.
P , if n is known to be a whole number of years less than or equal to 50?

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Solving for the interest rate (i)
Finding the interest rate is the hardest type of calculation, since the equation cannot usually be solved explicitly.

Example
Write down an equation for finding i , if P = 78.92 , I = 5 , R = 125 and n = 10 .

Solution
The equation here is:
78.92 = 5a10| + 125v 10 (calculated at rate i )
In terms of i , this is:

78.92 = 5 ×

1 − (1 + i ) −10
+ 125(1 + i ) −10 i The following approximation can be used to find a rough initial estimate:

Estimating an unknown interest rate (rough approximation)
To obtain an approximation for the interest rate you could combine the cashflows to make single payments based on average payment dates. This is illustrated in the following example.

Example
Find a rough estimate for the value of i that satisfies the previous equation.

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CT1-08: Equations of value

Solution
Five units are received at the end of each of years 1 to 10 (making a total of 50 paid on average at time 5½) plus 125 units at the end of year 10. Combining these (and weighting the timings by amounts) gives a single payment of 175 (ie 50 + 125 ) at time
8.7 (ie (50 × 5½ + 125 × 10) / 175 ). This gives an equation we can solve easily:
78.92 ≈ 175v 8.7 (calculated at rate i ) ie 78.92 ≈ 175(1 + i ) −8.7

So:

Ê 78.92 ˆ
1+ i ª Á
Ë 175 ˜
¯

-

1
8.7

= 1.096

ie i ≈ 9.6%

This rough estimate is quite close to the exact value of 10% (which we know from the earlier examples).
An alternative method for finding a first guess is to use a first order binomial expansion, replacing (1 + i ) n by (1 + ni ) . This is better suited to equations of value where there are not any annuities in order to get a good first guess. For example using this method here we would have attained a first guess of 7.7%.
A more accurate solution can then be found from the exact equation by linear interpolation, using a rough initial estimate as a starting point.
Linear interpolation is explained fully in FAC (Foundation ActEd Course), a course produced by ActEd. However we will quickly revise the technique here.

Estimating an unknown interest rate (linear interpolation)
If the present values calculated at interest rates i1 and i2 are P1 and P2 , then the interest rate corresponding to a present value of P can be approximated by:

i ≈ i1 +

P − P1
× (i2 − i1 )
P2 − P1

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Proof

P
1

P

P2

i1

i

i2

Yield

If the present value is a linear function of the interest rate, then the proportionate change in the interest rates will equal the proportionate change in the present values:

i − i1
P − P1
=
i2 − i1 P2 − P1
Rearranging this relationship gives the stated formula for i .
This approximation works best if the trial values are close to the true value, say values that are 1% apart. Note that this formula also works even if the true value does not lie between the two trial values, but we would not recommend this approach in the exam.
We recommend that you interpolate between values that are either side of the true value and are a maximum of 1% apart. It is even better if the two values are 0.5% apart.

Example
Find a rough value for i , if P = 75 , I = 5 , R = 125 and n = 10 .

Solution
Since the price is close to 78.92 , we know that the interest rate will be close to 10%. At
10%:
5 a10| + 125 v 10 = 78.92
The price paid (75) is lower than this. So the value of i must be greater than 10%. If we try the next higher rate included in the Tables, we find at 12%:
5 a10| + 125 v 10 = 5 × 5.6502 + 125 × 0.32197 = 68.50
So, approximating i by interpolating linearly using these two values, we get:

i ≈ 10% +

75 − 78.92
× (12% − 10%) = 10.8%
68.50 − 78.92

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CT1-08: Equations of value

Question 8.8
Calculate i if P = 70 , I = 5 , R = 120 and n = 10 .

1.3

Example applications
To illustrate how equations of value can be used in relation to some of the most common types of investment, we will briefly look at some examples involving equities
(= shares) and property (= buildings). We will revisit these topics again in more detail later in the course.

Example
An investor owns a block of shares that is expected to pay a dividend of amount D in one year’s time; dividends in each future year are expected to be 100 j % higher than in the previous year. Derive an expression for the present value of the proceeds from this investment, calculated using an interest rate i and assuming the shares will be held for
10 years.

Solution
The proceeds from this holding will be the future dividends, which will be:


D

in 1 year’s time



D(1 + j )

in 2 years’ time



D(1 + j ) 2

in 3 years’ time

D (1 + j )9

in 10 years’ time

to:


So the present value will be:
P = Dv + D (1 + j )v 2 + D (1 + j )2 v3 +

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Summing this series using the formula for a geometric series, or by multiplying by
(1 + j ) and subtracting, gives:

P=

(

Dv 1 - (1 + j )10 v10
1 - (1 + j )v

)

If we multiply the numerator and denominator by (1 + i ) , this simplifies to give:

P=

(

D 1 - (1 + j )10 v10
(1 + i ) - (1 + j )

) = D (1 - (1 + j)

10 10

i- j

v

)

Question 8.9
A company has just bought an office block for £5m, which it will rent out to a number of small businesses. The total rent for the first year will be £100,000, increasing by
4% pa compound in each future year. It will be sold after 20 years for £7.5m.
Assuming that rent is paid in the middle of each year, calculate the yield the company will obtain on this investment. Ignore tax.

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2

CT1-08: Equations of value

Uncertain payment or receipt
If there is uncertainty about the payment or receipt of a cashflow at a particular time, allowance can be made in one of two ways:



2.1

apply a probability of payment/receipt to the cashflow at each time use a higher rate of discount.

Probability of cashflow
The probability of payment/receipt can be allowed for by adapting the earlier equations. For example, Equation (1.4) can be revised to produce: n ∞

∑ pt ct (1 + i )−t + ∫0 r r =1

r

r

p(t ) ρ (t )(1 + i )−t dt = 0

(2.1)

where pt r and p(t ) represent the probability of a cashflow at time t.
Where the force of interest is constant, and we can say that the probability is itself in the form of a discounting function, then Equation (1.3) can be generalised as: n ∞

∑ ct e −δ t e − µt + ∫0 r r =1

r

r

ρ (t )e −δ t e − µt dt = 0

(2.2)

where µ is a constant force, rather than rate, of the probability of a cashflow at time t.
These probabilities of cashflows may often be estimated by consideration of the past experience of similar cashflows. For example, this approach is used to assess the probabilities of cashflows that are dependent on the survival of a life
– this is the theme of Subjects CT4, Models, and CT5, Contingencies.
In other cases, there may be lack of data from which to determine an accurate probability for a cashflow. Instead a more approximate probability, or likelihood, may be determined after careful consideration of the risks.
In some cases, it may be spurious to attempt to determine the probability of each cashflow and so more approximate methods may be justified.

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Wherever the uncertainty about the probability of the amount or timing of a cashflow could have significant financial effect, a sensitivity analysis may be performed. This involves calculations performed using different possible values for the likelihood and the amounts of the cashflows. Alternatively a stochastic approach could be used to indicate possible outcomes (see Chapter 15 and
Subject CT4, Models).

A stochastic approach involves setting up some of the key assumptions as random variables. For example, the likelihood and the amounts of payments might be modelled.
The model will then be run many times, whereby the values for the various parameters are determined according to random selection, given the probability distribution functions. This approach will indicate not just the likely outcome but also the possibility and likelihood of alternative outcomes.

Question 8.10
You are thinking about buying a lottery ticket that will cost you £1. The following table shows the amount of money that you could win together with the chance of winning and the delay before you will receive the prize money.
Prize
£20
£200
£2,000
£200,000
£2 million

Probability of winning
1 in 50
1 in 1,000
1 in 50,000
1 in 2 million
1 in 14 million

Time before payment
1 day
1 day
1 week
2 weeks
4 weeks

What is the expected present value of the prize money assuming an effective rate of interest of 0.016% per day?

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2.2

CT1-08: Equations of value

Higher discount rate
As the discounting functions and the probability functions in Equations (2.1) and
(2.2) are both dependent on time, they can be combined into a single time dependent function. In cases where there is insufficient information to objectively produce the probability functions, this combined function can be viewed as an adjusted discounting function that makes an implicit allowance for the probability of the cashflow.
Where the probability of the cashflow is a function that is of similar form to the discounting function, the combination can be treated as if a different discount rate were being used. For example, Equation (2.2) becomes: n ∞

∑ ct e −δ ′t + ∫0 r r =1

r

ρ (t )e −δ ′t dt = 0

where δ ′ = δ + µ . The revised force of discount is therefore greater than the actual force of discount, as µ must be positive in order to give a probability between 0 and 1. It can therefore be shown that the rate of discount that is effectively used is greater than the actual rate of discount before the implicit allowance for the probability of the cashflow.

Question 8.11
A woman has invested some money in a company run by some ex-criminals. In return for the investment she expects to receive £100 at the end of each of the next ten years.
Interest rates are 5% pa effective.
Calculate the present value of her investment by:
(i)

ignoring the possibility that the payments might not be made.

(ii)

assuming the probability of receiving the first payment is 0.95, the second payment is 0.9, the third payment is 0.85 etc.

(iii)

increasing the force of interest by 0.04652.

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Chapter 8 Summary
An equation of value equates the present value of money received to the present value of money paid out. You will need to use the following formulae for equations of value:
PV income – PV outgo = 0 n •

 ct e-d t + Ú0 r r =1 n  ct r =1

r

r (t )e -d t dt = 0



(1 + i ) -tr + Ú r (t )(1 + i ) - t dt = 0 r 0

Linear interpolation is often needed to find the yield from an equation of value. The formula is:

i ≈ i1 +

P − P1
× (i2 − i1 )
P2 − P1

If there is uncertainty about the payment or receipt of a cashflow at a particular time, allowance can be made in one of two ways: apply a probability of payment/receipt to the cashflow at each time use a higher rate of discount such as a new force of interest δ ′ , where δ ′ = δ + µ

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CT1-08: Equations of value

This page has been left blank so that you can keep the chapter summaries together for revision purposes.

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CT1-08: Equations of value

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Chapter 8 Solutions
Solution 8.1 i0 = eδ 0 − 1 , and since eδ 0 > 0 then i0 > −1 .

Solution 8.2
The equation of value is:
100 = 60v5 + 60v10
This is a quadratic in v5 , which can be solved to give:

v5 =

−60 ± 602 + 4 × 60 ×100
= 0.8844 or − 1.884
120

Rearranging, this gives us i to be 2.49% or –188%. Since i must be greater than –1, we have the annual effective rate of interest as 2.49%.

Solution 8.3
The price P can be calculated directly:
P = 5a 20| + 125v 20 = 5 × 8.5136 + 125 × 014864 = £61.15
.

Solution 8.4
The value of P is:
P = 10a10| + 125v 10 = 10 × 6.7101 + 125 × 0.46319 = £125.00
This calculation verifies the result just proved, since here we have:
I = 10 = 0.08 × 125 = iR and we find that P = 125 = R

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CT1-08: Equations of value

Solution 8.5
The equation here is (using a calculator to find the value of functions at 7.75% interest):
127.12 = Ia10| + 125v 10 ie 127.12 = I × 6.7864 + 125 × 0.47405

This can be rearranged to find I :

I=

127.12 − 125 × 0.47405
= 10.00
6.7864

Solution 8.6
If the amount of the rent payable in the first year is X , the equation of value is:
800 = Xa 33| + 2 Xv 33a 33| + 3 Xv 66 a 33| + 250,000v 99 ie 800 = X (1 + 2v 33 + 3v 66 )a 33| + 250,000v 99

So:
.
800 = X (1 + 2 × 0.07889 + 3 × 0.0062235) × 115139 + 250,000 × 0.00049096
This can be rearranged to find X :

X=

800 − 122.74
= £50.00
13545
.

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Solution 8.7
The equation of value is:

P = 5an + 125v n @10%
The greatest possible value for P is when n = 1, which gives:
P = 5a1| + 125v = 5 × 0.9091 + 125 × 0.90909 = 11818
.
The smallest possible value for P is when n = 50 , which gives:
P = 5a50| + 125v50 = 5 ¥ 9.9148 + 125 ¥ 0.00852 = 50.64
So P must be in the range 50.64 £ P £ 118.18 .

Solution 8.8
To find the yield we must solve the equation of value:
70 = 5a10| + 120v 10
At 12%, RHS = 66.89

At 10%, RHS = 76.99

Interpolating, we find that i ª 0.10 +

76.99 - 70
(0.12 - 0.10) = 0.1138 .
76.99 - 66.89

So i is approximately 11.4% pa.

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CT1-08: Equations of value

Solution 8.9
The equation of value here is (working in £000s):
5, 000 = 100(v½ + 1.04v1½ + 1.042 v 2½ +

ie

5, 000 =

(

100v½ 1 - (1.04v)20
1 - 1.04v

) + 7,500v

We can solve this by trial and error. expression) is:
-1
-1
Ê
100 Á1.04 2 + 1.04 2 +
Ë

+ 1.04

+ 1.0419 v19.5 ) + 7,500v 20

20

provided v π

1
1.04

At 4% the right hand side (using the first

7,500
7,500
-1
=2,000 ¥ 1.04 2 +
=5,384.06
˜+
¯ 1.0420
1.0420

-1 ˆ
2

At 5% the right hand side (using the second expression) is 4,611.57. Interpolating between these two values, we obtain: i-5 5, 000 - 4, 611.57
=
fi i = 4.5%
5 - 4 4, 611.57 - 5,384.06

Solution 8.10
1
1
1
PV = 20 ¥ 50 v + 200 ¥ 1,000 v + 2, 000 ¥ 50,000 v 7

+200, 000 ¥

1 v14 2,000,000

1
+ 2, 000, 000 ¥ 14,000,000 ¥ v 28

= £0.88

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Solution 8.11
(i)
(ii)

@5%
PV = 100a10| = 100 × 7.7217 = £772.17

This part could be calculated by adding up the series using your calculator but it is easier to express the series in terms of actuarial functions.
PV = 100v ¥ 0.95 + 100v 2 ¥ 0.9 +

= 95v + 90v 2 +

+ 100v10 ¥ 0.5

+ 50v10

= 100a10| - 5( Ia )10|
= 772.17 - 196.87 = £575.31
(iii)

.
.
The new force of interest is ln(105) + 0.04652 = ln(11) . Therefore we can use an effective rate of 10% pa.
@
PV = 100a10|10% = 100 × 61446 = £614.46
.

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CT1-09: Loan schedules

Page 1

Chapter 9
Loan schedules
Syllabus objective
(viii)

Describe how a loan may be repaid by regular instalments of interest and capital. 1.
2.

0

Describe flat rates and annual effective rates.
Calculate a schedule of repayments under a loan and identify the interest and capital components of annuity payments where the annuity is used to repay a loan for the case where annuity payments are made once per effective time period or p times per effective time period and identify the capital outstanding at any time.

Introduction
A very common transaction involving compound interest is a loan that is repaid by regular instalments, at a fixed rate of interest, for a predetermined term.

Loans are mostly used by companies or individuals to raise funds, usually to buy buildings or equipment.
Most loans operate like a repayment mortgage, ie the initial capital is repaid during the term of the loan. This is done by making repayments that are greater than the amount of interest due. The remainder of the payment is used to repay part of the capital.
Some loans operate like endowment mortgages, ie the repayments represent interest only.
This means that at the end of the term of the loan, the borrower will need to repay the capital using money from elsewhere.

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1

CT1-09: Loan schedules

An example
This topic is best approached through an example. We will consider a more general case afterwards.
Consider a very simple example. Assume a bank lends an individual £1,000 for three years, in return for three payments of £X, say, one at the end of each year.
The bank will charge an effective rate of interest of 7% per annum.
The equation of value for the transaction gives:
1,000 = Xa3| fi X = 381.05

Question 9.1
Confirm the value of X in this example.
So the borrower pays £381.05 at times t = 1,2 and 3 in return for the loan of
£1,000 at time 0. These three payments cover both the interest due and the
£1,000 capital.
It is helpful to see how this works in detail.

Each payment will be used to first, pay any interest due since the last payment, and then to reduce the amount of capital outstanding.
The initial amount of capital outstanding is £1,000 and the first payment is at time 1.
Interest will accrue before the first payment at 7% pa.
At time 1 the interest due on the loan of £1,000 is £70. The total payment made is
£381.05. This leaves £311.05 that is available to repay some of the capital. The capital outstanding after this is then £(1,000 - 311.05) = £688.95 .
At time 2 the interest due is now only 7% of £688.95 = £48.22 , as the borrower does not pay interest on the capital that is already repaid, only on the amount outstanding. This leaves £(38105 − 48.22) = £332.83 available to repay capital.
.
The capital outstanding after this is then £(688.95 − 332.83) = £356.12 .
Finally, at time 3 the interest due is 7% of £356.12 = £24.93 , leaving
£(381.05 − 24.93) = £356.12 available to pay the outstanding sum of £356.12, and the capital is precisely repaid.

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One important point is that each repayment must pay first for interest due on the outstanding capital. The balance is then used to repay some of the capital outstanding. Each payment therefore comprises both interest and capital repayment. It may be necessary to identify the separate elements of the payments – for example if the tax treatment of interest and capital differs. Notice also that, where repayments are level, the interest component of the repayment instalments will decrease as capital is repaid, with the consequence that the capital payment will increase.

In this example the interest payments reduce from £70 to £24.93 and the capital payments increase from £311.05 to £356.12. All payments, including the first payment and the last payment, include an interest element and a capital element.

Question 9.2
A bank lends a company £5,000 at a fixed rate of interest of 10% pa. The loan is to be repaid by five level annual payments. Calculate the interest and capital payments at each repayment date.

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CT1-09: Loan schedules

2

Calculating the capital outstanding

2.1

Introduction
In the example in the last section, we calculated the capital outstanding just after each payment by rolling forward the loan contract year by year. This was fine for a shortterm loan but if the term of the loan had been, say, 20 years then it would have been very time consuming. There are two much quicker ways to find the capital outstanding at points in time:
(a)

by calculating the accumulated value of the original loan less the accumulated value of the payments to date.

(b)

by calculating the present value of future payments.

For example, the capital outstanding just after the first payment is:
(a)

1,000(1 + i ) − 38105 = 688.95
.

(b)

38105a2| = 38105 × 180802 = 688.95
.
.
.

This agrees with the amount calculated above.

Question 9.3
Use both methods to check that the capital outstanding just after the second payment is
£356.12.
These results will now be proved for the more general case when the loan payments are not necessarily level.

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2.2

Page 5

The theory
Let Lt be the amount of the loan outstanding at time t = 0, 1, … , n , immediately after the repayment at t. The repayments are assumed to be in regular instalments, of amount X t at time t, t = 1, 2, 3, … , n . (Note that we are not assuming all instalments are the same amount.) Let i be the effective rate of interest, per time unit, charged on the loan. Let ft be the capital repaid at t, and let bt be the interest paid at t, so that X t = ft + bt .
The equation of value for the loan at time 0 is:

L0 = X 1v + X 2 v 2 +

+ X nv n

(2.1)

Question 9.4
Write down a simplification of the above equation if the loan is repaid by level regular instalments so that X t = X for all t.
We can find the loan outstanding at t prospectively or retrospectively.

Prospective loan calculation
Calculating the loan prospectively involves looking forward and calculating the present value of future cashflows.
Consider the loan transactions at time n, which is the end of the contract term.
After the final instalment of capital and interest the loan is exactly repaid. So the final instalment, X n , must exactly cover the capital that remains outstanding after the instalment paid at n − 1 , together with the interest due on that capital.
That is:

bn = iLn −1 ; fn = Ln −1 so that

X n = iLn − 1 + Ln − 1 = (1 + i )Ln − 1 ⇒ Ln − 1 = X nv

So the capital outstanding at time n − 1 , ( Ln−1 ), is equal to the present value of the future payment at time n.

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CT1-09: Loan schedules

Similarly, at any time t + 1 , t ≤ n − 2 we know that the capital repaid is Lt − Lt +1 , so that the instalment X t +1 is:

X t + 1 = iLt + (Lt − Lt + 1) ⇒ Lt = (Lt + 1 + X t + 1)v
Similarly, Lt + 1 = (Lt + 2 + X t + 2 )v , and working forward, successively substituting for Lt + r until we get to Ln = 0 , we get:

Lt = (Lt +1 + X t +1)v
= ((Lt + 2 + X t + 2 )v + X t +1)v = X t +1v + X t + 2v 2 + Lt + 2v 2
= X t +1v + X t + 2v 2 + X t + 3v 3 + Lt + 3v 3
=
= X t +1v + X t + 2v 2 + X t + 3v 3 +

+ X nv n - t

This gives the “prospective method” for calculating the loan outstanding. What this equation tells us is that, for calculating the loan outstanding immediately after the repayment at t, say, we have:
Prospective Method: The loan outstanding at time t is the present (or discounted) value at time t of the future repayment instalments.
Note carefully the condition for this method – the present value must be calculated at a repayment date.

Question 9.5
Write down a formula for Lt if the loan is repaid by level regular instalments so that
X t = X for all t.

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Retrospective loan calculation
Calculating the loan retrospectively involves looking backwards and calculating the accumulated value of past cashflows.
At t = 1 the interest due is b1 = iL0 , so the capital repaid is f1 = X 1 − iL0 , leaving a loan outstanding of:

L1 = L0 − ( X 1 − iL0 ) = L0 (1 + i ) − X 1
In general, at time t ≥ 1 the interest due is bt = iLt −1 , leaving capital repaid at t of

X t − iLt −1 , giving:
Lt = Lt -1(1 + i ) - X t
Similarly, Lt − 1 = Lt − 2 (1 + i ) − X t − 1 and, working back from t to 0 we have:

Lt = Lt -1(1 + i ) - X t
= (Lt - 2 (1 + i ) - X t -1)(1 + i ) - X t = Lt - 2 (1 + i )2 - X t -1(1 + i ) - X t
= Lo (1 + i )t - ( X 1(1 + i )t -1 + X 2 (1 + i )t - 2 +

+ X t -1(1 + i ) + X t )

This gives the “retrospective method” of calculating the outstanding loan. This may be described in words as:
Retrospective Method: The loan outstanding at time t is the accumulated value at time t of the original loan less the accumulated value at time t of the repayments to date.
Both of these approaches are very useful in calculating the capital outstanding at any time. Neither result actually depends on the interest rate being constant.
It may be useful to work through the equations assuming the interest charged on the loan in year r − 1 to r is i r , say.

Since both methods calculate the loan outstanding at time t, they must both give the same result. This is fairly easy to show algebraically.
Consider Equation (2.1) and multiply it by (1+ i ) t giving:

L0 (1 + i ) t = X 1 (1 + i ) t −1 + X 2 (1 + i ) t −2 +

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+ X t −1 (1 + i ) + X t + X t +1v +

+ X n v n−t

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CT1-09: Loan schedules

Rearranging gives:
L0 (1 + i )t − ( X1 (1 + i )t −1 + X 2 (1 + i )t −2 +

+ X t −1 (1 + i ) + X t )

= X t +1v + X t + 2 v 2 + X t + 3v 3 +

+ X nv n−t

which shows that the retrospective result equals the prospective result.

Question 9.6
Write down a formula for Lt using the retrospective method if the loan is repaid by level regular instalments so that X t = X for all t.
It is not necessary to memorise any of the formulae given above, but you should be able to derive them if necessary. When looking at questions about loans you are probably least likely to make a mistake if you apply the principles given in this chapter to the question, rather than trying to apply general formulae.

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Example
A loan of $50,000 is repayable by equal annual payments at the end of each of the next
5 years. Interest is 8% pa for the first three years and 12% pa thereafter.
Calculate the loan outstanding immediately after the second repayment.

Solution
Let the amount of each repayment be X so that:
3
50, 000 = X (a3|,8% + v8% a2|,12% )

fiX =

50, 000
2.5771 + 0.79383 ¥ 1.6901

fi X = $12, 759.16
We can calculate the loan outstanding immediately after the second payment either prospectively or retrospectively.
Prospectively:
12,759.16v8% (1 + a2|,12% ) = 12,759.16 × 0.92593 × 2.6901 = $31,781
Retrospectively:
50,000(108) 2 − 12,759.16((108) + 1) = $31,781
.
.
Although unless specifically asked, you shouldn’t need to perform the calculation both ways, but it is a good way to check your answer if you have time.

Question 9.7
A loan of £80,000 is repayable by eight annual payments, starting in one year’s time, with interest payable at 4½% pa. Payments one to three are half as much as payments four to eight. Calculate the loan outstanding one year before the loan is completely repaid. The Actuarial Education Company

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3

CT1-09: Loan schedules

Calculating the interest and capital elements
Given the outstanding capital at any time we can calculate the interest and capital element of any instalment.
For example, consider the instalment X t at time t. We can calculate the interest element contained in this payment by calculating the loan outstanding immediately after the previous instalment, at t − 1 , Lt −1 . The interest due on capital of Lt −1 for one unit of time at effective rate i per time unit is iLt −1 , and this is the interest paid at t. The capital repaid may be found using X t − iLt −1 , or by Lt − 1 − Lt .

This is precisely what we were doing in the example on page 2 of this chapter. The capital repaid was calculated by deducting the interest due from each instalment.
Alternatively the capital repaid can be calculated by taking the difference between the capital outstanding before and after the instalment.
Similarly, it is a simple matter to calculate the interest paid and capital repaid over several instalments. For example, consider the five instalments from t + 1 to t + 5 , inclusive. The loan outstanding immediately before the first instalment is Lt . The loan outstanding after the fifth instalment is Lt + 5 . The total capital repaid is therefore

X t +1 + X t + 2 + t +5

∑ bk

k = t +1

Lt − Lt + 5 .

The total capital and interest paid is

+ X t + 5 . Hence, the total interest paid is:
= (X t +1 + X t + 2 +

+ X t + 5 ) − ( Lt − Lt + 5 )

Note the key differences between working out interest and capital in a single payment or in a series of payments.
Single payment:


Calculate the loan outstanding after the previous payment



Calculate the interest by multiplying by the effective interest rate



Calculate the capital by subtracting the interest from the repayment

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Series of payments:


Calculate the capital by subtracting the loan outstanding after the payments from the loan outstanding before the payments



Calculate the interest by subtracting the capital from the total of the payments made Example
Look back at the example on page 9. We can easily calculate the total interest paid by the final three instalments.
Each instalment was $12,759.16 and so the last three payments totalled $38,277.48.
The capital repaid by these three instalments equals the loan outstanding immediately after the second payment which we calculated to be $31,781.08.
Therefore the interest paid was 38,277.48 − 31,78108 = $6,496 (4SF).
.

Question 9.8
A loan of £16,000 is repayable by ten equal annual payments. The annual effective rate of interest is 4%. Calculate:
(i)

the interest element of the 4th payment

(ii)

the capital element of the 7th payment

(iii)

the capital repaid in the last five years of the loan

(iv)

the total interest paid over the whole loan.

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CT1-09: Loan schedules

The loan schedule
In the solution to Question 9.2 we set out the loan outstanding and the capital and interest part of each payment in a table. This type of table is called a “loan schedule”.
It will now be defined in a more general context.
The loan payments can be expressed in the form of a table, or “schedule”, as follows. Year r → r +1

Loan outstanding at r

Instalment at r + 1

Interest due at r +1

Capital repaid at r +1

0→1

L0

X1

iL0

X 1 − iL0

t →t +1

Lt

X t +1

iLt

X t + 1 − iLt

n − 1→ n

Ln − 1

Xn

iLn − 1

X n − iLn − 1

Loan outstanding at r +1
L1
= L0 − ( X 1 − iL0 )
Lt + 1
= Lt − ( X t + 1 − iLt )
0

With spreadsheet software it is a simple matter to construct the entire schedule for any loan.

Without a spreadsheet it is quite time consuming to have to construct a complete schedule for a loan with more than say four repayments. In an exam you would probably only be asked to construct part of a schedule, or a schedule for a short-term loan. Question 9.9
For a loan of amount L where interest and capital are repaid by equal annual payments at the end of each of the next n years, construct a loan schedule in terms of the interest rate i and the annual repayment P = L / an| . Simplify your answer where possible.

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Page 13

Instalments payable more frequently than annually
Most loans will be repaid in quarterly, monthly or weekly instalments. No new principles are involved where payments are made more frequently than annually, but care needs to be taken in calculating the interest due at any instalment date.
If the rate of interest used is effective over the same time unit as the frequency of the repayment instalments, then the calculations proceed exactly as above, with the time unit redefined appropriately.
For the case where the interest is expressed as an effective annual rate, with repayment instalments payable pthly, we have the equation of value for the loan,
1 2 3 given repayments of X t at time t = p , p , p , … , n :

L0 = X 1/ pv 1/ p + X 2 / pv 2 / p + X 3 / pv 3 / p +

+ X nv n

In the case where the loan is repaid by level instalments of amount X payable pthly, the loan equation simplifies to:
(
L0 = pXan|p )

Example
A loan of £900 is repayable by equal monthly payments for 3 years, with interest payable at 18½% pa effective. Calculate the amount of each monthly payment.

Solution
Let M equal the monthly payment. Then:
900

(12)
= 12 Ma3|

fiM =

900i (12)
12(1 - v3 )

= 12 M
=

1 - v3 i (12)

900 ¥ 12(1.1851/12 - 1)
12(1 - 1.185-3 )

= £32.13

It is easy to show that the two basic principles for calculating the loan outstanding hold when repayments are more frequent than annual. That is, the loan outstanding at any repayment date, immediately after an instalment has been paid, may still be calculated as the present value of the remaining repayment instalments, or as the accumulated value of the original loan less the repayments made to date.

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CT1-09: Loan schedules

Prospectively:

Lt = X t + 1/ pv 1/ p + X t + 2 / pv 2 / p +

+ X nv n − t

Retrospectively:

Lt = L0 (1 + i )t − ( X 1/ p (1 + i )t − 1/ p + X 2 / p (1 + i )t − 2 / p
+ X 3 / p (1 + i )t − 3 / p +

+ X t − 1/ p (1 + i )1/ p + X t )

Question 9.10
Simplify the above two formulae for the case where the loan is repaid by level instalments of amount X payable pthly.
Given an annual effective rate of interest of i, the effective rate of interest over a
1
1 period p is (1 + i )1/ p − 1 , which is equal to i ( p ) p . The interest due at t + p , given capital outstanding of Lt at some repayment date t, is therefore
1
bt + 1/ p = ((1 + i )1/ p − 1)Lt . The capital repaid at t + p is then:

ft + 1/ p = X t + 1/ p − ((1 + i )1/ p − 1)Lt = X t + 1/ p −

which is just the total payment at time t + capital repaid at t +

1 p outstanding at time t +

1 p i ( p)
Lt
p

less the interest due at time t +

1 p . The

is also equal to the capital outstanding at time t less the capital
1
p

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, ie Lt − Lt +1/ p .

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Example
Calculate the interest and capital portions of the thirteenth payment of the loan given in the example on page 13.

Solution
The loan outstanding immediately after the twelfth payment is:
(
12 × 32.13a212 ) = 38556 ×
.
|

1− v2 i (12 )

= 38556 × 16839 = £649.25
.
.

The interest portion of the thirteenth payment is therefore:

i (12)
649.25 ×
= £9.25
12
The capital portion is:
32.13 − 9.25 = £22.88

Question 9.11
A loan of £120,000 is repayable by equal quarterly payments for 25 years. effective rate of interest is 6% pa. Find the interest portion of the first payment.

The

Question 9.12
A loan of £4,000 is repayable by equal monthly payments for 5 years. Interest is payable at a rate of 7% pa effective. Calculate the interest paid and the capital repaid in the 4th year.

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6

CT1-09: Loan schedules

Exam-style question
Try the following question yourself before turning the page and working through the solution. Question
A woman takes out a home improvement loan for £11,000 over 5 years. She makes monthly repayments in arrears and the bank charges an effective rate of interest of 6% pa. (i)

What is the monthly repayment?

[2]

(ii)

How much interest does she pay in the third year?

[2]

(iii)

How much capital is repaid in the 20th instalment?

[2]

(iv)

At the end of the fourth year she decides to make further improvements to her house and wants to borrow another £4,000 at that stage. If her total balance is to be repaid over 3 years by level monthly payments and there is no alteration to the interest rate, how much is each payment?
[3]
[Total 9]

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CT1-09: Loan schedules

Page 17

Solution
(i)

1

If we work in months, i = 106 12 − 1 = 0.4867551% .
.
repayment, then:

If P is the monthly

Pa60| = 11, 000
We can then calculate P:

P=

(ii)

11, 000 11, 000
=
= £211.85 a60| 51.924

The total repayment she makes in the third year is 12 ¥ 211.85 = 2,542.20 .
The capital outstanding at the start of the year with 3 years or 36 months left to go is:
Pa36| = 211.85 ¥ 32.94896 = 6,980.20
The capital outstanding at the end of the year is:
Pa24| = 211.85 ¥ 22.59937 = 4, 787.70
The capital repaid is therefore 6,980.20 - 4, 787.70 = 2,192.5 .
The interest paid is the difference between the total repayment and the capital repaid, ie 2,542.20 - 2,192.50 = £349.70 .

(iii)

The capital outstanding immediately after the 19th payment is:
Pa60-19| = Pa41| = 211.85 ¥ 37.08644 = 7,856.76
The interest paid in the 20th payment is:

0.004867551 ¥ 7,856.76 = 38.24
The capital element is therefore:
211.85 - 38.24 = £173.61

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CT1-09: Loan schedules

We need to know how much money is outstanding on her original loan:

Pa60- 48| = Pa12| = 211.85 ¥ 11.6288 = 2, 463.60
She now has £6,463.60 to pay off over 3 years, so we can calculate the repayment: Ra36| = 6, 463.60 , where R is the repayment
So R =

6, 463.60 6, 463.60
=
= £196.17 . a36| 32.9490

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CT1-09: Loan schedules

7

Page 19

Consumer credit: flat rates and APRs
Where the borrower is an individual, borrowing from an institution such as a bank, it is common to use the flat rate of interest as a measure of the interest charge. The flat rate of interest is defined as the total interest paid over the whole transaction, per unit of initial loan, per year of the loan. For example, if a loan of L0 is repaid over two years by level monthly instalments of amount X, then the total capital and interest paid is 24X. The total capital must be the amount of the original loan, so the total interest paid is 24 X − L0 . This gives the flat rate of interest per annum:

F=

24 X − L0
2L0

The flat rate is a very simple calculation that ignores the details of the gradual repayment of capital over the term of a loan. Flat rates are only useful for comparing loans of equal term. Two loans of different terms calculated using the same effective rate of interest will have different flat rates. Since the flat rate ignores the repayment of capital over the term of the loan, it will be considerably lower than the true effective rate of interest charged on the loan.

Example
Calculate the flat rate paid on a loan of £48,000 that is repaid over 25 years. Payments are made monthly. Each payment is £278.

Solution
The total amount of money paid out is 12 × 278 × 25 = £83, 400 .
The total interest paid is 83, 400 − 48, 000 = £35, 400 .
The flat rate is the total amount of interest divided by the loan, divided by the number of
35, 400 years of the loan. Here it is
= 2.95% .
48, 000 × 25
In general: flat rate =

total interest total repayment - original loan
=
original loan ¥ term in years original loan ¥ term in years

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CT1-09: Loan schedules

To ensure that consumers can make informed judgements about the interest rates charged, lenders are required (in most circumstances) to give information about the effective rate of interest charged. In the UK this is in the form of the
Annual Percentage Rate of charge, or APR, which is defined as the effective annual rate of interest, rounded to the nearest 1/10th of 1%.

The APR is the rate of interest at which the present value of the amount borrowed equals the present value of the repayments (including all other charges). It is the smallest positive solution (or the most negative solution, if there aren’t any positive solutions) and is rounded to the nearest 0.1%.
Very roughly, the APR is twice the flat rate. We will justify this shortly.

Example
A motorist buys a car costing £5,000 using a loan with a flat rate of interest of 10%, and repayments at the end of each of the next 12 months.
Calculate the APR paid by the borrower.

Solution
The calculation of the payments is as follows:
Amount borrowed:

£5,000

Total repayments:

Interest due (@10%): £500

£5,500

Monthly repayments: £458.33 (ie 5,500 12 )
To find the annual effective rate of interest, we need to solve the equation of value:
12 × 458.33a1(12) = 5, 000
This gives a1(12) = 0.90910 .
Since the APR is roughly twice the flat rate, we can try 20% as a first guess:
At 20%, a1(12) = 0.90721 . At 19.5%, a1(12) = 0.90921 . At 19.6%, a1(12) = 0.90881 .
Since the 19.5% value is closer to 0.90910 than the 19.6% value, the APR is 19.5%.

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CT1-09: Loan schedules

Page 21

Question 9.13
Explain why the effective annual interest rate is roughly twice the flat rate of interest.

Question 9.14
A loan of £3,000 is repayable by 36 monthly instalments, payable in arrears. The flat rate of interest charged on the loan is 8% pa.
(i)

What is the monthly repayment?

(ii)

What is the APR on this transaction?

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CT1-09: Loan schedules

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Chapter 9 Summary
A very common transaction involving compound interest is a loan that is repaid by regular instalments, at a fixed rate of interest, for a predetermined term.
Each repayment must pay first for interest due on the outstanding capital; the balance is then used to repay some of the capital outstanding.
We can find the loan outstanding prospectively or retrospectively.
A prospective method involves finding the present value of future cashflows.
A retrospective method involves calculating the accumulated value of the initial loan less the accumulated value of the repayments to date.
We can calculate the interest element contained in a single payment by calculating the loan outstanding immediately after the previous instalment and multiplying it by the rate of interest. The capital element is the total payment less the interest payment.
We can calculate the capital repaid in a period where there is more than one payment by subtracting the capital outstanding at the end of the period from the capital outstanding at the start of the period. The interest paid in this period is then the total payment less the capital repaid.
The interest and capital components in the repayments for a loan can be set out in the form of a loan schedule. The components for any period can be calculated directly using actuarial functions.
No new principles are involved where payments are made more frequently than annually, but care needs to be taken in calculating the interest due at any instalment date. The flat rate of interest is defined as the total interest paid over the whole transaction, per unit of initial loan, per year of the loan.
Lenders are usually required to give information about the effective rate of interest charged. In the UK this is in the form of the Annual Percentage Rate of charge, or APR, which is defined as the effective annual rate of interest, rounded to the nearest 1/10th of 1%.

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CT1-09: Loan schedules

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Chapter 9 Solutions
Solution 9.1 a3| at 7% is given in the Tables as 2.6243.
Therefore X =

1, 000
= 381.05 .
2.6243

Solution 9.2
First calculate the total amount of each payment Y.

Ya5| = 5,000 ⇒ Y =

5,000
= £1,318.98
3.7908

The following table shows how each payment of £1,318.98 is split between interest and capital payments.

Year
1
2
3
4
5

Loan outstanding at start of the year
(L)
5,000
4,181.02
3,280.14
2,289.17
1,199.11

Interest due at the end of the year (I = 10% of L)
500
418.10
328.01
228.92
119.91

Capital repaid at the end of the year
(C = 1,318.98 − I )
818.98
900.88
990.97
1,090.06
1,199.07

Loan outstanding at end of the year
(L–C)
4,181.02
3,280.14
2,289.17
1,199.11
0.04*

* This is non-zero due to rounding error.

Solution 9.3
(a)

1,000(1 + i ) 2 − 38105s2| = 1,000 × 114490 − 38105 × 2.0700 = 35613
.
.
.
.

(b)

38105v =
.

38105
.
= 35612
.
107
.

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CT1-09: Loan schedules

Solution 9.4
L0 = Xan|

Solution 9.5
Lt = X (v + v 2 + v3 +

+ v n -t )

= Xan -t|

Solution 9.6
Lt = L0 (1 + i ) t − Xst |

Solution 9.7
Let X equal the amount of the first instalment, then:
80,000 = Xa3| + 2 Xv 3a5| or equivalently:

Ê 1 - 1.045-8
1 - 1.045-5 ˆ
80, 000 = Xa8| + Xv3a5| fi 80, 000 = X Á
+ 1.045-3 ¥
0.045 ˜
Ë 0.045
¯
Therefore:

X =

80, 000
= 7, 660.74
6.5959 + 0.8763 ¥ 4.3900

The loan outstanding one year before the end of the term equals (prospectively) the present value of the final repayment, which equals:
2 × 7,660.75v = £14,662
We

can

check

this

answer

using

the

retrospective

method,

namely,

80,000(1 + i ) 7 − Xs7| − Xs4| = £14,662 .

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CT1-09: Loan schedules

Page 27

Solution 9.8
If the annual repayment is X then:

Xa10 = 16, 000 fi (i)

X =

16, 000 16, 000
=
= 1,972.66 a10 8.1109

The capital outstanding after the 3rd payment is:
1,972.66a7 = 1,972.66 ¥ 6.0021 = 11,840.01
The interest element of the 4th payment is:

0.04 ¥ 11,840.01 = 473.60
(ii)

The capital outstanding after the 6th payment is:

1,972.66a4 = 1,972.66 ¥ 3.6299 = 7,160.55
The interest element of the 7th payment is:

0.04 ¥ 7,160.55 = 286.42
Therefore, the capital element of the 7th payment is:

1,972.66 - 286.42 = 1, 686.24
(iii)

The capital repaid over the last five years of the loan must be the capital outstanding after the 5th payment, ie:
1,972.66a5 = 1,972.66 ¥ 4.4518 = 8, 781.93

(iv)

The total interest payable over the whole loan is the total payment made less the capital borrowed, ie:

10 ¥ 1,972.66 - 16, 000 = 3, 726.60

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CT1-09: Loan schedules

Solution 9.9
Year
r → r +1

Loan o/s at r

Instalment at r + 1

0→1

L = Pan|

P = L / an |

1→ 2

Pan −1|

P = L / an |

t → t +1

Pan − t |

P = L / an |

n −1→ n

Pa1| = Pv

P = L / an |

Capital repaid at r +1

Loan outstanding at r +1

n

Pv n

P(an| − v n ) = Pan −1|

n −1

Pv n−1

Pan−2|

P(1 − v n − t )

Pv n −t

Pan −t −1|

P(1 − v )

Pv

0

Interest due at r + 1 iL = iPan|
= P(1 − v ) iPan −1|
= P(1 − v

)

Solution 9.10
Prospectively:

( p)
Lt = pXan −t |

Retrospectively:

Lt = L0 (1 + i ) t − pXst(| p )

Solution 9.11
The amount of each instalment is not needed because we are given the loan outstanding at the start. The interest portion of the first payment equals:
120,000(1061/ 4 − 1) = £1,760.86
.

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Solution 9.12
Calculating the monthly payment, X:

12 Xa (12) = 4, 000
5



X =

4, 000
12a (12)

= 78.80

5

The capital outstanding at the start of the 4th year is:
12 ¥ 78.80 ¥ a (12) = 12 ¥ 78.80 ¥ 1.8653 = 1, 763.84
2

The capital outstanding at the end of the 4th year is:
12 ¥ 78.80 ¥ a (12) = 12 ¥ 78.80 ¥ 0.9642 = 911.75
1

So the capital repaid in the 4th year is:

1, 763.84 - 911.75 = 852.09
The interest paid is the difference between the total payment and the capital repaid, ie:
12 ¥ 78.80 - 852.09 = 945.60 - 852.09 = 93.51

Solution 9.13
The initial capital (ie the £5,000 borrowed) is repaid during the year. So the borrower
“owes” £5,000 at the beginning of the year and nothing at the end of the year. The average amount “owed” during the year is therefore only £2,500, ie half the amount borrowed. However, the interest payments are calculated as 1 year’s interest on the whole of the initial amount borrowed. So this approximately doubles the true interest rate. Alternatively, we can estimate the effective annual interest rate here by dividing the total interest payments by the average loan outstanding, which gives: i ≈ £500 / £2,500 = 20% .

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CT1-09: Loan schedules

Solution 9.14
3, 000 + 3 ¥ 0.08 ¥ 3, 000
= 103.33 .
36

(i)

The monthly repayment will be

(ii)

To find the APR we need to solve the equation of value:

3,000 = 12 × 103.33a (12 )
|
3

ie

a (12 ) = 2.4194
|
3

We know the APR will be close to twice the flat rate, so try i = 16% to start with. (12)
@ i = 16% , i (12 ) = 0149342 and a3| = 2.4062 .
.
(12)
@ i = 15% , i (12) = 0.140579 and a3| = 2.4362 .
(12)
@ i = 15.5% , i (12) = 0.144969 and a3| = 2.4211 .
(12)
@ i = 15.6% , i (12) = 0.145844 and a3| = 2.4181 .

Since the value for 15.6% is closer than that for 15.5%, the APR is 15.6%.

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CT1-10: Project appraisal

Page 1

Chapter 10
Project appraisal
Syllabus objective
(ix)

Show how discounted cashflow techniques can be used in investment project appraisal. 1.

2.

Calculate the internal rate of return implied by the receipts and payments from an investment project.

3.

Describe payback period and discounted payback period and discuss their suitability for assessing the suitability of an investment project.

4.

Determine the payback period and discounted payback period implied by the receipts and payments from an investment project.

5.

0

Calculate the net present value and accumulated profit of the receipts and payments from an investment project at given rates of interest.

Calculate the money-weighted rate of return, the time-weighted rate of return and the linked internal rate of return on an investment or a fund.

Introduction
This chapter starts by looking at methods that can be used to decide between alternative investment projects. We consider the following criteria:


net present value and accumulated profit



internal rate of return



payback period



discounted payback period

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CT1-10: Project appraisal

We then look at three measures of investment performance:


money-weighted rate of return



time-weighted rate of return



linked internal rate of return

Estimating cashflows
Suppose an investor is considering the merits of an investment or business project. The investment or project will normally require an initial outlay and possibly other outlays in future, which will be followed by receipts, although in some cases the pattern of income and outgo is more complicated. The cashflows associated with the investment or business venture may be completely fixed (as in the case of a secure fixed interest security maturing at a given date) or they may have to be estimated.

Question 10.1

(Revision)

Describe the cashflows for an organisation that issues a fixed interest security.
The estimation of the cash inflows and outflows associated with a business project usually requires considerable experience and judgement. All the relevant factors (such as taxation and investment grants) and risks (such as construction delays) should be considered by the actuary, with assistance from experts in the relevant field (eg Civil Engineering for building projects). The identification and assessment of the risks may be done using the Risk Analysis and Management for Projects (RAMP) approach for risk analysis and management that has been developed by, and published on behalf of, the actuarial and civil engineering professions. You don’t need to know any more about RAMP than this for the Subject CT1 exam.
However, if you are interested to find out more then there is a manual entitled “Risk
Analysis and Management for Projects” available from the Institute or Faculty.
Considerable uncertainty will exist in the assessment of many of the risks, so it is prudent to perform calculations on more than one set of assumptions, eg on the basis of “optimistic”, “average”, and “pessimistic” forecasts respectively.

A set of optimistic assumptions is often called a “weak” basis and a pessimistic set of assumptions is often called a “strong”, “prudent” or “cautious” basis. Average assumptions are also called “best-estimate” or “realistic” assumptions.

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More complicated techniques (using statistical theory) are available to deal with this kind of uncertainty. Precision is not attainable in the estimation of cashflows for many business projects and hence extreme accuracy is out of place in many calculations.

For example, there is no point in quoting a final answer to a calculation to eight decimal places if the figures you input into the calculation are only given to the nearest million.
Net cashflow ct at time t (measured in suitable time units) is:

ct = cash inflow at time t − cash outflow at time t
If any payments may be regarded as continuous then ρ (t ) , the net rate of cashflow per unit time at time t, is defined as:

ρ (t ) = ρ 1(t ) − ρ 2 (t ) where ρ 1(t ) and ρ 2 (t ) denote the rates of inflow and outflow at time t respectively. We now start looking at methods that can be used to decide between alternative investment projects. We will use the following two hypothetical projects as examples.
Both relate to a small software company that has been asked to set up a new computer system for a major client.
Project R
Project R delegates all the development work to outside companies. The estimated cashflows for Project R are (where brackets indicate expenditure):
Beginning of Year 1 (£150,000)

(contractors’ fees)

Beginning of Year 2 (£250,000)

(contractors’ fees)

Beginning of Year 3 (£250,000)

(contractors’ fees)

End of Year 3 £1,000,000

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CT1-10: Project appraisal

Project S
Project S carries out all the development work in-house by purchasing the necessary equipment and using the company’s own staff. The estimated cashflows for Project S are: Beginning of Year 1 (£325,000)

(new equipment)

Throughout Year 1

(£75,000)

(staff costs)

Throughout Year 2

(£90,000)

(staff costs)

Throughout Year 3

(£120,000)

(staff costs)

End of Year 3 £1,000,000

(sales)

The staff costs can be assumed to be paid uniformly throughout the year.
The next few sections will refer back to these two examples.

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1

Page 5

Fixed interest rates
Having ascertained or estimated the net cashflows of the investment or project under scrutiny, the investor will wish to measure its profitability in relation to other possible investments or projects. In particular, the investor may wish to determine whether or not it is prudent to borrow money to finance the venture.
Assume for the moment that the investor may borrow or lend money at a fixed rate of interest i per unit time. The investor could accumulate the net cashflows connected with the project in a separate account in which interest is payable or credited at this fixed rate. By the time the project ends (at time T, say), the balance in this account will be:
T

∑ ct (1 + i )T − t + ∫0 ρ (t )(1 + i )T −t dt

(1.1)

where the summation extends over all t such that ct ≠ 0 .

1.1

Accumulated value
So one criterion that can be used to assess an investment project involves calculating the
“accumulated profit” at the end of the project. This is the accumulated value of the net cashflows (as at the time of the last payment).
The accumulated value, at time T, of a cashflow can be expressed as:

A(T ) =

T

∑ ct (1 + i )T −t + ∫0 ρ (t )(1 + i )T −t dt

The accumulated profit for a project has the intuitive appeal that it represents the final amount you would have “left over” if all the payments for the project were transacted through a bank account that earned interest at a rate i.
However, accumulated profit calculations suffer from the disadvantage that they can only be used in situations where there is a definite fixed time horizon for the project.
This will not be the case if the time horizon (ie the time until the last cashflow payment) is unlimited or the timing of the payments is uncertain.

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CT1-10: Project appraisal

Example
Give two examples to illustrate the problems of using accumulated profit to assess the suitability of an investment project.
Solution
Example 1
Suppose an investor is considering purchasing a fixed interest security with coupon payments that continue forever. In order to determine a value for the accumulated profit, the investor would have to (arbitrarily) pick a date and assume that the holding would be sold on that date. (Otherwise, the accumulated profit would be infinite.) The value calculated for the accumulated profit will then depend crucially on which date is selected.
Example 2
Suppose a retired man (in good health) is considering using a lump sum to buy a pension payable for the rest of his life. Since he does not know when he will die, he cannot know what date to accumulate the payments to.

Question 10.2
Find the accumulated profit after 20 years of a project that pays out $20,000 at time 0 and then receives $5,000 at times 5 to 15, inclusive. Assume an annual effective rate of interest of 3%.
Another problem associated with accumulated profit calculations is that the accumulated profits for two different projects cannot be compared directly if they have different time horizons, since the calculated values will relate to different dates.
However, this problem can be avoided by accumulating all the profits to the date of the last payment for the longest project.
These problems can be avoided by calculating the “net present value” instead.

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1.2

Page 7

Net present values
The present value at rate of interest i of the net cashflows is called the “net present value” at rate of interest i of the investment or business project, and is usually denoted by NPV (i ) . Hence:

NPV (i ) =

T

∑ ct (1 + i )−t + ∫0 ρ (t )(1 + i )−t dt

(1.2)

The rate of interest i used to calculate the net present value is often referred to as the risk discount rate. So note that the risk discount rate is an i not a d rate.
The net present value is equivalent to the accumulated profit, the only difference being that we are now looking at the value at the outset (which, by definition, is a fixed date), rather than the value at the end of the project. A higher net present value indicates a more “profitable” project.
(If the project continues indefinitely, the accumulation (1.1) is not defined, but the net present value may be defined by Equation (1.2) with T = ∞ .) If ρ (t ) = 0 , we obtain the simpler formula:

NPV (i ) =

∑ ct v t

where v = (1 + i ) −1 .
Since the equation:

NPV (i ) = 0 is the equation of value for the project at the present time, the yield i 0 on the transaction is the solution of this equation, provided that a unique solution exists. It may readily be shown that NPV (i ) is a smooth function of the rate of interest i and that NPV (i ) → c0 as i → ∞ .

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Example
Calculate the net present value for Project R and Project S using a risk discount rate of
20% per annum. Using net present values as a criterion, which project is preferable?
Solution
For Project R, the net present value (in £000s) is:
NPVR = -150 - 250v - 250v 2 + 1, 000v3 @ 20%
= -150 - 250 (0.83333) - 250 (0.69444) + 1, 000 (0.57870) = 46.8
For Project S, the net present value (in £000s) is:

NPVS = -325 - 75a1| - 90v a1| - 120v 2 a1| + 1, 000v3 @ 20%
= -325 - (75 + 90v + 120v 2 )

d

d

+ 1, 000v3 @ 20%

= -325 - [75 + 90 (0.83333) + 120 (0.69444)] ¥

0.166667
+ 1, 000 (0.57870)
0.182322

= 40.4
The net present values are: £46,800 for Project R and £40,400 for Project S.
So, using a risk discount rate of 20%, Project R appears more favourable.
Note that the net present value will depend on the risk discount rate used.

Question 10.3
Calculate the net present value of Projects R and S using a risk discount rate of 10% pa.
The following graph shows the net present values for Projects R and S for different risk discount rates. There is a crossover point corresponding to about 16.8%. If the risk discount rate used exceeds 16.8%, the net present value is greater for Project R than for
Project S. If the risk discount rate is less than 16.8%, the net present value is greater for
Project S.

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110
100

cross-over point i
= 16.8%

NPV

90
80
70
60
50
15

16

17

18

19

20

i

In the graph, the dotted line represents Project R and the solid line represents Project S.

Question 10.4
The cashflows Ct (where the time t is measured in years and the amounts are in £000) for two business ventures are as follows:
Venture 1:

C0 = −100 , C1 = −40 , C2 = +50 , C3 = +120

Venture 2:

C1 = −45 , C3 = +25 , C4 = +25 , C5 = +25

Calculate the accumulated profit at time 5 and the net present value for each of these ventures using a risk discount rate of 15% per annum.

1.3

Internal rate of return
In economics and accountancy the yield per annum is often referred to as the
“internal rate of return” (IRR) or the “yield to redemption”. The latter term is frequently used when dealing with fixed interest securities, for which the
“running” (or “flat”) yield is also considered.

We will leave the definition of the running yield until later in the course.
The internal rate of return for an investment project is the effective rate of interest that equates the present value of income and outgo, ie it makes the net present value of the cashflows equal to zero.

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If all the payments for the project were transacted through a bank account that earned interest at the same rate as the internal rate of return, the net proceeds at the end of the project (ie the accumulated profit) would be zero. A higher internal rate of return indicates a more “profitable” project.
For projects where the payments going out all precede the payments coming in, there will be a unique solution to the equation defining the internal rate of return, since the quantity “ PV payments in − PV payments out ” will always decrease as i increases.
The internal rate of return need not be positive. A zero return implies that the investor received no return on investment and if the yield is negative then the investor lost money on investment. It is difficult, however, to find a practical interpretation for a yield less than –1, and so if there is not a solution to the equation greater than –1, the yield is undefined.
In some cases, it is possible for there to be more than one solution. In such cases the smallest positive solution is usually used (or the most negative solution greater than
–1, if there aren’t any positive solutions). Some authors however have stated that the yield is undefined if there is not a unique solution greater than –1. You are unlikely to come across many such examples whilst studying Subject CT1. Also, if there are only inflows of cash (ie no outflow), the internal rate of return will be infinite.
Usually, the equation of value cannot be solved directly to find the interest rate. In these cases, an approximate solution can be found by first estimating an approximate value using the methods we looked at earlier. A more accurate value can then be found using linear interpolation by calculating the net present value for interest rates close to the initial estimate.

Example
Find the internal rate of return for Project R.

Solution
We need to find the interest rate i that satisfies the equation of value:

−150 − 250v − 250v 2 + 1,000v 3 = 0
We already know that:
At 20%:

−150 − 250v − 250v 2 + 1,000v 3 = 46.8

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Since the outgo precedes the income, the value of i must be greater than this. So we need to try a higher rate:
At 25%:

−150 − 250v − 250v 2 + 1,000v 3 = 2.00
46.8

2.00

0

20%

25%

i

We can approximate i by extrapolating (linearly) using these two values:

i ≈ 20% +

0 − 46.8
× (25% − 20%) = 25.2%
2.00 − 46.8

Question 10.5
Find the internal rate of return for Project S and hence determine which project is more favourable using this criterion.
The practical interpretation of the net present value function NPV (i ) and the yield is as follows. Suppose that the investor may lend or borrow money at a fixed rate of interest i1 . Since, from Equation (1.2), NPV (i1) is the present value at rate of interest i1 of the net cashflows associated with the project, we conclude that the project will be profitable if and only if:

NPV (i1) > 0
Also, if the project ends at time T, then the profit (or, if negative, loss) at that time is (by Expression (1.1)):

NPV (i1 )(1 + i1)T
Let us now assume that, as is usually the case in practice, the yield i 0 exists and
NPV (i ) changes from positive to negative when i = i 0 . Under these conditions it is clear that the project is profitable if and only if:

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Many projects will need to provide a return to shareholders and so there will not be a specific fixed rate of interest that has to be exceeded. Instead a target, or hurdle, rate of return may be set for assessing whether a project is likely to be sufficiently profitable.

1.4

The comparison of two investment projects
Suppose now that an investor is comparing the merits of two possible investments or business ventures, which we call projects A and B respectively.
We assume that the borrowing powers of the investor are not limited.

There are therefore no restrictions on how much the investor can borrow.
Let NPVA (i ) and NPVB (i ) denote the respective net present value functions and let i A and i B denote the yields (which we shall assume to exist). It might be thought that the investor should always select the project with the higher yield, but this is not invariably the best policy. A better criterion to use is the profit at time T (the date when the later of the two projects ends) or, equivalently, the net present value, calculated at the rate of interest i1 at which the investor may lend or borrow money. This is because A is the more profitable venture if:

NPVA (i1) > NPVB (i1)
The fact that i A > i B may not imply that NPVA (i1) > NPVB (i1) is illustrated in the following diagram. Although i A is larger than i B , the NPV(i) functions “crossover” at i ′ . It follows that NPVB (i1) > NPVA (i1) for any i1 < i ′ , where i ′ is the cross-over rate. There may even be more than one cross-over point, in which case the range of interest rates for which project A is more profitable than project B is more complicated.

The following graph is similar to the graph shown on page 9, although it has been extended to show the yields for the two projects.

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We now give a final example for this section.

Worked Example
An investor is considering whether to invest in either or both of the following loans: Loan X:
For a purchase price of £10,000, the investor will receive £1,000 per annum payable quarterly in arrears for 15 years.
Loan Y:
For a purchase price of £11,000, the investor will receive an income of £605 per annum, payable annually in arrears for 18 years, and a return of his outlay at the end of this period.
The investor may lend or borrow money at 4% per annum. Would you advise the investor to invest in either loan, and, if so, which would be the more profitable?

Solution
We first consider loan X:
NPV X (i ) = −10,000 + 1,000a( 4 )
|
15

and the yield is found by solving the equation NPV X (i ) = 0 , or a( 4 ) = 10 , which
15|
gives i X ≈ 5.88% .

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CT1-10: Project appraisal

(4
This is easily checked by calculating a15|) @ 5.88%:

(4 a15|) =

1 − v15 i ( 4)

=

1 − 10588−15
.
4(105881/ 4 − 1)
.

= 10.00

as required

For loan Y we have:

NPVY (i ) = −11,000 + 605a18| + 11000v 18
,
and the yield (ie the solution of NPVY (i ) = 0 ) is iY = 5.5% .

Question 10.6
Check that iY = 55% solves the equation of value.
.
The rate of interest at which the investor may lend or borrow money is 4% per annum, which is less than both i X and iY , so we compare NPV X (0.04) and

NPVY (0.04) .
Now NPV X (0.04) = £1,284 and NPVY (0.04) = £2,089 , so it follows that, although the yield on loan Y is less than on loan X, the investor will make a larger profit from loan Y. We should therefore advise him that an investment in either loan would be profitable, but that, if only one of them is to be chosen, then loan Y will give the higher profit.
The above example illustrates the fact that the choice of investment depends very much on the rate of interest i1 at which the investor may lend or borrow money. If this rate of interest were 5¾%, say, then loan Y would produce a loss to the investor, while loan X would give a profit.

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Different interest rates for lending and borrowing
We have assumed so far that the investor may borrow or lend money at the same rate of interest i1 . In practice, however, the investor will probably have to pay a higher rate of interest ( j1 , say) on borrowings than the rate ( j 2 , say) he receives on investments.

This is because banks “make their money” by borrowing money from savers at one rate of interest and lending it out for mortgages, business loans etc at a higher rate.
The difference j1 − j 2 between these rates of interest depends on various factors, including the credit-worthiness of the investor and the expense of raising a loan.
The concepts of net present value and yield are in general no longer meaningful in these circumstances. We must calculate the accumulation of net cashflows from first principles, the rate of interest depending on whether or not the investor’s account is in credit. In many practical problems the balance in the investor’s account (ie the accumulation of net cashflows) will be negative until a certain time t1 and positive afterwards, except perhaps when the project ends.
In some cases the investor must finance his investment or business project by means of a fixed-term loan without an early repayment option. In these circumstances the investor cannot use a positive cashflow to repay the loan gradually, but must accumulate this money at the rate of interest applicable on lending, ie j 2 .

Example
The cashflows for Project C are:
Outlay
£100,000

(initially)

Proceeds
£140,000

(end of year 5)

The cashflows for Project E are:
Outlay
£80,000
£20,000

(initially)

Income
£10,000

(end of year 1)

(start of year 2)

£30,000

(end of year 2)

£87,000

(end of year 3)

£5,000 (start of year 3)

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A company must choose between Projects C and E, both of which would be financed by a loan, repayable only at the end of the project. The company must pay 6.25% per annum on money borrowed, but can earn only 4% on money invested in its deposit account. Calculate the accumulated profit at the end of 5 years for each project.

Solution
Project C
Since Project C does not generate any income, the company will be relying on the loan throughout. The accumulated profit will be:

140,000 − 100,000 × 106255 = £4,592
.
Project E
Since Project E does generate income, we need to consider the company’s net assets at the end of each year to see whether there are any excess funds available to invest.
At the end of year 1, there will be an income payment of £10,000, but interest of
80,000 × 0.0625 = £5,000 is required and a further outlay of £20,000 must be made.
So further borrowing of £15,000 is required (taking the total borrowing to £95,000).
At the end of year 2, there will be an income payment of £30,000, but interest of
95,000 × 0.0625 = £5,937.5 is required and a further outlay of £5,000 must be made.
This leaves 30,000 − 5,937.5 − 5,000 = £19,062.5 available for investment.
At the end of year 3, the company receives an income of £87,000. It pays interest of
95,000 × 0.0625 = £5,937.5 , leaving £81,062.5. This, together with the proceeds from year 2 (plus interest), will be used to pay off the loan. The remaining money is invested for the last two years of the project.
So the accumulated profit at the end of year 5 will be:
(19,062.5 × 104 + 87,000 − 95,000 × 10625) × 104 2 = £6,368
.
.
.

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Question 10.7
If spare funds can be used to repay part of the loan at any time, calculate the revised accumulated profits for Projects C and E, and comment on your results.

Question 10.8
Explain why banks and other organisations lending money to people buying homes with mortgages often impose restrictions and/or penalties if all or part of the mortgage is repaid during the first 3 years.

2.1

Payback periods
Another quantity that is useful to calculate when an investment project is financed by outside borrowing is the discounted payback period.
“The discounted payback period is the number of years before the project gets out of the red.” In many practical problems the net cashflow changes sign only once, this change being from negative to positive. In these circumstances the balance in the investor’s account will change from negative to positive at a unique time t1 , or it will always be negative, in which case the project is not viable. If this time t1 exists, it is referred to as the “discounted payback period” (DPP). It is the smallest value of t such that A(t ) ≥ 0 , where:

A(t ) =

t

∑ cs (1 + j1)t − s + ∫0 ρ (s )(1 + j1)t − s ds

(2.1)

s ≤t

Note that t1 does not depend on j 2 but only on j1 , the rate of interest applicable to the investor’s borrowings. Suppose that the project ends at time T. If
A(T ) < 0 (or, equivalently, if NPV ( j1) < 0 ) the project has no discounted payback period and is not profitable. If the project is viable (ie there is a discounted payback period t1 ) the accumulated profit when the project ends at time T is:

P = A(t1)(1 + j2 )T − t1 +

T

∑ ct (1 + j2 )T −t + ∫t

t > t1

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ρ (t )(1 + j2 )T − t dt

1

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CT1-10: Project appraisal

This follows since the net cashflow is accumulated at rate

j 2 after the

discounted payback period has elapsed.

Other things being equal, a project with a shorter discounted payback period is preferable to a project with a longer discounted payback period because it will start producing profits earlier.

Example
Derive a formula for the accumulated value of an investment project which has cashflows at times t1 , t 2 , … , t10 (where t1 < t2 < < t10 ), given that the first 3 cashflows are negative while the remainder are positive, and that the discounted payback period is t 7 . Assume that the project is financed by flexible borrowing at rate j and that excess funds can be invested at rate i ( i < j ).

Solution
“Flexible borrowing” implies that the borrowings can be repaid at any time. Up to time t 7 the project will have to be funded by borrowing (at rate j ).
So the accumulated value at this time will be:
7

Acc.V (t7 ) = ∑ Ctk (1 + j ) t7 −tk k =1

Thereafter, there will be excess funds which can be invested at rate i up to time t10 .
So the accumulated value at the end of the project will be:
7

10

k =1

k =8

Acc.V (t10 ) = ∑ Ctk (1 + j ) t7 −t k (1 + i ) t10 −t7 + ∑ Ctk (1 + i ) t10 −t k

If interest is ignored in formula (2.1) (ie if we put j1 = 0 ), the resulting period is called the “payback period”. However, its use instead of the discounted payback period often leads to erroneous results and is therefore not to be recommended.

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Question 10.9
The business plan for a new company that has obtained a 5 year lease for operating a local bus service is shown in the table below. Items marked with an asterisk represent continuous cashflows.
Cashflow item

Timing

Amount (£000)

Initial set up costs

Immediate

–100

Fees from advertising contracts

1 month

+200

Purchase of vehicles

3 months

–2,000

Fares from passengers*

From 3 months onwards

+1,000 pa

Staff costs and other operating costs* From 3 months onwards

– 400 pa

Resale value of assets

+500

5 years

Determine the discounted payback period for this project assuming that it will be financed by a flexible loan facility based on an effective annual interest rate of 10% per annum. Sketch a graph showing the accumulated profit as a function of time.
The discounted payback period is often employed when considering a single investment of C, say, in return for a series of payments each of R, say, payable annually in arrears for n years. The discounted payback period t1 years is clearly the smallest integer t such that A∗ (t ) ≥ 0 , where:

A∗ (t ) = −C (1 + j1 )t + Rst |

at rate j1

ie the smallest integer t such that:

Rat | ≥ C

at rate j1

The project is therefore viable if t1 ≤ n , in which case the accumulated profit after n years is clearly:

P = A∗ (t1 )(1 + j 2 )n − t1 + Rsn − t |
1

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at rate j 2

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Question 10.10
A speculator borrows £50,000 at an effective interest rate of 8% per annum to finance a project that is expected to generate £7,500 at the end of each year for the next 15 years.
Find the discounted payback period for this investment.

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Other considerations
At the simplest level, for projects involving similar amounts of money and with similar time horizons, the project that results in the highest accumulated profit will be the most favourable. Where the project can be funded without the need for external funds, this is equivalent to selecting the project with the highest net present value. The internal rate of return will provide a useful secondary criterion.
Where external borrowing is involved, the accumulated profit must be calculated directly by looking at the cashflows and taking into account the precise conditions of the loan. The discounted payback period will provide a useful secondary criterion.
However, it may not be a straightforward decision for the owners of a business to decide between alternative investment projects purely on the basis of net present values or internal rates of return or discounted payback periods. In many cases a comparison of net present values or internal rates of return for alternative projects will not lead to a decisive conclusion, since the values may be very close or they may conflict. So other considerations will have to be brought into the decision.

Example
Compare the following alternative investment opportunities:
(1)

An initial investment of £1 is made in return for a payment of £200 at the end of the year.

(2)

An initial investment of £10,000 is made in return for a payment of £12,000 at the end of the year.

Solution
Since the “growth factor” (a factor of 200) is so much higher for Investment 1, the internal rate of return is much higher for Investment 1.
However, since the actual amount of the profit (£2,000) is so much higher for
Investment 2, the net present value is much higher for Investment 2.
Continued…

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CT1-10: Project appraisal

If the initial capital needed for Investment 2 (ie the £10,000) was readily available, then
Investment 2 would be the natural choice. However, if the extra £9,999 required was not available and would have to be borrowed, the decision would depend on the rate of interest charged on the loan.
This is an extreme example, since it would be unlikely in practice for a business to have to decide between such dissimilar alternatives. However, it illustrates that usually a comparison of net present values can be considered to be the basic criterion for comparing projects, since for projects spanning the same period, this is equivalent to maximising the accumulated profit. However, the decision will be influenced by other criteria as well.

Question 10.11
What would be the approximate rate of interest charged on borrowed funds at which the decision in the above example would change, assuming the decision was made on the basis of net present values?
The investors will also need to take into account a number of other factors, such as:

Cashflow
Are the cashflow requirements for the project consistent with the business’s other needs? What will the accumulated profit at the end of the project be, ie how much cash will it generate? Over what period will the profits be produced and how will the profits be used?
Is it worth carrying out the project if the potential profit is very small in money terms?

Borrowing requirements
Can the business raise the necessary cash at the times required?
What rate of interest will the business have to pay on borrowed funds?
Are time limits or other restrictions imposed on borrowing?

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Resources
Are the other resources required for the project available?
Does the business have the necessary staff, technical expertise and equipment?

Risk
What are the financial risks involved in going ahead with the project (and in doing nothing)? How certain is the business about the appropriate risk discount rate to use?
Is it possible that the project might make an unacceptably large loss?
Can suppliers be relied on to fulfil their contracts according to the agreed timetable and budget? Investment conditions
What is the economic climate?
Are interest rates likely to rise or fall?

Cost vs benefit
Is the project worth doing at all?
Do the costs outweigh the benefits?

Indirect benefits
Will the project bring any additional benefits?
Will the equipment purchased and the skills developed be of value to the business in the future? The following example illustrates these points.

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CT1-10: Project appraisal

Example
For each of the projects outlined below, calculate:
(i)

the internal rate of return.

(ii)

the range of interest rates at which money can be borrowed in order for the projects to be viable.

(iii)

the accumulated profit at the end of 5 years, assuming that the projects are financed by a loan subject to interest at 6.25%.

Project C
Initial outlay £100,000
Proceeds (at the end of 5 years)

£140,000

Project D
Initial outlay £100,000
Proceeds (at the end of each of the next 3 years)

£38,850

Solution
(i)

The internal rates of return iC and i D are given by:
100,000(1 + iC )5 = 140,000
38,850a3| = 100,000

(ii)

⇒ iC = 7.0%
⇒ a 3| = 2.574

⇒ i D = 81%
.

If the borrowing rate is less than 7%, both projects will be profitable.
If the borrowing rate is between 7% and 8.1%, Project D will be profitable, but
Project C will not be.
If the borrowing rate exceeds 8.1%, neither project will be profitable.

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(iii)

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The accumulated value of the profits at the end of 5 years, using a rate of interest of 6.25%, are:

140,000 − 100,000 × 106255 = £4,592
.
38,850 × 106252 s3| − 100,000 × 106255 = £4,561
.
.

Question 10.12
Indicate other considerations that might be taken into account when deciding between
Project C and Project D.

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CT1-10: Project appraisal

Measurement of investment performance
It is often necessary to be able to measure the investment performance of a fund
(for example a pension fund, or the funds of an insurance company) over a period. This is important for those responsible for investment funds, eg the trustees of a pension fund, to monitor how the fund is performing, ie to find out the rate of return the fund is achieving, and to compare this with the performance of other funds.
In this section we will look at three measures of investment performance.
In most cases our calculations will be based on the market value of the fund, ie how much the fund’s assets would realise if they were sold on the day in question. It is helpful to realise that the value of the fund will go up or down as a result of changes in the following components:

Income generated by the fund
Changes in market value – capital gains/losses “New money”

This includes interest payments, dividends and rental payments “earned” by the fund’s assets.
The price investors are prepared to pay for assets in the fund will vary from day to day, in the same way that the price of apples may vary over time.
This includes all “extra” money paid into the fund that was not generated by the fund itself.
Withdrawals from the fund correspond to negative new money.

In investment performance calculations it is important to distinguish between money generated by the fund (the first two items in the table) and “new money”. Otherwise there is the danger of double counting or under counting. The cashflows in this section all refer to new money only.

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Money-weighted rate of return
One measure of the performance is the yield earned on the fund over the period.
The yield earned on the fund is also called the “money-weighted rate of return”
(MWRR).

The money-weighted rate of return is the interest rate satisfying the equation of value incorporating the initial and final fund values and the intermediate net cashflows.
Normally we discount each of the cashflows in the equation of value, however here we will accumulate the initial value of the fund and the cashflows and equate it to the final value of the fund.
Note that the equation of value used in calculating the MWRR only takes account of new money. Any cashflows generated by the fund itself must be ignored.
For example, consider a fund with value F0 at time 0, with net cashflows Ct k at times t1, t 2 , … , t n and fund value FT at time T ≥ t n , then the equation of value, equating values at time T, is:

F0 (1 + i )T + Ct1 (1 + i )T − t1 + Ct 2 (1 + i )T − t 2 +

+ Ct n (1 + i )T − t n = FT

where i is the effective annual rate of interest earned by the fund in the interval
[0, T].
In this equation of value the left-hand side is the value at time T of the fund at the start of the period plus or minus all the cashflows received or paid out in the interval. Example
The market value of a small pension fund’s assets was £2.7m on 1 January 2006 and
£3.1m on 31 December 2006. During 2006 the only cashflows were:


bank interest and dividends totalling £125,000 received on 30 June



a cash payment of £100,000 received on 1 August when a block of shares was sold •

a lump sum retirement benefit of £75,000 paid on 1 May



a contribution of £50,000 paid by the company on 31 December.

Show that the money-weighted rate of return is 16.0%.

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Solution
Only the last two payments represent new money. So the equation of value (working in
£000s) is:
2,700(1 + i ) − 75(1 + i ) 8/12 + 50 = 3,100
Evaluating the LHS at interest rates either side of 16.0%, we find: i = 15.95%

⇒ LHS = 3,097.9 < 3,100

i = 16.05%

⇒ LHS = 3,100.5 > 3,100

So the MWRR lies in the range (15.95%, 16.05%) ie it is 16.0% (to the nearest 0.1%).

Question 10.13
Explain why cashflows such as interest, dividends and capital gains, which are generated by the fund itself, are ignored in the equation of value, whereas cashflows in respect of new money must be included.
The equation of value for the MWRR will probably contain i several times and therefore there will probably not be an analytical solution to the equation. We must use a trial and error type approach or some numerical method to solve the equation. We can also try to find a good first guess. The following example illustrates this.

Example
The value of a fund’s assets was £10m on 1 December 2005 and £11m on 31 December
2006. The cashflows during this period were:


£10,000 received on 1 March



£50,000 paid out on 1 June



£75,000 received on 31 October

Calculate the money-weighted rate of return.

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Solution
The equation of value (working in £000s) is:
13

10

7

2

10, 000(1 + i ) 12 + 10(1 + i ) 12 - 50(1 + i ) 12 + 75(1 + i ) 12 = 11, 000
As a first guess we could use a first-order binomial expansion, replacing (1 + i )n with
(1 + ni ) :
7 ˆ
2 ˆ
Ê 13 ˆ
Ê 10 ˆ
Ê
Ê
10, 000 Á1 + i ˜ + 10 Á1 + i ˜ - 50 Á1 + i ˜ + 75 Á1 + i ˜
Ë 12 ¯
Ë 12 ¯
Ë 12 ¯
Ë 12 ¯

fi 10, 000 + 10 - 50 + 75 + fi 129,900 i 12

11, 000

130, 000 + 100 - 350 + 150 i 11, 000
12

965 fi i

8.9%

Evaluating the LHS of the equation of value at interest rates close to 9%:

i = 9% fi LHS = 11, 012.8 i = 8% fi LHS = 10,903.8
Interpolating between these two values, we get:
11, 000 - 11, 012.8 i -9
=
fi i = 8.9%
8 - 9 10,903.8 - 11, 012.8
So the MWRR is approximately 8.9%.
Notice that you don’t have to use the first guess method, but it may help you to get to the answer more quickly.

4.2

Time-weighted rate of return
As a measure of investment performance the money-weighted rate of return is not entirely satisfactory, as it is sensitive to the amounts and timing of the net cashflows. If, say, we are assessing the skill of the fund manager, this is not ideal, as the fund manager does not control the timing or amount of the cashflows – he or she is merely responsible for investing the positive net cashflows and realising cash to meet the negative net cashflows.

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Another measure which tries to eliminate this effect is the time-weighted rate of return.
The rationale here is to calculate the “growth factors” reflecting the change in the value of the fund between the times of consecutive cashflows (ie during periods when
“nothing happened”), then to combine these growth factors to come up with an overall rate of return for the whole period.
The time-weighted rate of return is found from the product of the growth factors between consecutive cashflows.
Define F0 , FT , and Ct k as above, and let C0 be the cashflow (if any) at time

t = 0 ; in addition let Ft k − be the amount of the fund just before the cashflow due at time t k , so that the amount of the fund just after the receipt of the net cashflow due at time t k is Ft k − + Ct k . Then the “Time-Weighted Rate of Return”
(TWRR) is i per annum, where:

(1 + i )T =

Ft1 −

Ft 2 −

Ft 3 −

F0 + C0 Ft1 − + Ct1 Ft 2 − + Ct 2

FT
Ft n − + Ct n

Each factor on the right hand side gives the proportionate increase in the fund between cashflows.
The product of these factors gives the notional accumulation factor for a single investment of 1 at time t = 0 , invested until time
T.

Again, note that the cashflows in the formula for calculating the TWRR only include those relating to new money. Any cashflows generated by the fund itself must be taken into account in the figures for the fund value.
Using the TWRR eliminates the effect of the cashflow amounts and timing, and therefore gives a fairer basis for assessing the investment performance for the fund. © IFE: 2009 Examinations

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Example
Calculate the TWRR for the fund in the example on page 27 given the extra information that the fund value (including all accrued interest and capital gains) was £3.0m on 30
April 2006.

Solution
The progress of the fund (again working in £000s) was as follows:
1 January to 30 April

Fund value increased from £2,700 to £3,000

1 May

Cashflow of –£75

1 May to 30 December

Fund value increased from £2,925 to £3,050

31 December

Cashflow of +£50, taking fund value to £3,100

So, during the period from 1 January to 30 April, there were no cashflows and the fund value grew by a factor of:
3,000
.
= 1111
2,700
During the period from 1 May to 30 December, the fund value grew by a factor of:
3,100 − 50
= 1043
.
3,000 − 75
So the growth factor for the whole year is 1111 × 1043 = 1159 and the TWRR for 2006
.
.
.
is 15.9%.

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Question 10.14
The table below shows the progress of a lottery winner’s investment portfolio for the
2006 calendar year:
Fund value
1 January

£75,000

31 March

Cashflow

£90,000
Won Lottery. Winnings invested:

1 April
30 September
1 October
31 December

£2,700,000

£2,600,000
Funds withdrawn to pay off gambling debt:

£2,500,000

£125,000

Calculate (to the nearest %) the MWRR and the TWRR for 2006 for this portfolio and comment on your answers.
The disadvantages of both the time-weighted and money-weighted rates of return are that the calculation requires information about all the cashflows of the fund during the period of interest. In addition, the TWRR requires the fund values at all the cashflow dates. A disadvantage of the MWRR is that the equation may not have a unique solution – or indeed any solution. If the fund performance is reasonably stable in the period of assessment then the TWRR and the MWRR will give similar results.

4.3

Linked internal rate of return
Linked rates of return provide a way of combining rates of return for successive subperiods to obtain an approximate rate of return over a longer period. The linked
(internal) rate of return is found from the product of the yields for the subperiods.
If the rate of return on a fund is measured over a series of intervals
(0, t1),(t1, t 2 ),(t 2 , t 3 ), … ,(t n − 1, t n ) , such that the annual effective rate of interest earned by a fund in the interval (t r − 1, t r ) is i r (where i1 is the annual rate earned in (0, t1) ) then the “Linked Internal Rate of Return” is i per annum, where:
(1 + i )t n = (1 + i1 )t1 (1 + i 2 )t 2 − t1 (1 + i 3 )t 3 − t 2 … (1 + i n )t n − t n − 1

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The linked internal rate of return will be equal to the TWRR if the subintervals
(t r − 1, t r ) are the same in each calculation. In practice, the yields i r may be calculated by approximate methods, and then, if the subintervals used are sufficiently short, the linked internal rate of return will be close to the TWRR.

Notice that here, the interest rates are annual rates. So for quarterly subintervals over one year, the linked internal rate of return would be:
1

1

1

1

(1 + i ) = (1 + i1 ) 4 (1 + i2 ) 4 (1 + i3 ) 4 (1 + i4 ) 4
However, if we calculated jk to be the quarterly effective return in quarter k for one year, then the linked internal rate of return would be given by:
(1 + i ) = (1 + j1 )(1 + j2 )(1 + j3 )(1 + j4 )
The calculation is similar to the calculation used for the TWRR, except that the subperiods no longer correspond to the intervals between cashflows, but correspond to
“convenient” times in the calendar year.
A common situation is for an investment manager to have estimates of the quarterly returns of a fund available from computer printouts, which can then be combined to calculate linked rates of return for longer periods. The quarterly rates of return in this situation may be calculated “exactly” using MWRRs, or they may be calculated on an approximate basis.

Example
A life office operates a Far East fund which achieved quarterly money-weighted rates of return of 4.1%, 2.8%, 1.7% and 2.1% during the four quarters of 2004 and halfyearly rates of 2.5% and 3.8% during the two halves of 2005. Calculate the linked rate of return for period under consideration.

Solution
The growth factor for the two-year period is:
1.041 ¥ 1.028 ¥ 1.017 ¥ 1.021 ¥ 1.025 ¥ 1.038 = 1.182257

So the annual linked rate of return for the period is 1.182257 - 1 = 8.73% .

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Question 10.15
Another life office, which also operates a Far East fund, achieved rates of return which underperformed the first life office by ½% per quarter in each of the first two quarters of 2004, outperformed it by 1% per quarter in each of the last two quarters of 2004 and equalled the performance in 2005. Calculate the linked rate of return for the rival life office over the same period.

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Exam-style question
The following question is a good example of the type of long question that you may encounter in your exam. This questions focuses on the first half of this chapter. Try this question yourself before turning the page, where you will find the solution. You are just as likely to be examined on money-weighted, time-weighted and linked internal rates of return as well. (Note that not all of the questions will be as long as this one.)
You can find many more example questions in the Q&A Bank, the assignments and past exam papers. Recent exam papers and Examiners’ Reports can be found on the
Profession’s website or in ASET (ActEd’s Solutions with Exam Technique).

Question
An investor is considering making an investment in one or both of two projects. The cashflows associated with the projects are as follows. The unit of time is years.
Project A:
Initial payments of £2 million at time zero and £4 million at time 2 are made. In return a sum of £900,000 per annum is paid continuously from time 5 to time
25.
Project B:
Regular payments of £100,000 are made at the start of each year for 10 years. In return, amounts of X , 2 X , 3X and so on are made annually for 10 years, the first payment being made at time 11.
(i)

Find the net present value of Project A at an effective annual interest rate of
10%.
[2]

(ii)

Show that the internal rate of return for Project A is 9.38% pa.

(iii)

Find the value of X if the internal rate of return for Project B is the same as that for Project A.
[3]

(iv)

Find the value of X if both projects are to have the same net present value at
10% pa.
[3]

(v)

The investor proposes to borrow all the money needed for the project. Funds are available at an interest rate of 7% per annum effective. Repayments can be made at any time, and positive cash balances can be invested to yield 3% per annum. If X = £45,000 , find the accumulated value of each project at the end of the 25 year period.
[12]
[Total 22]

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[2]

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Solution
(i)

The present value of Project A is given by (working in millions of pounds):
NPV A = −2 − 4v 2 + 0.9v 5a20|
Evaluating these functions at 10%, we get a net present value of
-£0.314 million.

(ii)

We need to evaluate the compound interest functions in the expression for the net present value above at i = 0.0938 . If we do so we get a net present value of
£ 0.0011 million, which is close to zero. So the IRR is 9.38%.

(iii)

The equation for the present value of the cashflows arising from Project B is:

NPVB = −01 a10| + Xv10 v + 2v 2 + 3v 3 +
.

+ 10v10 = −01 a10| + Xv10 ( Ia )10|
.

This must equal zero at a rate of interest of 9.38%. So evaluating the compound interest functions at this interest rate, we get:
−01 × 6.90373 + X × 0.407963 × 30107691 = 0
.
.
Solving this equation gives X = 0.056206 . So the value of X is £56,200.
(iv)

If both projects are to have the same net present value at 10%, we need to solve the equation:

-0.1 a10| + X v10 ( Ia )10| = -0.31406

@10%

fi -0.1 ¥ 6.7590 + X ¥ 0.38554 ¥ 29.0359 = -0.31406
Solving this equation for X , we find that X = 0.032323 . So the value of X is
£32,320.

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(v)

Page 37

We first need to find the discounted payback period for each project, so that we know when the loan is completely repaid. For Project A, we want the solution, at 7%, of the equation:
−2 − 4v 2 + 0.9v 5an| = 0 where n is the number of years from time 5 to the time when the loan is repaid.
Rearranging this equation we get: an| = 8.5614172
So:
1− vn
= 8.5614172 log 107
.



v n = 0.420746

Solving this equation either by trial and error or by using logs, we find that: n= log e 0.420746 log e (107) −1
.

= 12.795

So the project breaks even after 12.795 years of continuous payments, ie 17.795 years after the start of the project.
To find the accumulated value at the end of 25 years, we need to accumulate the cashflows occurring after time 17.795 at 3%. So:
0.9 s7.2045072| = 7.226201 and the accumulated profit at the end of 25 years is £7.226 million.
For Project B, the discounted payback period must be a whole number of years, since cashflows only occur at annual intervals. We need to find the smallest integer value of n for which:
−01 a10| + 0.045 v10 ( Ia ) n| > 0
.
at i = 7% . Rearranging this, we find that we want ( Ia ) n| > 32.852 at 7%.

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CT1-10: Project appraisal

Trying out some values for n , we find that:
( Ia ) 9| = 29.656

and ( Ia )10| = 34.739

So the project only becomes profitable at the moment when the last payment is received at time 20.
So the net present value of all the cashflows at 7% is:
−01 a10| + 0.045 v10 ( Ia )10| = 0.0431592
.
So the accumulated profit at the end of 20 years will be:
0.0431592 ¥ 1.07 20
We are using 7% because the loan is not paid off until time 20 and so there will not be a positive cash balance to invest. After time 20, the cash is invested to yield 3% pa and so the accumulated profit at the end of 25 years will be:
0.0431592 × 107 20 × 1035 = 01936
.
.
.

ie £193,600.

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End of Part 2
You have now completed Part 2 of the CT1 Notes.

Review
Before looking at the Question and Answer Bank we recommend that you briefly review the key areas of Part 2, or maybe re-read the summaries at the end of Chapters 8 to 10.

Question and Answer Bank
You should now be able to answer the questions in Part 2 of the Question and Answer
Bank. We recommend that you work through several of these questions now and save the remainder for use as part of your revision.

Assignments
On completing this part, you should be able to attempt the questions in Assignment X2.

Reminder
If you have not booked a tutorial, then maybe now is the time to do so.

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This page has been left blank so that you can keep the chapter summaries together for revision purposes.

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Chapter 10 Summary
The profitability of an investment project can be assessed by calculating the accumulated profit:

Acc. profit = Acc.V income − Acc.V outgo
For projects that are self-financing, the net present values or internal rates of return can be compared.
The net present value (NPV) of an investment project is the present value of the net cashflows, calculated at the risk discount rate.

Net present value = PV income − PV outgo

(@ risk discount rate)

The internal rate of return for an investment project is the effective rate of interest that equates the present value of income and outgo, ie it makes the net present value of the cashflows equal to zero.
The discounted payback period for an investment project is the earliest time after the start of the project when the accumulated value of the past cashflows (positive and negative), calculated using the borrowing rate, becomes greater or equal to zero.

Acc. profit at discounted payback period ≥ 0

(@ borrowing rate)

There are a variety of other factors that must be taken into account when comparing alternative investment projects. These include: the cashflow requirements of the business, the borrowing requirements, other resources required, risks involved, investment conditions, cost/benefit considerations and indirect benefits.
Investment performance can be measured using money-weighted, time-weighted or linked internal rates of return.
The money-weighted rate of return is the interest rate satisfying the equation of value incorporating the initial and final fund values and the intermediate net cashflows. The equation of value only takes account of new money. Any cashflows generated by the fund itself must be ignored. The formula for MWRR is:

F0 (1 + i ) T + Ct1 (1 + i ) T −t1 + Ct2 (1 + i ) T −t2 +

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CT1-10: Project appraisal

The time-weighted rate of return is found from the product of the growth factors between consecutive cashflows. The formula for TWRR is:
(1 + i ) T =

Ft1 −

Ft2 −

Ft3 −

F0 + C0 Ft1 − + Ct1 Ft2 − + Ct2



FT
Ftn − + Ctn

The linked (internal) rate of return is found from the product of the yields for the subperiods. The formula for the linked internal rate of return is:
(1 + i ) tn = (1 + i1 ) t1 (1 + i2 ) t2 − t1 (1 + i3 ) t3 − t2 … (1 + in ) tn −tn −1

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Chapter 10 Solutions
Solution 10.1
The organisation has an initial positive cashflow, a single known negative cashflow on a specified future date, and a series of smaller known negative cashflows on a regular set of specified future dates.

Solution 10.2
The accumulated profit is:

-20, 000 ¥ 1.0320 + 5, 000s11 ¥ 1.035
= -36,122 + 74, 239 = $38,117

Solution 10.3
For Project R, the net present value (in £000s) is:
NPVR = -150 - 250v - 250v 2 + 1, 000v3

@10%

= -150 - 250(0.90909) - 250(0.82645) + 1, 000(0.75131) = 167.4

For Project S, the net present value (in £000s) is:

NPVS = -325 - 75a1| - 90va1| - 120v 2 a1| + 1, 000v3
= -325 - (75 + 90v + 120v 2 )

d

d

+ 1, 000v3

@10%
@10%

= -325 - [75 + 90(0.90909) + 120(0.82645)] ¥

0.090909
+ 1, 000(0.75131)
0.095310

= 182.1
So the net present values are: £167,400 for Project R and £182,100 for Project S.
So, using a risk discount rate of 10%, Project S appears more favourable – a reversal of the conclusion we arrived at using a risk discount rate of 20%.

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Solution 10.4
The accumulated profit for Venture 1 at time 5 is:

Acc.Pr1 = -100(1 + i )5 - 40(1 + i ) 4 + 50(1 + i )3 + 120(1 + i ) 2 @15%
= -36.352
The accumulated profit for Venture 2 at time 5 is:
Acc Pr2 = -45(1 + i ) 4 + 25(1 + i ) 2 + 25(1 + i) + 25 @15%
= 8.107
So the accumulated profits are –£36,352 (ie a loss) for Venture 1 and £8,107 for
Venture 2.
The net present values are:
NPV1 = -100 - 40v + 50v 2 + 120v3 @15%
= -100 - 40(0.86957) + 50(0.75614) + 120(0.65752) = -18.073
NPV2 = -45v + 25v3 + 25v 4 + 25v5 @15%
= -45(0.86957) + 25(0.65752) + 25(0.57175) + 25(0.49718) = 4.031
So the NPVs are –£18,073 for Venture 1 and £4,031 for Venture 2.
Note that you could have worked out the NPVs first, then multiplied by 1.155 to find the accumulated profit figures.

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Solution 10.5
We need to find the interest rate i that satisfies the equation of value:

−325 − (75 + 90v + 120v 2 )

d

δ

+ 1,000v 3 = 0

We already know that:
At 20%:

−325 − (75 + 90v + 120v 2 )

d

δ

+ 1,000v 3 = 40.41

Since the outgo precedes the income, the value of i must be greater than this. So we need to try a higher rate:
At 25%:

−325 − (75 + 90v + 120v 2 )

d

δ

+ 1,000v 3 = −1359
.

We can approximate i by interpolating (linearly) using these two values: i ≈ 25% +

0 − ( −1359)
.
× (20% − 25%) = 23.7%
40.41 − ( −1359)
.

So, using this criterion, Project R appears more favourable.

Solution 10.6
NPVY (5.5%) = -11, 000 + 605a18| + 11, 000v18
= -11, 000 + 605

1 - 1.055-18
+ 11, 000 ¥ 1.055-18
0.055

=0

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Solution 10.7
Project C
The accumulated profit for Project C would be unchanged, since there is no opportunity for repaying the loan.

Project E
The calculation for the first year would be unchanged, since there are no excess funds.
At the end of year 2, £19,062.5 of the loan could be repaid.
The accumulated profit at the end of year 5 would then become:
[87,000 − (95,000 − 19,062.5) × 10625] × 104 2 = £6,832
.
.
Comment: the accumulated profit is greater if loans can be repaid early. This is because the expense of interest payments will be reduced, and the excess funds can be invested for a longer period.

Solution 10.8
The rate of interest charged to the mortgage holder will be set a level that generates profits for the bank. If the mortgage is repaid prematurely the bank will lose out on this “extra money” and its profits will be reduced. In the extreme case where the mortgage was repaid immediately, the lender would make a loss because it would not be able to cover the expenses involved in setting up the loan.

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Solution 10.9
Applying the definition strictly would indicate that the discounted payback period occurs after 1 month, ie at the time when the fees from the advertising contracts are received. However, this interpretation would be rather silly here because, at this point, the company has not yet made the main item of investment, buying the vehicles. So a more sensible interpretation would be to treat the positive cashflow after 1 month as just a blip and to look for the first time when the accumulated value becomes positive again after the vehicles have been purchased.
Remembering that the starred cashflows are continuous, we find that the accumulated value to time t (where 3 months < t < 5 years ) is:
Acc.V (t ) = -100 ¥ 1.10t + 200 ¥ 1.10t -1/12 - 2, 000 ¥ 1.10t -3/12 t +

Ú

(1, 000 - 400) ¥ 1.10t - s ds

3/12

Using the substitution u = t − s in the integral, we can write this equation in the form:
3/12

Acc.V (t ) = ( -100 ¥ 1.10

+ 200 ¥ 1.10

2 /12

t -3/12

- 2, 000) ¥ 1.10

t -3/12

+ 600

Ú

1.10u du

0

= -1,899.21 ¥ 1.10t -3/12 + 600 st -3/12|
Using this formula, we find that Acc.V (3 / 12) = −1,899.21 and Acc.V (5) = +617.91 .
So the accumulated value will be zero for some value of t in the range
3 months < t < 5 years .
In fact, this will happen when:
−1,899.21 × 110 t −3/12 + 600st −3/12| = 0
.
.
Dividing through by 110 t − 3/12 and rearranging:

a t −3/12| =

1,899.21
= 31654
.
600

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ie

CT1-10: Project appraisal

1 − 110 − ( t − 3/12)
.
.
= 31654 log 110
.

⇒ t − 3 / 12 = 3.77

⇒ t = 4.02

So the discounted payback period is just over 4 years and the graph of accumulated profit looks like this:

Solution 10.10
The accumulated profit at the end of year t will be:
Acc.Profit(t ) = −50,000(1 + i ) t + 7,500st |
This will be positive when:
−50,000(1 + i ) t + 7,500st | ≥ 0

Dividing through by 7,500(1 + i ) t : at | ≥

50,000
= 6.6666
7,500

Looking at the Tables (at 8%), we see that a 9| = 6.2469 and a10| = 6.7101 .
So the discounted payback period is 10 years.

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Solution 10.11
If the interest rate charged on borrowing was 18%, the profit from Investment 2 would be 2,000 − 1,800 = £200 , which is almost the same as the profit of £199 on
Investment 1.
So the critical rate of interest is approximately 18%.

Solution 10.12
Other considerations would include:
1.

Although i D > iC , the internal rates of return are quite close. So the decision based on this criterion is not clear cut.

2.

The higher investment yield available under Project D applies for only 3 years, whereas Project C will produce profits over a 5 year period. An investor in
Project C is “locked into” a yield of 7.0% for the full 5 years. So, if interest rates turn out to be lower in years 4 and 5, Project C may prove to be a better investment. 3.

Project C may not be desirable if it will leave the investor short of cash during the next 5 years.

4.

If the investor is not able to borrow more than £100,000 and the investor is required to make regular repayments on money borrowed, then Project C will not be viable, since it does not produce any income with which to pay the interest, and the investor would be unable to borrow the money required.

Solution 10.13
It is the interest, dividends and capital gains that lead to the growth in the value of the fund, which is what the MWRR i is measuring. These payments are already
“absorbed” in the value of i . Including them as cashflows in the equation would result in double counting.
Cashflows in respect of new money, on the other hand, are not reflected in the value of i , and so these must be included as extra terms in the equation of value.
Note that capital gains aren’t really cashflows but increases to the value of the fund.

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Solution 10.14
The MWRR is the value of i for which:
75(1 + i ) + 2,700(1 + i ) 9 /12 − 2,500(1 + i ) 3/12 = 125
Establishing a first guess:
9 ˆ
3 ˆ
Ê
Ê
75(1 + i ) + 2, 700 Á1 + i ˜ - 2,500 Á1 + i ˜
Ë 12 ¯
Ë 12 ¯ fi 75 + 2, 700 - 2,500 + fi 17, 700 i 12

125

900 + 24,300 - 7,500 i 125
12

-150 fi i

-10.2%

Trying some values close to this first guess:

i = -10% fi LHS = 127.35 i = -11% fi LHS = 112.57
Interpolating between these values:
125 - 127.35 i - ( -10)
=
fi i = -10.2%
( -11) - ( -10) 112.57 - 127.35
So the MWRR is approximately -10% .
The TWRR is given by:
1+ i =

90
2,600
125
.
.
.
×
×
= 1200 × 0.932 × 1250 = 1398
75 90 + 2,700 2,600 − 2,500

fi i = 39.8%
So the TWRR is +40% (to the nearest %).
Looking at the component factors in the TWRR, we see that the strong positive returns in the first and third subperiods outstripped the smaller percentage loss in the middle subperiod, leading to a positive TWRR for the year as a whole.

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The MWRR, however, puts almost all the weight on the middle subperiod when the fund size was largest. This was the period when the lottery winner was rich and the returns on the fund were strongly negative. It is the performance for this period that leads to the negative MWRR for the year as a whole.

Solution 10.15
The growth factor for the period for the rival life office is:
1.036 ¥ 1.023 ¥ 1.027 ¥ 1.031 ¥ 1.025 ¥ 1.038 = 1.1939 .
So the linked rate of return for the period is 1.1939 - 1 = 9.27% .

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Chapter 11
Investments
Syllabus objective
(x)

Describe the investment and risk characteristics of the following types of asset available for investment purposes:



fixed interest borrowing by other bodies



shares and other equity-type finance



0

fixed interest government borrowings

derivatives

Introduction
When a private individual has excess cash (eg immediately after each month’s pay cheque), the individual will lend the money for a short period to a bank (ie the cash is left in a bank account). When an individual is short of cash (eg just before each month’s pay cheque arrives), the individual may borrow from the bank for a short time
(ie through an overdraft).
In a similar way, big corporations will have short-term surpluses or deficits of cash.
This chapter considers the types of asset that might be used by corporations as a home for this “spare cash”. However, this chapter is only an introduction to investment. You will study the topic in more detail in the Finance and Financial Reporting Subject.

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Fixed interest government borrowings
Governments will often wish to raise or borrow money from investors to fund their spending plans. They may wish to repay the money in the near future by issuing shortterm loans or at a later date by issuing medium-term or long-term loans.

1.1

Fixed interest government bonds
Cashflows
The cashflows associated with fixed interest bonds have already been described to you earlier in the course. This section quickly recaps this information and gives some other investment features of fixed interest bonds.
A government or government body may raise money by floating a loan on a stock exchange. The terms of the issue are set out by the borrower and investors may be invited to subscribe to the loan at a given price (called the
“issue price”), or the issue may be by tender, in which case investors are invited to nominate the price that they are prepared to pay and the loan is then issued to the highest bidders, subject to certain rules of allocation.

An issue by tender is just like an auction but be careful because the word tender is often used to describe a special form of auction. In this method a minimum price is set and bidders may tender at or above this price. The government will then determine the price at which it will sell the stock and all successful bidders pay the same price.

Question 11.1
What are the main advantages and disadvantages, to the government and to investors, of issuing bonds at a set price as opposed to by tender?
The term “nominal” refers to an amount of stock. It is the amount specified on the stock certificate. Dealings in bonds and calculations for bonds are carried out in amounts of nominal.
The annual interest payable to each holder, which is often but not invariably payable half-yearly, is found by multiplying the nominal amount of his holding N by the rate of interest per annum D, which is generally called the “coupon rate”.

The term “coupon” comes from the days when fixed interest bonds were bearer documents and the holder had to cut out a coupon in exchange for the interest.

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The money payable at redemption is calculated by multiplying the nominal amount held N by the redemption price R per unit nominal (which is often quoted
“per cent” in practice). If R = 1 the stock is said to be redeemable at par; if R > 1 the stock is said to be “redeemable above par” or at “a premium”; and if R < 1 the stock is said to be “redeemable below par” or at “a discount”. The redemption date is the set date on which the redemption money is due to be paid. The redemption payment will usually correspond with the final coupon payment.
Example
An investor purchases £2,000 nominal of a bond redeemable at 110% in exactly six years’ time. Coupons at a rate of 8% pa are payable half yearly in arrear.
The investor will receive £80 at the end of each half-year for six years together with
£2,200 in six years’ time.

Variations
Some bonds have variable redemption dates, in which case the redemption date may be chosen by the borrower (or perhaps the lender) as any interest date within a certain period, or any interest date on or after a given date. In the latter case the stock is said to have no final redemption date, or to be undated.

Undated stocks are also known as “irredeemable” stocks. Investors tend to prefer stocks with fixed redemption dates rather than spread redemption dates as they give more certainty.
If the borrower can choose the redemption date then he will choose the date that gives it the greatest return, or the lowest cost of borrowing. We will discuss this idea in more detail later in the course.
Some banks allow the interest and redemption proceeds to be bought and sold separately, effectively creating bonds with no coupon and bonds redeemable at zero, ie with no redemption payment. These are known as strips.

So, for example, a 20-year bond with a half-yearly coupon becomes 40 separate coupon payments and one capital payment. Each payment can be owned, bought and sold by different investors. Additionally, the appropriate stripped coupon and redemption payments can usually be combined to reconstruct a particular bond.

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Tax
The coupon rate, redemption price and term to redemption of a fixed interest security serve to define the cash payments promised to a tax-free investor in return for his purchase price. If the investor is subject to taxation, appropriate deductions from the cashflow must be made. For example, if an investor is liable to income tax at rate t1 on the interest payments, the annual income after tax will be (1 − t1 )DN .

The rates of taxation will have a large impact on the price that investors will be prepared to pay for bonds.

Question 11.2
A government is issuing bonds by an auction. Two investors who are bidding for the issue are both aiming for the same net rate of return on their investments. One of the investors does not pay tax whilst the other pays tax on income of 25%.
Which investor will make the highest bid?

Security, marketability and return
In most developed economies, bonds issued by the government form the largest, most important and most liquid part of the bond market. This means that investors can deal in large quantities with little (or no) impact on the price.
Bonds issued by the governments of developed countries in their domestic currency are the most secure long-term investment available: there is virtually no

default risk.
However, this security together with the low volatility of return relative to other long-term investments should lead to a low expected return, though this will be compensated for to an extent by very low dealing costs.
Relative to inflation, however, the income stream may be volatile.

If fixed interest bonds are held until redemption, the monetary amounts of income and capital are known and fixed. To this extent, the returns are known at the outset.

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The return achieved over the period up to the maturity date of the bond will, however, be uncertain for various reasons:
(a)

The investor may need to reinvest the coupon payments. The terms that will be available for reinvestment are not known at outset.

(b)

For an investor who plans to sell before redemption, the ultimate sale price is not known at outset.

(c)

The real return (ie in excess of inflation) is uncertain. If inflation turns out to be higher than expected at outset, the real returns from fixed interest bonds will be lower than originally anticipated.

(d)

Tax rates may change, affecting the income and capital proceeds received by the investor. Some governments therefore issue index-linked bonds that provide interest and redemption payments that are linked to an inflation index.
However, indexation will need to be based on the movement of the inflation index with a time lag to allow for publication of the index figure and the need to calculate monetary amounts of coupons in advance. There is effectively no inflation protection during the lag period.

Later in the course we will look at the methods used to calculate nominal and real yields obtainable from fixed interest and index-linked bonds.

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Example
Consider a 3½% coupon stock issued in February 2000 and redeemed in February 2005.
The coupon payments are made each year and are linked to an inflation index with a one-year time lag. The index values each February from 1999 to 2005 are given in the table below.
Year 1999 2000 2001 2002 2003 2004 2005
Index 540 562 584 607 632 657 788
The base month for indexation is February 1999 because of the time lag.
The coupon payments per £100 nominal are:
February 2001:
February 2002:

2000 index
562
= 3.5 ¥
= 3.64
1999index
540
2001 index
584
3.5 ¥
= 3.5 ¥
= 3.79
1999index
540
3.5 ¥

and so on until:
February 2005:

3.5 ¥

2004 index
657
= 3.5 ¥
= 4.26
1999index
540

The real value of the payments in February 2000 terms are:
February 2001:
February 2002:

2000 index
562
= 3.64 ¥
= 3.50
2001index
584
2000 index
562
3.79 ¥
= 3.79 ¥
= 3.51
2002index
607
3.64 ¥

and so on until:
February 2005:

4.26 ¥

2000 index
562
= 4.26 ¥
= 3.04
2005index
788

The inflation rates implied by the table of indices are:
Year
Inflation

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99–00 00–01 01–02 02–03 03–04 04–05
4.07% 3.91% 3.94% 4.12% 3.96% 19.94%

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Inflation has been roughly 4% from 1999–2004 but it then leapt to nearly 20% for the last year. Because of the time lag there is no protection against inflation during this last year and so the real value of the last coupon payment is much lower than the real value of the other coupon payments.

Question 11.3
Using the information in the previous example, calculate the price that an investor should have paid in February 2000 in order to achieve a total return of 6%. The redemption payment is index linked in the same way as the coupon payments.

Question 11.4
Fixed interest government bonds provide a known and fixed stream of income. Why then is the rate of return achieved over 10 years by an investor buying a 10-year bond not known at outset?

1.2

Government bills
Government bills are short-dated securities issued by governments to fund their short-term spending requirements. They are issued at a discount and redeemed at par with no coupon.

They are issued at a price P and on maturity are redeemed at the face value or “par value” of the loan. There is no explicit coupon or interest. The reward for investors is the difference between P and the par value. Three months is a typical term for a government bill.
They are mostly denominated in the domestic currency, although issues can be made in other currencies.
The yield on government bills is typically quoted as a simple rate of discount for the term of the bill. For example, if a 3-month bill may be quoted as being offered at a discount of 2%. This would mean that the initial investment required to buy the bill would be 2% less than the payment 3 months later.

The yield may also be quoted as a simple annual rate of discount. This makes it easier to compare the yields on bills of different terms.

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For example, if a 91-day $100 bill is issued in the US at a price of $97, the “discount rate” could be quoted as 12% ie there is a $3 discount on an investment of value $100 made over a quarter of a year. Hence, if the discount rate is quoted as d, then the price of a $100 bill with 91 days until redemption is:

100 × (1 −

91
d)
365

The effective annual interest rate is quite simple to calculate. For example, a bill with n days to maturity bought for P gives an effective annual interest rate i given by:
P × (1 + i ) n / 365 = 100
So, the effective rate of interest on a 91-day bill quoted at a discount rate of 12% is
12.96%.
Government bills are absolutely secure and often highly marketable, despite not being quoted. They are often used as a benchmark risk free short-term investment. “Not being quoted” means that the price of the bill is not quoted on any stock exchange.
An investment is marketable if it can be sold quickly at low cost and without affecting the market price.

Question 11.5
A company wishes to invest £10,000 in 182-day bills from the British government. The bills are currently issued at an annual discount of 10%. Calculate the par value of the bills that could be purchased.

Question 11.6
Summarise the main features of:
(i)

government bills

(ii)

fixed interest government bonds

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2

Fixed interest borrowing by other bodies

2.1

Characteristics of corporate debt
In the same way that governments raise money by issuing bonds, companies borrow from investors by issuing corporate bonds.
Corporate bonds are, in many ways, similar to conventional government bonds in their characteristics. Here the debt is issued by a company rather than a government. The cashflows are almost identical. The investor will pay or lend a lump sum of money to the company. In return the company will pay regular interest payments and a final payment representing a return of capital at the end of the term of the contract.
The major differences between corporate bonds and government bonds are:
Corporate bonds are usually less secure than government bonds. The level of security depends on the type of bond, the company which has issued it, and the term. Corporate bonds are usually less marketable than government bonds, mainly because the sizes of issues are much smaller.

Question 11.7
The two main differences between corporate bonds and government bonds make the latter sound like a better investment. It is more secure and more marketable. Why then might investors choose to invest in corporate bonds?
The lower security and marketability offered by corporate bonds means that investors require a yield greater than on the corresponding government bonds, ie a yield margin, in order to induce them to hold them. The size of this yield margin depends on both the security and the marketability of the debt. Relatively low security and marketability will mean a large margin whereas a large secure issue will trade at a small yield margin to the closest equivalent government bond.

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The major difference between the different types of corporate bonds relates to the degree of security offered.




2.2

Loans may be secured on some or all of the assets of the issuing company. This means that if the company fails to make one of the coupon payments or the redemption payment, the stockholders may take possession of the asset.
With unsecured loan stock, there is no specific security for the loan. If the company defaults, the loan stockholders must sue the company. They therefore rank equally with other creditors and after the holders of the secured loan stock.

Debentures
Debentures are part of the loan capital of companies. The term loan capital usually refers to long-term borrowings rather than short-term. The issuing company provides some form of security to holders of the debenture.
Debenture stocks are considered more risky than government bonds and are usually less marketable. Accordingly the yield required by investors will be higher than that for a comparable government bond.

The yield on debentures increases for less sound companies as the perceived riskiness of the debenture stock increases.
The security guaranteeing the debenture can either be a fixed charge or a floating charge. A fixed charge means that a specific asset, or group of assets, (often property) will act as security for the debt. If the company winds up, the lenders will have the right to sell the assets to repay the loan. The company is not allowed to sell the specific asset and hence the security is “fixed”. A floating charge means that lenders have a prior right in the event of a wind up over all the assets of the company.

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2.3

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Unsecured loan stocks
Unsecured loan stocks are issued by various companies. They are unsecured – holders rank alongside other unsecured creditors (eg customers awaiting delivery

of goods, suppliers awaiting payment).
If a company fails to make interest or capital repayments when they are due, the loan stockholders can apply to the courts to have the company wound up.
The main difference between unsecured loan stocks and debentures is the different level of security (ie the priority should something go wrong). An unsecured loan stock is less secure than a debenture.
Yields will be higher than on comparable debentures issued by the same company, to reflect the higher default risk.

2.4

Eurobonds
Eurobonds are a form of unsecured medium or long-term borrowing made by issuing bonds which pay regular interest payments and a final capital repayment at par.
Eurobonds are issued by large companies, governments and supra-national organisations. They are usually unsecured. Yields depend upon the issuer (and hence risk) and issue size (and hence marketability), but will typically be slightly lower than for the conventional unsecured loan stocks of the same issuer.

The Eurobond market is an international market, which is not controlled by any particular country. To illustrate just how international the Eurobond market is, a new
Eurobond issue might involve, say:





a Swedish company, borrowing from ... a Belgian investor, in Japanese currency (ie Yen) and arranged through a bank in London!

Eurobonds are issued in many different currencies so we have Euroyen, Euromark,
Eurosterling and Euroeuro issues. The prefix “Euro” no longer refers to Europe. It now tends to be used for any financial arrangements in a currency external to the country where the arrangement is taking place.

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The features of Eurobonds vary a lot more than traditional bond issues. In the absence of any full-blown government control, issuers have been free to add novel features to their issues. They do this to make them appeal to different investors. One example of such a novel feature is to give the holder of the bond the right to select the currency in which interest and/or capital is paid.

Question 11.8
An investor is trying to decide whether to invest in one company’s debentures, unsecured loans or Eurobonds. Which is likely to give the highest yield and which will give the lowest yield?

2.5

Certificates of deposit
Unlike debentures, unsecured loans and Eurobonds, certificates of deposits are shortterm investments.
A certificate of deposit is a certificate stating that some money has been deposited. They are issued by banks and building societies. Terms to maturity are usually in the range 28 days to 6 months. Interest is payable on maturity.
The degree of security and marketability will depend on the issuing bank. There is an active secondary market in certificates of deposit.

Certificates of deposit are issued at 100, with a fixed coupon rate. Both the coupon rate, used to determine the interest payment, and the dealing rate i are quoted as annual simple rates of interest. For a sterling certificate of deposit with t days between settlement and maturity, the price P is determined by the formula:

t ˆ
Ê
P Á1 + i
˜ =C+D
Ë
365 ¯ or: P=

C+D t ˆ
Ê
Á1 + i
˜
Ë
365 ¯

where C is the nominal value, D is the interest on maturity.

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Question 11.9
For a particular corporate bond, which two main factors will determine the yield margin over an equivalent conventional government bond?

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Shares and other equity-type borrowing
This section looks at ordinary shares, preference shares and convertibles.
Unlike fixed interest investments, the returns from shares are largely unknown when the investor buys them. Dividends and share prices will vary according to the performance of the company. This means that returns from shares are potentially volatile. However, because of this uncertainty, we would expect to get a higher return overall from shares than from, say, fixed interest government bonds.

3.1

Ordinary shares
You have already come across equities or ordinary shares in Chapter 1. We now consider the characteristics of these securities that are particularly relevant to investors.
Ordinary shares – also called equities – are securities, issued by commercial undertakings and other bodies, which entitle their holders to receive all the net profits of the company after interest on loans and fixed interest stocks has been paid. The cash paid out each year is called the dividend, the remaining profits (if any) being retained as reserves or to finance the company’s activities.
Ordinary shares are the principal way in which companies in many countries are financed. They offer investors high potential returns for high risk, particularly risk of capital losses.
Ordinary shares are the lowest ranking form of finance issued by companies.
Dividends are not a legal obligation of the company but are paid at the discretion of the directors.

However, if the directors fail to declare sufficient dividends, the shareholders may vote for the directors to be replaced! In principle, the level of dividends can fluctuate widely as a company’s profits change. In practice directors try to pay a steadily increasing stream of dividends. But companies do have to cut, or even “pass” (ie not pay) dividends from time to time, and some companies do get wound up which may result in the investor losing his initial capital investment.
The initial running yield on ordinary shares is low but dividends should increase with inflation and real growth in a company’s earnings.

The running yield is defined as the dividend divided by the market price. It will be discussed again in the next chapter.

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For many good quality companies, dividends have grown at a rate well in excess of long-term inflation. But, dividends are not a precise match for any form of inflation.
Price inflation of 15% in a given year does not mean that dividends will increase by exactly 15% in that year.
The expected overall future return on ordinary shares ought to be higher than for most other classes of security to compensate for the greater risk of default, and for the variability of returns.

Whether the returns actually achieved will be higher or lower than the returns on more secure investments is a matter of chance. For example, investors in Polly Peck achieved a return of over 1,000% in the 1980s. When the company got into trouble at the start of the 1990s the return was –100%! In neither case was the market expecting such extreme results.
The return on ordinary shares is made up of two components, the dividends received and any increase in the market price of the shares.
Marketability of ordinary shares varies according to the size of the company but will be better than for the loan capital of the same company if:


the bulk of the company’s capital is in the form of ordinary shares;



the loan capital is fragmented into several different issues;



investors buy and sell ordinary shares more frequently than they trade in loan capital, perhaps because the residual nature of ordinary shares makes them more sensitive to changes in investors’ views about a company. The level of marketability varies from being excellent (although still not as good as for fixed interest government bonds) for very large companies like Glaxo Wellcome or BP
Amoco, to almost non-existent for small unquoted companies.
Ordinary shareholders get voting rights in proportion to the number of shares held, so shareholders may have the ability to influence the decisions taken by the directors and managers of the company.

Ordinary shares are normally irredeemable, in contrast to most loan capital issues.

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As well as having the right to residual profits and assets, ordinary shareholders will probably have the right to:


attend and speak at company meetings, or appoint a proxy to attend and vote on their behalf (a proxy is not allowed to speak)



vote to reduce, but not increase, dividends



vote to appoint directors



vote to forgo the “pre-emptive” right to be offered any new shares to be issued



vote to change the company’s borrowing powers



receive the annual report and accounts

Question 11.10
A friend of yours says: “Ordinary shares are the most risky form of investment so a prudent investor should avoid investing in them.”
Comment briefly on this statement.

Question 11.11
List the main features of the income paid to shareholders.

3.2

Preference shares
Preference shares are less common than ordinary shares. Assuming that the company makes sufficient profits, they offer a fixed stream of investment income. The investment characteristics are often more like those of unsecured loan stocks than ordinary shares.
The crucial difference between preference shares and ordinary shares is that preference share dividends are limited to a set amount which is almost always paid. Preference shareholders rank above ordinary shareholders (both for dividends and, usually, on winding up), and only get voting rights if dividends are unpaid or if there is a matter which directly affects the rights of preference shareholders.

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Preference dividends, like ordinary dividends, are only paid at the directors’ discretion, but no ordinary dividend can be paid if there are any outstanding preference dividends. In most cases preference shares are cumulative, which means that unpaid dividends are carried forward. In a given company, the risk of preference shareholders not getting their dividends is greater than the risk of loan stockholders not being paid, but less than the risk of ordinary shareholders not being paid.
For all investors, the expected return on preference shares is likely to be lower than on ordinary shares because the risk of holding preference shares is lower.
Preference shares rank higher on a winding-up, and the level of income payments is more certain.

The variability of return will only be a little greater than for loan capital. However, the variability of return will be significantly less than that of ordinary shares, because a preference share’s capital value will fluctuate much less than the capital value of an ordinary share.
Marketability of preference shares is likely to be similar to loan capital marketability, ie worse than for ordinary shares and a lot worse than for fixed interest

government bonds.

3.3

Convertibles
Convertible forms of company securities are, almost invariably, unsecured loan stocks or preference shares that convert into ordinary shares of the issuing company. The convertible will have a stated annual interest payment. The date of conversion might be a single date or, at the option of the holder, one of a series of specified dates.

The period during which conversion may take place is, not surprisingly, known as “the conversion period”.
The dates and terms of conversion are specified at the time of issue. For example, each convertible preference share may be able to convert into 2 ordinary shares on any 1
January between 2005 and 2010. It is up to the investor to choose when (and whether) to convert between these dates.
Occasionally, the conversion terms may change according to the date of conversion.
For example, each convertible preference share may convert into 2 shares if conversion takes place on 1 January 2005, 2.1 shares if conversion takes place on 1 January 2006,
2.2 shares if conversion takes place on 1 January 2007 etc.

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The characteristics of a convertible security in the period prior to conversion are a cross between those of fixed interest stock and ordinary shares. As the likely date of conversion (or not) gets nearer, it becomes clearer whether the convertible will stay as loan stock or become ordinary shares. As this happens, its behaviour becomes closer to that of the security into which it converts.
Convertibles generally provide higher income than ordinary shares and lower income than conventional loan stock or preference shares.

Question 11.12
Why do convertibles give lower income than conventional loan stock?
There will generally be less volatility in the price of the convertible than in the share price of the underlying equity.

The risk involved depends on the type of stock (loan stocks will rank ahead of preference shares) but most importantly on the quality of the company. Convertibles tend to be unsecured because they convert into the least secure type of investment ie ordinary shares. It would be inappropriate, for example, to have a debenture, which is one of the most highly secured forms of borrowing, converting into ordinary shares, which are the least secured.
From the investor’s point of view, convertible securities offer the opportunity to combine the lower risk of a debt security with the potential for large gains of an equity. The price paid for this is a lower running yield than on a normal loan stock or preference share. The option to convert will have time value, which will be reflected in the price of the stock.

Question 11.13
Describe the income and return on ordinary shares. Compare this with the income and return on preference shares and convertibles.

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Property
Within investment terminology property means a legal title to the use of land and buildings. This section is therefore about investing in land and/or buildings.
There are many different types of properties available for investment, for example: offices, shops and industrial properties (eg warehouses, factories).
The return from investing in property comes from rental income and from capital gains, which may be realised on sale. Property is a real investment and as such rents and capital values might be expected to increase broadly with inflation in the long term, which makes the returns from property similar in nature to those from ordinary shares. However, neither rental income nor capital values are guaranteed and there can be considerable fluctuations in capital values in particular, in real and nominal terms.
Rental terms are specified in lease agreements.

A lease is an agreement that allows one party, the leaseholder, the use of a specified portion of a property for a specified period of time in return for some payment. The term of the lease may range from about five to over 100 years.
Typically, it is agreed that rents are reviewed at specific intervals such as every three or five years. The rent is changed, at a review time, to be more or less equal to the market rent on similar properties at the time of the review. Some leases have clauses which specify upward-only adjustments of rents.

Consequently, if the level of market or rack rents decreases between reviews, the rent will not be reduced at the next review, but must remain fixed at its existing level.

Question 11.14
Why are rent reviews usually not annual?
The following characteristics are particular to property investments:
(a)

large unit sizes, leading to less flexibility than investment in shares. This

is in contrast to most other securities, which may be purchased in small quantities. Indivisibility may prevent smaller investment funds from investing in property, or lead them to invest in property indirectly – for example, via property company shares.
(b)

each property is unique, so can be difficult to value. Valuation is expensive, because of the need to employ an experienced surveyor.

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the actual value obtainable on sale is uncertain: values in property markets can fluctuate just as stock markets can. Property valuation is both

subjective and expensive and therefore the “true” market value of a property may be known only when a sale occurs. In addition, as sales are infrequent and prices agreed are normally treated with a degree of confidentiality, it may be difficult to place a certain value upon a particular property. This difficulty could again reduce the appeal of direct property investment to certain investors.
(d)

buying and selling expenses are higher than for shares and bonds.

(e)

net rental income may be reduced by maintenance expenses, although the

tenant is often responsible for building maintenance and insurance. However, there are still high management costs including the costs of rent collection and review. (f)

there may be periods when the property is unoccupied, and no income is received. A property in this situation is often referred to as “void”.

Marketability is poor because each property is unique and because buying and selling incur high costs.

The running yield on property is defined as the rental income net of all management expenses divided by the cost of buying the property gross of all purchase expenses. As with other asset classes, it varies between different types of property according to factors such as marketability, the level of expenses involved and, most importantly, risk.
Property types considered more risky will generally offer a higher running yield.
The running yield from property investments will normally be higher than that for ordinary shares. The reasons for this are:
1.

dividends usually increase annually, whereas rents are reviewed less often 2.

property is much less marketable

3.

expenses associated with property investment are much higher

4.

large, indivisible units of property are much less flexible

5.

on average, dividends will tend to increase more rapidly than rents, as dividends benefit from returns arising from the retention of profits and their reinvestment within the company.

Also, property running yields will often be lower than for fixed interest bonds because of the prospect of a capital gain, reflecting growth of rental levels.

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Question 11.15
Define in one sentence exactly what a lease means.

Question 11.16
List the characteristics that are particular to property investments.

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5

CT1-11: Investments

Derivatives
A derivative is a financial instrument with a value dependent on the value of some other, underlying asset.

The distinguishing feature of derivative instruments (or derivative contracts) is that they are legal agreements between two parties to trade an underlying asset at a date in the future. This is in contrast to most investments, which are issued by a borrower to an investor in order to raise money. Derivatives are mainly used to control risk. They can be used to reduce risk (a process known as “hedging”), or to increase risk (known as
“speculation”) in order to enhance returns.
This section looks at the three most common types of “derivative” instrument: futures, options and swaps. The syllabus item for this Subject requires you to be able to
“describe the investment and risk characteristics of ” futures, options and swaps. We will not be going into the finer details of the pricing of these contracts, or the way in which they are used, as these aspects come into the Finance and Financial Reporting
Subject.
Because some of the concepts in this section may be very new to you, you might expect to spend proportionately slightly more time here than on other units.

5.1

Futures
If you go to buy a car you may well order it a few weeks in advance. You will agree the price (to be paid when you take delivery), the model, specification and colour etc. You may also have to make a small deposit payment. In agreeing to buy the car on a future date at a specified price, you have entered into what is known as a “forward” contract.
A futures contract is a standardised, exchange tradable contract between two parties to trade a specified asset on a set date in the future at a specified price.

This might sound just like our example of a forward contract, defined above, so what is the difference?
Say you wanted to go and buy a pair of curtains for your kitchen window. You might be lucky and be able to buy some ready-made curtains that are just the right size and colour that you were looking for. Alternatively, you may decide to go for home-made curtains, buy some material and make the curtains yourself (or pay someone to do it).

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Futures are a bit like ready-made curtains! All of the details are predetermined. You only get futures in specific goods, with specific delivery details, ie they are standardised, and all futures are arranged with a clearing house (see below). Forward contracts are more like home-made curtains, coming in all shapes and sizes.
Because futures are standardised, lots of identical futures are arranged between lots of different parties. The result of this is that futures are tradable, ie you can buy or sell futures in the open market.

Long and short
If you “buy” a futures contract, you have declared that you will pay an agreed price in, say, 3 months’ time and in return you will receive delivery of the asset in 3 months’ time. The purchaser of a future is said to be the “long” party (because when the contract is settled the party will have much of the asset).
If you “sell” a futures contract, you have declared that you will receive the agreed price in, say, 3 months’ time and in return you will have to hand over the underlying asset in
3 months’ time. The seller of the future is said to be the “short” party (because when the contract is settled the party will be short of the asset, ie have little of the asset).

Range of futures
Futures are such a wonderful idea that they exist on many different underlying assets.
Originally futures contracts were all based on commodities (eg sugar, cocoa, wheat, pigs, gold). Commodity futures go back as far as 1848.
Financial futures are based on an underlying financial instrument, rather than a physical commodity. They exist in four main categories:


Bond futures



Short interest rate futures



Stock index futures



Currency futures

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CT1-11: Investments

The clearing house
When two traders agree to deal, the contract is created and the clearing house is informed. A clearing house is a self-contained institution whose only function is to clear futures trades and settle margin payments (described below). The clearing house checks that the buy and sell orders match each other. It then acts as “a party to every trade”. In other words it simultaneously acts as if it had sold to the buyer, and bought from the seller. Following registration, each party has a contractual obligation to the clearing house. In turn the clearing house guarantees each side of the original bargain.

Margin
Each party to a futures contract must deposit a sum of money known as
“margin” with the clearing house. Margin payments act as a cushion against potential losses which the parties may suffer from future adverse price movements. The potential losses referred to here are those that would arise if one party to the trade defaults on the agreement. The risk that you default on the agreement to trade may increase if the market moves against you. For example, if you agree to buy a future and then the price of the underlying asset, and hence the future, goes down, you face a larger potential loss and consequently may be more likely to default.
The essential feature of a future is that the money changes hand on the delivery date, rather than on the date that the deal is agreed. However, in practice, clearing houses require a small good faith deposit – known as “margin” – to be deposited soon after the deal is agreed. This money is held by the clearing house. Margin will be much smaller than the value of the underlying asset being dealt in.
As well as being small, margin earns interest so that most traders shouldn’t be too concerned by the need to deposit margin.
When the contract is first struck, “initial margin” is deposited with the clearing house. Additional payments of “variation margin” are made daily to ensure that the clearing house’s exposure to credit risk is controlled. This exposure can increase after the contract is struck through subsequent adverse price movements. For example, suppose that Party A agrees to pay Party B £100 in 6 months’ time for a bond that is currently worth £100.

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At the start of an agreement, both parties to the future are required to deposit a returnable “initial margin”, say £5. Margin may be in the form of cash (on which the clearing house will pay interest) or in the form of acceptable securities (eg the underlying asset or Treasury bills).
As time progresses, the underlying asset price is likely to change. For example, suppose that the price of the underlying asset, the bond, goes up to £102. This is good news for the buyer of the future because he has agreed to buy the (now more valuable) asset at a specified fixed price. The seller of the future is in the opposite position. So, the seller of the future is facing a loss, it may have to sell an asset worth £102 for just
£100 and make a loss of £2.
This increase in the value of the underlying asset might make it more likely that the seller will fail to honour the contract. Accordingly, the clearing house may demand extra margin payments known as “variation margin” from the seller, say £2, to cover the expected loss. Conversely, the buyer is less likely to default and so will be able to withdraw some of her initial margin from the clearing house (ie variation margin is negative for the buyer).
Similarly, if the underlying asset price falls, variation margin payments may be required from the buyer, while the seller will be able to withdraw some margin money.

Question 11.17
Give a definition of a financial futures contract.

Bond futures
Rather than being based on a single bond, most bond futures allow the seller to deliver one out of a range of government bonds.
For delivery, the contract requires physical delivery of a bond. If the contract were specified in terms of a particular bond then it would be possible simply to deliver the required amount of that stock. If the contract is specified in terms of a notional stock then there needs to be a linkage between it and the cash market.
The bonds which are eligible for delivery are listed by the exchange.
The party delivering the bond will choose the stock from the list which is cheapest to deliver. The price paid by the receiving party is adjusted to allow for the fact that the coupon may not be equal to that of the notional bond which underlies the contract settlement price.

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CT1-11: Investments

For example, say we are about to settle a futures contract, defined in terms of a notional
10-year fixed interest bond with 9% coupon. There will be a list of actual stocks that qualify to be delivered. These stocks are allowable. Say we have two allowable stocks, a 9-year stock with a coupon of 5% and an 11-year stock with a coupon of 10%.
The exchange might specify that I must deliver £120 nominal of the 9-year stock, for every £100 nominal of the notional stock, but only £80 nominal of the 11-year stock.
Although the conversion ratios work such that the deliverable stocks are roughly equivalent, the conversion will not be exact. There will still be a “cheapest deliverable” stock at any one time.
If the price of the 9-year stock is £82 per £100 nominal and the price of the 11-year stock is £118 per £100 nominal then it will be cheaper to deliver the 11-year stock
(since 0.8 ¥ £118 < 1.2 ¥ £82 ).
Which of the deliverable stocks is cheapest to deliver may change from time to time, although in practice the cheapest deliverable stock tends to be quite stable as long as yields do not change greatly.

Short interest rate futures
Short interest rate futures are based on a benchmark interest rate and settled for cash.
The price of these futures is quoted in a slightly unusual way.
The way that the quotation is structured means that as interest rates fall the price rises, and vice versa. The price is stated as 100 minus the 3-month interest rate. For example, with an interest rate of 6.25% the future is priced as 93.75.

The reason for this complex structure is that you end up with the normal position that if interest rates rise then the price (of the future) falls.
Note that the interest rate is quoted as four times the effective three-month rate of interest (ie convertible quarterly). So 6.25% pa in the example really means 1.5625% over three months (equivalent to an effective annual rate of interest of 6.398%).
The contract is based on the interest paid on a notional deposit for a specified period from the expiry of the future. However no principal or interest changes hands. The contract is cash settled. On expiry the purchaser will have made a profit (or loss) related to the difference between the final settlement price and the original dealing price.
The party delivering the contract will have made a corresponding loss (or profit).

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For example, if the market value of a short interest rate future position increases from
£70,000 to £74,500 (ie interest rates fall) then the purchaser will make a profit of
£4,500.

Question 11.18
If you think that interest rates are going to fall would you “buy” or “sell” the short interest rate future?

Stock index futures
With stock index futures, the asset is a notional portfolio of shares as represented by the particular index. The seller of a futures contract based on a stock market index is theoretically supposed to deliver a portfolio of shares made up in the same proportions as the index. This is totally impractical, so stock index futures are always settled for cash. The cash amount on settlement is determined by the difference between the future price, that the investor agreed, and the index value on the day of settlement.
The contract provides for a notional transfer of assets underlying a stock index at a specified price on a specified date.

Examples of indices on which futures contracts are based are the FTSE 100 index in the
UK and the Standard & Poors (S&P) 500 index in the US.

Currency futures
The contract requires the delivery of a set amount of a given currency on the specified date.

Question 11.19
Outline the role of margin and the clearing house in reducing counterparty credit risk.

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5.2

CT1-11: Investments

Options
An option gives an investor the right, but not the obligation, to buy or sell a specified asset on a specified future date.

Options are contracts agreed between investors to trade in an underlying security at a given date at a set price. The difference between options and futures is that the holder of the option is not obliged to trade hence the name “option”. The other party (the
“writer”) is obliged to trade if the holder of the option wants to.
Like futures, the first options arose from commercial transactions. For example,
Air France might buy an option on a new aeroplane from Boeing. This would give Air
France the right to order the aeroplane at a specified price. The contract would not be legally binding on Air France (but it would be legally binding on Boeing). Air France would pay Boeing a small premium for this option.
There are two basic types of options:
A call option gives the right, but not the obligation, to buy a specified asset on a set date in the future for a specified price.
A put option gives the right, but not the obligation, to sell a specified asset on a set date in the future for a specified price.

From these two types of option, there are four positions that an investor could hold:
1.

buying a call option

2.

buying a put option

3.

“writing” (ie selling) a call option

4.

“writing” (ie selling) a put option

Question 11.20
Explain why buying a put is not the same as selling a call.
When you write (ie sell) an option, you collect a premium for giving the holder the right to exercise (or not) the option. We will use the terms “write” and “sell” interchangeably. © IFE: 2009 Examinations

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Timing
Some options can be exercised only on the specified “expiry” or “contract” date, others can be exercised on any working day prior to expiry. They are respectively known as
“European” style and “American” style options.
An American style option is an option that can be exercised on any date before its expiry.
A European style option is an option that can be exercised only at expiry.

Question 11.21
State precisely what the writer of an American style put option contract with an exercise price of 20p and an October expiry date has done.

Margin
Because a call writer’s liability can be unlimited, and a put option writer’s liability can be very large relative to the premium, margin is required from writers of options.
However, the buyer’s maximum loss is the premium that is paid at the outset of the contract. Therefore, margin is not required from holders of options.

5.3

Swaps
A swap is a contract between two parties under which they agree to exchange a series of payments according to a prearranged formula.

The swapped payments are normally either:


two different types of interest payments (an interest rate swap)



two different currencies (a currency swap).

Interest rate swaps
In the most common form of interest rate swap, one party agrees to pay to the other a regular series of fixed amounts for a certain term. In exchange, the second party agrees to pay a series of variable amounts based on the level of a short-term interest rate. Both sets of payments are in the same currency. These

are known as coupon swaps.

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CT1-11: Investments

The fixed payments can be thought of as interest payments on a deposit at a fixed rate, while the variable payments are the interest on the same deposit at a floating rate. The deposit is purely a notional one and no exchange of principal takes place.

For example, a company that has too much variable rate borrowing may reduce its exposure to changes in short-term interest rates using an interest rate swap. The company might agree to pay a bank 9% pa fixed interest and in return receive a particular 6-month variable interest rate.

9% fixed
Bank

Company
6 month variable

The payments might be based on a notional principal amount of, say, £10m. The company would then pay the bank £450,000 every six months (assuming twice yearly payments). In return the bank would pay the company an amount based on the 6-month variable rate. If the 6-month variable rate at the time of a particular payment were 7% pa then the bank would pay the company £350,000. (Often the deal will be arranged so that only the net payment of £100,000, ie £450,000 – £350,000, is made.)
The overall position for the company may then look like this:
9% fixed
Creditor

6-month variable

Company

6-month variable

Bank

Although it borrowed from its creditors on variable rate interest its net cashflow is now fixed. The company has removed its exposure to increases (and decreases) in shortterm interest rates.

Currency swaps
A currency swap is an agreement to exchange a fixed series of interest payments and a capital sum in one currency for a fixed series of interest payments and a capital sum in another.

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For example you might simultaneously arrange to:


sell £10m in exchange for $15m as a spot transaction



buy £10m for $15m at the end of a set term



make or receive payments for the net difference between sterling and dollar interest rates.

The initial exchange of principal would usually be based on current spot rates. This means that it happens at a fair value and so, in a sense, it doesn’t really matter if it happens or not (because either party could easily carry out the transaction in the spot market instead). However, most currency swaps do in fact consist of all three steps set out above.
Unlike an interest rate swap the principal amount is no longer notional. Because of exchange rate movements over the life of the swap, the £10m nominal may be worth more or less than the $15m nominal by the end of the swap’s life. Exchange of the principal amount at the end (or at least the payment of the net difference in value) is an integral and important part of the deal.
The exchange of principal under the swap at the end of the contract will be in the same direction as the interest flows. The initial exchange of principal under a currency swap is in the opposite direction to the interest payments and terminal exchange of principal.

Characteristics
Swaps are not conventional investments. They are financial tools, which allow institutions to change the nature of their assets and liabilities. Usually one party to a swap agreement will be a market making bank (often referred to as the market maker) and the other will be a company. The parties involved in a swap are often called the counterparties. The swap will be priced so that the present value of the cashflows is slightly negative for the investor and positive for the issuing organisation. The difference represents the price that the investor is prepared to pay for the advantages brought by the swap on the one hand, and the issuer’s expected profit margin on the other.

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Each counterparty to a swap faces two kinds of risk:
(a)

Market risk is the risk that market conditions will change so that the present value of the net outgo under the agreement increases. The market maker will often attempt to hedge market risk by entering into an offsetting agreement.

In other words, the market maker would enter into a second agreement, which worked in exactly the opposite direction, so that the potential loss could be cancelled out.
(b)

Credit risk is the risk that the other counterparty will default on its payments. This will only occur if the swap has a negative value to the defaulting party so the risk is not the same as the risk that the counterparty would default on a loan of comparable maturity.

As well as credit risk only being a problem when the swap has a positive value to you, the problem may be quite small on an interest rate swap because:
1.

the principal is not at stake

2.

the loss of future interest income will largely be compensated by the fact that you will no longer need to make the interest outgo payments if your counterparty defaults.

Question 11.22
Give a definition of a swap.

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Chapter 11 Summary
Government bills are short-dated securities issued by governments to fund their shortterm spending requirements. They are issued at a discount and redeemed at par with no coupon. They are often highly marketable and secure.
There are two types of government bonds: fixed interest bonds which provide payments that are fixed in monetary terms, and index-linked bonds which provide payments that increase in line with changes in an inflation index.
Some bonds have variable redemption dates and some bonds are undated.
Bonds issued by the governments of developed countries in their domestic currency are the most secure long-term investment available but this should lead to a low expected return. Corporate bonds are usually less secure and less marketable than government bonds.
Investors will however require a greater yield as compensation.
Debentures are loans that are secured on some or all of the assets of the company.
With unsecured loan stock there is no specific security for the loan. Convertible unsecured loan stocks give their holders the right to convert into ordinary shares of the company at a later date.
Eurobonds are a form of unsecured loan capital issued by large companies, governments and supra-national organisations.
Certificates of deposit are short-term investments issued by banks and building societies. Ordinary shares are the most common type of share capital. They give rights to a share of the residual profits of the company, and to the residual capital value if the company is wound up, together with voting rights and various other rights.
Preference shares give their holders a preferential right to dividends and return of capital, compared to ordinary shareholders. Preference shares pay a fixed dividend.
They normally give voting rights only when dividends are not declared.
Convertible preference shares are one variety of preference shares. They give holders the right to convert their preference shares into ordinary shares on fixed terms on certain dates.

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CT1-11: Investments

Property characteristics include:


large unit sizes



uniqueness



uncertain sale value



higher dealing costs



high maintenance expenses



void periods

A derivative is a financial instrument with a value dependent on the value of some other, underlying asset.
A futures contract is a standardised, exchange tradable contract between two parties to trade a specified asset on a set date in the future at a specified price.
An option gives an investor the right, but not the obligation to buy or sell a specified asset on a specified future date.
A call option gives the right, but not the obligation, to buy a specified asset on a set date in the future for a specified price.
A put option gives the right, but not the obligation, to sell a specified asset on a set date in the future for a specified price.
An American style option is an option that can be exercised on any date before its expiry. A European style option is an option that can only be exercised at expiry.
A swap is a contract between two parties under which they agree to exchange a series of payments according to a prearranged formula.
Each counterparty to a swap faces two kinds of risk: market risk and credit risk.

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Chapter 11 Solutions
Solution 11.1
Advantages of issuing bonds at a fixed price
The government will know the price that investors will pay at outset and so it will know the cost of borrowing money.
It will be administratively less difficult than by tender.

Disadvantages of issuing bonds at a fixed price
Investors may be willing to pay more for the bond than the set price and so the money could be borrowed more cheaply if a tender is used.
Insufficient investors may be prepared to pay the set price causing only part of the offer to be sold. The government may not meet its finance requirements.

Solution 11.2
The tax-free investor will make the highest bid because it will receive higher net payment than the taxpayer and hence will be prepared to pay more for them.

Solution 11.3
The coupon payments per £100 nominal are:

Year of coupon
Coupon

2001
3.64

2002
3.79

The redemption payment is 100 ×

2003
3.93

2004
4.10

2005
4.26

657
= £121.67 .
540

The total payment in 2005 is therefore 4.26 + 12167 = £125.93 .
.
The price is given by the equation:

P = £3.64v + 3.79v 2 + 3.93v3 + 4.10v 4 + 125.93v5

@ 6%

= £107.46%

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CT1-11: Investments

Solution 11.4
The rate of return is not known at outset because:





the coupons will be reinvested on unknown terms sale price on sale before redemption is unknown at outset real return not known because inflation is not known tax rates (and systems) may change

Solution 11.5
PV =

10,000
= £10,524.80
1 − 01 × 182
. 365

Solution 11.6
(i)

Government bills







Short-term
Issued at a discount, redeemed at par
No coupons
Very marketable and secure
Yield quoted as simple rate of discount

(ii)

Fixed interest government bonds




Issued at a price or by tender
Investors receive coupons (usually half-yearly) plus redemption payment
(usually at par)
Some redemption dates are variable and some stocks are undated
Very secure, liquid and marketable (in developed countries)
Low expected return
Low dealing costs
Real returns are uncertain unless index-linked bond







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Solution 11.7
They invest in corporate bonds because they expect to receive a higher investment return to compensate for the lower security and marketability.

Solution 11.8
The unsecured loan will give the highest yield because it is probably the least secure and least marketable.
The debenture will give the lowest yield because it is the most secure investment.

Solution 11.9
The yield margin will depend on:
(a)

The security of the bond (based on the quality of the company)

(b)

Marketability (based largely on the size of issue)

Solution 11.10
It is true that ordinary shares are potentially risky in that the return from them may be very volatile and uncertain. However, an investor would expect, on average over the very long term, to gain a higher return from shares than from other investments, to compensate for the higher level of risk.
In the long term, dividends and share prices should increase in line with inflation and real economic growth. Therefore shares are a suitable investment for an institution which requires a return linked to inflation.

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CT1-11: Investments

Solution 11.11
The income paid to shareholders (ie dividend income) has the following features:


the payments are uncertain



they may be made forever (provided the company does not go bust)



the initial running yield is low



the income level should increase over time

Solution 11.12
Convertibles generally give a lower income than conventional unsecured loan stock because of the option to convert. The investor is expecting to receive a higher income from dividends after conversion than the stock is currently paying. The value of this expected future income makes the stock more attractive than a conventional stock, and means that the company can pay a lower income now.

Solution 11.13
When we talk about income we are talking about the dividend payment. The dividend payment on an ordinary share can sometimes be quite low. Sometimes dividends grow as the company becomes more successful. The overall return on a share consists of the regular dividend payments and the capital gain when the share is sold. The overall return on an ordinary share is likely to be greater than the return on any other financial asset because the holder takes the greatest risk.
Imagine that you are considering buying a preference share, an ordinary share and a convertible. The preference share gives a steady income but doesn’t offer the prospect of high rewards; the convertible gives a steady income plus the potential high rewards of an ordinary share (therefore is more attractive than a preference share and can be rewarded with a lower income); the ordinary share might offer low and possibly volatile income now but potentially high income and capital growth in the future. The ordinary share is riskier. If the ordinary share were providing higher, more stable payments than a convertible there would be no point in buying the convertible since you would buy the ordinary share.

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Solution 11.14
Although annual rent reviews might lead to more frequent increases in the level of rent, the extra rental income would probably be more than offset by the additional costs incurred re-negotiating the rent each year.

Solution 11.15
A lease is an agreement that allows one party – the leaseholder – the use of a specified portion of a property for a specified period of time in return for some payment.

Solution 11.16


Large unit sizes, leading to less flexibility than investment in shares.



Each property is unique, so can be difficult to value. Valuation is expensive, because of the need to employ an experienced surveyor.



The actual value obtainable on sale is uncertain: property markets can crash just as stock markets can.



Buying and selling expenses are higher than for shares and bonds.



Maintenance expenses may reduce net rental income.



There may be periods when the property is unoccupied, and no income is received. Solution 11.17
A futures contract is a standardised, exchange tradable contract between two parties to trade a specified asset on a set date in the future at a specified price. The underlying asset/benchmark might be a bond, a currency, an index reflecting the level of a stock market, or an interest rate.

Solution 11.18
Buy. As with any investment and any future, the purchaser makes a profit if the price goes up (ie if interest rates fall).

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Solution 11.19
Once a contract is matched, the clearing house becomes the “party to every trade”. So, there is no need to rely on the financial soundness and integrity of the original counterparty. The investor has to pay margin to the clearing house:



initial margin (up-front payment for each open position) variation margin (on a daily basis, following price changes)

thus ensuring that the clearing house itself is not exposed to excessive counterparty risk.

Solution 11.20
The difference is between right and obligation. Buying a put costs you money and allows you to choose whether or not to sell the underlying asset. Selling a call means that you receive money and must sell the underlying asset if, and only if, the holder of the option wants to.
If you buy a put you are likely to choose to sell the underlying asset if the market price is less than the exercise price. If you sell a call you are likely to be forced to sell the underlying asset if the market price exceeds the exercise price.

Solution 11.21
He has given another person the right, but not the obligation, to sell the underlying asset to him at a price of 20p at any time up to a date in the following October. In return the writer has received a premium.

Solution 11.22
A swap is a contract between two parties under which they agree to exchange a series of payments according to a prearranged formula. The swapped payments are normally either two different types of interest payments or in two different currencies.

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CT1-12: Elementary compound interest problems

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Chapter 12
Elementary compound interest problems
Syllabus objective
(xi)

Analyse elementary compound interest problems.
1.
Calculate the present value of payments from a fixed interest security where the coupon rate is constant and the security is redeemed in one instalment.
2.
Calculate upper and lower bounds for the present value of a fixed interest security that is redeemable on a single date within a given range at the option of the borrower.
3.
Calculate the running yield and the redemption yield from a fixed interest security (as in 1.), given the price.
4.
Calculate the present value or yield from an ordinary share and a property, given simple (but not necessarily constant) assumptions about the growth of dividends and rents.
5.
Solve an equation of value for the real rate of interest implied by the equation in the presence of specified inflationary growth.
6.
Calculate the present value or real yield from an index-linked bond, given assumptions about the rate of inflation.
7.
Calculate the price of, or yield from, a fixed interest security where the investor is subject to deduction of income tax on coupon payments and redemption payments are subject to the deduction of capital gains tax.
8.
Calculate the value of an investment where capital gains tax is payable, in simple situations, where the rate of tax is constant, indexation allowance is taken into account using specified index movements and allowance is made for the case where an investor can offset capital losses against capital gains.

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CT1-12: Elementary compound interest problems

This Chapter also deals with real rates of interest as required in syllabus objective (iv).

This chapter is very long and so you may want to use two or three study sessions to cover the material.

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Fixed interest securities
We are now in a position to apply the theory we have developed to fixed interest securities, ie investments that make regular interest payments at a fixed rate.
As in other compound interest problems, one of two questions may be asked:
(1)

(2)

1.1

What price A, or P per unit nominal, should be paid by an investor to secure a net yield of i per annum?
Given that the investor pays a price A, or P per unit nominal, what net yield per annum will be obtained?

Calculating the price
The price, A, to be paid to achieve a yield of i per annum is equal to the present value, at rate of interest i per annum, of the interest and capital payments, less any taxes payable by the investor. That is:

⎛ Present value, at rate
⎞ ⎛ Present value, at rate


⎟ ⎜

A = ⎜ of interest i per annum, ⎟ + ⎜ of interest i per annum, ⎟
⎜ of net interest payments ⎟ ⎜ of net capital payments ⎟

⎠ ⎝


(1.1)

The price per unit nominal is of course P = A / N , where N is the nominal amount of stock to which the payments relate.

Prices are often calculated per £100 nominal and remember that the coupon payments and redemption amounts are expressed as a percentage of the nominal amount and not as a percentage of the purchase price.
For example, in the simple case where no taxes apply, the price to be paid for a fixed interest stock bearing interest, payable half-yearly, at D% per annum and redeemable at par in n years is calculated as:
A = [Da( 2) + 100v n ] at i%
|
n

(1.2)

“Redeemable at par” means that the redemption payment, ie the amount of capital returned to the investor at the end of the term, is equal to the nominal amount. In
Equation (1.2), because we are working in units of £100 nominal, the redemption payment, at time n is 100.

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CT1-12: Elementary compound interest problems

Example
What price should an investor who requires a yield of 10% per annum (and doesn’t pay tax) pay on 15 July 2007 for £100 nominal of a six-year stock with half-yearly coupon payments of 13% pa? The next coupon is due in six months’ time and the stock is redeemable at par.
Solution
The price P is the PV of the proceeds (valued at 10% interest):
(2)
P = 13a6| + 100v 6 = 13 ¥ 4.4615 + 100 ¥ 0.56447 = £114.45

Note: One could also work with a period of half a year. equation of value would then be:
A = [ D 2 a | + 100v 2n ]
2n

The corresponding

at rate i ′

where (1 + i ′ )2 = 1 + i .

Question 12.1
Confirm the answer to the example above by using a half-yearly interest rate.

Allowing for income tax
Since tax payments represent negative cashflows for an investor, the price an investor who pays tax should pay to obtain a given yield can be found by valuing the net payments received:
Price = PV receipts − PV tax

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Example
What price should an investor who requires a yield of 10% pa and pays income tax at
33% pay on 15 July 2007 for £100 nominal of a six-year stock with half-yearly coupon payments of 13% pa? The stock is redeemable at par.

Solution
The stock in this question is the same as the one in the example on the previous page.
However, the investor in this example has to pay tax on income. The price the investor is prepared to pay should turn out to be lower than £114.45.
The coupon payments (£13pa) are income and so will be subject to 33% income tax, payable at the same time as each payment is received. So the amount of tax payable on each year’s coupons will be 33% × £13 (and the net coupon payments received each year will be 67% × £13 ). No income tax is payable on the redemption payment since it is a return of capital.
Assuming the investor intends to hold the stock till redemption, the price P is the present value of the net proceeds (valued at 10% interest):
(2)
(2)
P = 13a6| + 100v 6 - 0.33 ¥ 13a6|
(2)
= 0.67 ¥ 13a6| + 100v 6 = 0.67 ¥ 13 ¥ 4.4615 + 100 ¥ 0.56447 = £95.31

Comparing this with the price of £114.45 we calculated in the earlier example for an investor not subject to tax, we see that, as expected, an investor subject to tax must buy at a lower price to obtain the same yield.

1.2

Calculating yields
The yield available on a stock that can be bought at a given price, A, can be found by solving equation (1.1) (or equation (1.2)) for the net yield i.
If the investor is not subject to taxation the yield i is referred to as a gross yield.
The yield on a security is sometimes referred to as the yield to redemption or the redemption yield to distinguish it from the flat (or running) yield, which is defined as D/P, the ratio of the coupon rate to the price per unit nominal of the stock.

Therefore, a redemption yield that is calculated without making any allowance for tax is called the gross redemption yield (GRY). If tax is incorporated in the calculation this gives the net redemption yield (NRY).

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CT1-12: Elementary compound interest problems

We can determine the gross redemption yield for a fixed interest stock by trial and error and then interpolation.
Gross redemption yields are frequently quoted as nominal rates convertible half-yearly because most coupons are payable half-yearly. However, we will quote yields as effective rates unless specified otherwise.

Example
Calculate the gross redemption yield obtained by an investor who pays £120 on
15 July 2007 for £100 nominal of a six-year stock with half-yearly coupon payments of
13% pa and is not subject to tax. The stock is redeemable at par.

Solution
We need to find the interest rate i that satisfies the equation of value:

120 = 13a ( 2 ) + 100v 6
|
6

If the purchase price was £100, the redemption yield, convertible half-yearly, would be
13% because an investment of £100 is receiving interest of £13 each year payable halfyearly. The effective yield is just greater than 13%. Since the investor has paid more than £100, the redemption yield will be less than 13%.
Also, we saw in the example on page 4 that if the yield is 10%, the price is £114.45.
The investor is paying even more than this. So we need to try a lower rate:
(2)
At 9%: 13a6| + 100v 6 = 13 ¥ 4.5847 + 100 ¥ 0.59627 = 119.23
(2)
At 8%: 13a6| + 100v 6 = 13 ¥ 4.7136 + 100 ¥ 0.63017 = 124.29

We can approximate i by interpolating (linearly) using these two values:

i -8
120 - 124.29
ª
fi i 8.8%
9 - 8 119.23 - 124.29

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Question 12.2
(i)

(ii)

1.3

Calculate the running yield for the investor in the above example at the time of purchase. Assuming that the investor pays income tax of 25% on all income, calculate the net redemption yield.

The effect of the term to redemption on the price
Consider first a loan of nominal amount N which has interest payable pthly at the annual rate of D per unit nominal. Suppose that the loan is redeemable after n years at a price of R per unit nominal. An investor, liable to income tax at rate t1 , wishes to purchase the loan at a price to obtain a net effective annual yield of i .
Let g = D / R and C = NR , so that gC = DN . The price to be paid by the investor is: Ï(1 - t )DNa ( p ) + Cv n
1
Ô n| Ô
Ô
A(n, i ) = Ì(1 - t1)gCa ( p ) + C[1 - i ( p )a ( p ) ] n| n|
Ô
Ô
( p)
( p)
ÔC + [(1 - t1)g - i ]Can|
Ó

¸
Ô
Ô
Ô
at rate i ˝
Ô
Ô
Ô
˛

(1.3)

The following (equations 1.4) are immediate consequences of equation (1.3):
(a)

If i ( p ) = (1 − t1)g , then for any value of n , A(n, i ) = C , ie the price paid is the same as the redemption value.

(b)

If i ( p ) < (1 − t1)g , then, A(n, i ) is an increasing function of n (ie if n2 > n1 , then A(n2 , i ) > A(n1, i ) ).

(c)

If i ( p ) > (1 − t1)g , then, , A(n, i ) is a decreasing function of n (ie if n2 > n1 , then A(n2 , i ) < A(n1, i ) ).

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1.4

CT1-12: Elementary compound interest problems

The effect of the term to redemption on the yield
Recall the equation of value for a security:

A(n, i ) = C + [(1 - t1)g - i ( p ) ]Ca ( p )
|
n

If A(n, i ) < C , the purchaser (ie the investor) will receive a capital gain when a security is redeemed.

It is called a capital gain because the redemption amount, C, is greater than the price of the security, B. The redemption amount is just a return of the initial capital, which has therefore increased from B to C, hence a capital gain.
We are now going to consider two securities, one redeemed after n1 years and the other after n2 years, where n2 > n1 .
(a)

If i ( p ) = (1 − t1)g , then A(n, i ) = C , and the yield obtained on the two securities will be the same.

(b)

If i ( p ) < (1 − t1)g , then, A(n, i ) > C . This means that there is a capital loss for the investor. The investor will receive a higher yield on the security which is redeemed later.

(c)

If i ( p ) > (1 − t1)g , then, A(n, i ) < C . This means that there is a capital gain for the investor. The investor will receive a higher yield on the security which is redeemed earlier.

The above results are intuitively obvious.
From the investor’s viewpoint, the sooner the capital gain is received the better.
The investor will therefore obtain a greater yield on the security which is redeemed first. On the other hand, if A(n, i ) > C there will be a capital loss when a security is redeemed. The investor will wish to defer this loss as long as possible, and will therefore obtain the greater yield on a security which is redeemed later.
In summary, if i ( p ) > (1 - t1)g =

D
(1 - t1) then there is a capital gain for the
R

D investor and if i ( p ) < (1 - t1)g = (1 - t1) there is a capital loss for the investor.
R

We will call this the capital gains test.

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Question 12.3
(i)

(ii)

1.5

An investor purchases a £100 zero-coupon bond for £80. Calculate the yield obtained if the bond is redeemed after (a) five years, and (b) ten years.
Repeat the question with a purchase price of £120.

Optional redemption dates
Sometimes a security is issued without a fixed redemption date. In such cases the terms of issue may provide that the borrower can redeem the security at the borrower’s option at any interest date on or after some specified date.
Alternatively, the issue terms may allow the borrower to redeem the security at the borrower’s option at any interest date on or between two specified dates (or possibly on any one of a series of dates between two specified dates).

In such cases, the loan will be redeemed at the time considered to be most favourable by the borrower. If the interest rate payable on the loan is high relative to market rates, it will be cheaper for the borrower to repay the loan and borrow from elsewhere.
Conversely, if the interest rate payable is relatively low, it will be cheaper to allow the loan to continue.
The latest possible redemption date is called the “final redemption date” of the stock, and if there is no such date, then the stock is said to be “undated”. It is also possible for a loan to be redeemable between two specified interest dates, or on or after a specified interest date, at the option of the lender, but this arrangement is less common than when the borrower chooses the redemption date. The loan will be redeemed at the time when the party with the choice of date will obtain the greatest yield.
An investor who wishes to purchase a loan with redemption dates at the option of the borrower cannot, at the time of purchase, know how the market will move in the future and hence when the borrower will repay the loan. The investor thus cannot know the yield which will be obtained. However, by using equation (1.4) the investor can determine either:
(1)

The maximum price to be paid, if the net yield is to be at least some specified value; or

(2)

The minimum net yield the investor will obtain, if the price is some specified value.

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CT1-12: Elementary compound interest problems

Suppose that the outstanding term of the loan, n years, may be chosen by the borrower subject to the restriction that n1 ≤ n ≤ n2 . (We assume that n1 and n2 are integer multiples of 1/p.) Using the notation of Section 1.3, we let A(n, i ) be the price to provide a net annual yield of i, if the loan is redeemed at time n.

From the previous section we know that:
1.

If there is a gain on redemption (ie i ( p ) > (1 - t1 ) g ), the minimum yield will be obtained if redemption takes place at the latest possible date.

2.

If there is a loss on redemption (ie i ( p ) < (1 - t1 ) g ), the minimum yield will be obtained if redemption takes place at the earliest possible date.

Suppose that the investor wishes to achieve a net annual yield of at least i. It follows from (1.2) that:
(a)

If i ( p ) < (1 − t1)g , (ie a capital loss) then A(n1, i ) < A(n, i ) for any value of n such that n1 ≤ n ≤ n2 . In this case, therefore, the investor should value the loan on the assumption that redemption will take place at the earliest possible date (ie the worst case scenario). If this does in fact occur, his net annual yield will be i . If redemption occurs at a later date, the net annual yield will exceed i .

(b)

If i ( p ) > (1 − t1)g , (ie a capital gain) then A(n2 , i ) < A(n, i ) for any value of n such that n1 ≤ n ≤ n2 . In this case, therefore, the investor should value the loan on the assumption that redemption will occur at the latest possible date (ie the worst case scenario). This strategy will ensure that the net annual yield will be at least i .

(c)

If i ( p ) = (1 − t1)g , the net annual yield will be i irrespective of the actual redemption date chosen.

Example
A fixed interest stock with a coupon of 8% per annum payable half yearly in arrears can be redeemed at the option of the lender (ie the investor) at any time between 10 and 15 years from the date of issue.
What price should an investor subject to tax at 25% on income, who wishes to obtain a net yield of at least 7% per annum, pay for £100 nominal of this stock?

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Solution
Carrying out the capital gains test:
(1 - t1 ) g = 0.75 ¥ 0.08 = 0.06

i (2) = 2(1.070.5 - 1) = 0.0688

Since i (2) > (1 - t1 ) g there is a gain on redemption. The worst case scenario is the latest possible redemption date. This will give the minimum yield. So:
P = 0.75 × 8a ( 2|) + 100v 15 @7% = 5559 + 36.24 = £91.83
.
15

Of course, since the investor has the option on when the bond is redeemed, if he behaves rationally, he should choose the earliest date, as there is a capital gain.
However, as we are asked for the price to give a yield of at least 7% pa, we must consider the worst case. If the investor pays £91.83 for £100 nominal of the bond, he is certain to achieve a return of at least 7% pa whether he redeems at the earliest or latest possible date (if he redeems at the earliest date, the return will exceed 7% pa).
Suppose, alternatively, that the price of the loan is given. What can be said about the yield which the investor will obtain? As before, let P and R be the purchase price per unit nominal and the redemption price per unit nominal respectively. The minimum net annual yield is obtained by solving an appropriate equation of value. For example if:
(a)

P < R , the investor should determine the net annual yield on the assumption that the loan will be repaid at the latest possible date. If this does in fact occur, the net annual yield will be that calculated. If redemption takes place at an earlier date, the net annual yield will be greater than that calculated.

(b)

P > R , the investor should determine the net annual yield on the assumption that the loan will be repaid at the earliest possible date. The actual yield obtained will be at least the value calculated on this basis.

(c)

P = R , the net annual yield is i, where i ( p ) = g (1 - t1) , irrespective of the actual redemption date chosen.

Question 12.4
If the price of the stock in the above example was £110, what would be the minimum net yield this investor could expect to obtain?

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1.6

CT1-12: Elementary compound interest problems

Deferred income tax
It has been assumed so far that income tax payable on coupons is due at the time that the coupons are paid. It is possible in some countries that the tax is paid at some later date, for example at the calendar year end.
This does not cause any particular problems as we follow the usual procedure.
Identify the cashflow amounts and dates and set out the equation of value.
As an example of this an n-year bond with annual coupons of DN and an income tax rate t1, then the annual tax due on each bond is t1DN (regardless of coupon frequency). Assume that this tax is paid in a single instalment, due, say, k years after the second half-yearly coupon payment each year. Then the equation of value for a given net yield i and price (or value) A is, immediately after a coupon payment: A = DNa( 2) + RNv n − t 1DNv k an| n| Other arrangements may be dealt with similarly from first principles.

Example
A ten-year bond with half yearly coupons of 6% pa has just been issued with a redemption yield of 9% pa effective. It is redeemable at par. What price would an investor paying 15% tax on income pay for the bond? Tax payments are due four months after each coupon is received.

Solution
The price can be calculated from the equation:
(2)
(2)
P = 6a10| + 100v10 - 6 ¥ 0.15 ¥ v 4 /12 a10| @ 9%

= 6 ¥ 6.5589 + 100 ¥ 1.09-10 - 0.9 ¥ 1.09-4 /12 ¥ 6.5589
= £75.86 per £100 nominal

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Uncertain income securities
Securities with uncertain income include:
1.

Equities, which have regular declarations of dividends. The dividends vary according to the performance of the company issuing the stocks and may be zero.

2.

Property which carries regular payments of rent, which may be subject to regular review.

3.

Index-linked bonds which carry regular coupon payments and a final redemption payment, all of which are increased in proportion to the increase in a relevant index of inflation.

For all of these investments investors may be interested in calculating the yield for a given price, or the price or value of the security for a given yield. In order to calculate the value or the yield it is necessary to make assumptions about the future income.
Given the uncertain nature of the future income, one method of modelling the cashflows is to assume statistical distributions for, say, the inflation or dividend growth rate. In this course however we will make simpler assumptions – for example that dividends increase at a constant rate. It is important to recognise that modelling random variables deterministically, ignoring the variability of the payments and the uncertainty about the expected growth rate, is not adequate for many purposes and stochastic methods will be required.
In all three cases, using this deterministic approach means that we estimate the future cashflows and then solve the equation of value using the estimated cashflows. Index-linked bonds differ slightly from the other two in that the income is certain in real terms. These are therefore covered separately, in Section 4.

2.1

Equities
In this section you must remember the work that we did in Chapter 6 on perpetuities.
You might want to review that before continuing.
Given deterministic assumptions about the growth of dividends, we can estimate the future dividends for any given equity, and then solve the equation of value using estimated cashflows for the yield or the price or value.

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CT1-12: Elementary compound interest problems

So, let the value of an equity just after a dividend payment be A, and let D be the amount of this dividend payment. Assume that dividends grow in such a way that the dividend due at time t is estimated to be Dt . We generally value the equity assuming dividends continue in perpetuity, and without explicit allowance for the possibility that the company will default and the dividend payments will cease. In this case, assuming annual dividends:

A=



∑ Dt v it

t =1

where i is the return on the share, given price A.
If we assume a constant dividend growth rate of g, say, then Dt = (1 + g )t D and:

Ê (1 + g ) (1 + g ) 2 (1 + g )3
+
+
+
A = DÁ
Ë (1 + i ) (1 + i ) 2 (1 + i )3

ˆ
˜
¯

This is just a compound increasing annuity. So:
A = Da |
∞ i′

⇒A=

since a∞|i ′ =

where i ′ =

1+ i
−1
1+ g

D(1 + g ) i−g 1
.
i′

At certain times close to the dividend payment date the equity may be offered for sale excluding the next dividend. This allows for the fact that there may not be time between the sale date and the dividend payment date for the company to adjust its records to ensure the buyer receives the dividend. An equity which is offered for sale without the next dividend is called ex-dividend or “xd”. The valuation of ex-dividend stocks requires no new principles.

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Example
A French investor, who is taxed at 35% on income, has just purchased 500 shares in a small education company ex-dividend. Dividends are paid annually and the next dividend is due in one month’s time. The last dividend was 8 euros per share and dividends are expected to rise by 4% pa. Calculate the price paid by the investor if the expected yield is 12% pa effective.

Solution
The investor does not receive the dividend due in one month because the shares are purchased ex-dividend. The present value of the shares in one month’s time is:

(

500 8 ¥ 1.042 ¥ (1 - 0.35)v + 8 ¥ 1.043 ¥ (1 - 0.35)v 2 +

)

So the present value of the shares now (one month earlier) is:
1

(

500v 12 8 ¥ 1.042 ¥ 0.65v + 8 ¥ 1.043 ¥ 0.65v 2 +
1

= 500v 12

)

8 ¥ 1.042 ¥ 0.65v
= £34,822
1 - 1.04v

This can also be calculated using annuities as follows:
1

(

P = 500 ¥ 8 ¥ 1.04 ¥ 0.65v 12 1.04v + 1.042 v 2 + where i ¢ =

) = 500 ¥ 5.408v

1
12

a•| i ¢

1.12
0.08
.
-1 =
1.04
1.04

fi P = 500 ¥ 5.408 ¥ 1.12-1/12

1.04
= £34,822
0.08

since a•| i ¢ =

1 1.04
=
i ¢ 0.08

Question 12.5
What yield would be expected by the investor if the purchase price had been 120 euros per share?

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2.2

CT1-12: Elementary compound interest problems

Property
The valuation of property by discounting future income follows very similar principles to the valuation of equities. Both require some assumption about the increase in future income; both have income which is related to the rate of inflation (both property rents and company profits will be broadly linked to inflation, over the long term); in both cases we use a deterministic approach.
The major differences between the approach to the property equation of value, compared with the equity equation of value, are:
(1)

property rents are generally fixed for a number of years at a time and

(2)

some property contracts may be fixed term, so that after a certain period the property income ceases and ownership passes back to the original owner (or another investor) with no further payments.

Let A be the price immediately after receipt of the periodic rental payment. Let m be the frequency of the rental payments each year. We estimate the future
1 2 cashflows, such that Dt m is the rental income at time t, t = m , m , … . If the rents cease after some time n then clearly Dt = 0 for t > n .

Then the equation of value is:

A=



1
∑ m Dk / m v k / m

k =1

It will usually be much easier to work from first principles than try and apply this formula to every question about property values.

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Example
The rent for the next five years of an eighty-year property contract is set at £4,000 per month. Thereafter the rent will increase by 20% compound every five years. What price would be paid for the contract in order to achieve a yield of 7% per annum effective? Solution
The price is given by:
(
(
P = 48,000a512 ) + 48,000 × 12 × v 5a512 ) +
.
|
|

(
+ 48,000 × 1215 × v 75a512 )
.
|

Note that the monthly rent is £4,000 and so the annual rent is £48,000.
Next, you either get your calculator out and start tapping away or, preferably, notice that this formula can be simplified quite easily.

(

(12)
P = 48, 000a5| 1 + 1.2v5 + 1.22 v10 +

+ 1.215 v 75

)

(12)
= 48, 000a5|7% a16| j %

1075
.
where 1 + j =
⇒ j = 16.879% .
12
.
Therefore:
P = 48, 000 ¥ 4.2301 ¥

1 - 1.16879 -16
= £1, 290, 000
0.16879 1.16879

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3

CT1-12: Elementary compound interest problems

Real rates of interest
An investor’s basic objective is to maximise the rate of return. So, if an investor has to choose between a number of possible investment opportunities then, other things being equal, the higher the interest rate the better.
However, this is not the whole story. In free market economies, price inflation is present ie there is a tendency for the prices of goods to increase gradually (or rapidly) over time.
This means that the “purchasing power” of a specified sum of money tends to be eroded as time passes.

Example
A pensioner has just invested £3,000 in a government savings account that guarantees to provide a rate of return of 7¼% per annum over the next 5 years.
(i)

How much will the pensioner have at the end of the 5 years?

(ii)

If toasters currently cost £30 each and are expected to increase in price by 2½% per annum over the next 5 years, how many toasters could be bought now with the initial investment and with the proceeds at the end of the 5 years?

(iii)

Comment on your answers to (i) and (ii).

Solution
(i)

By the end of the 5 years the account will have accumulated to:
.
.
3,000 × (1 + i ) 5 = 3,000 × 107255 = 3,000 × 14190 = £4,257

(ii)

The £3,000 invested now could buy 3,000 / 30 = 100 toasters. At the end of the 5 years toasters will cost £30 × 10255 = £33.94 each.
.
So the proceeds will be able to buy 4,257 / 33.94 = 125 toasters.

(iii)

In money terms, the fund has grown by 41.9%, but in terms of the number of toasters it will buy it has only grown by 25% ie there has been an erosion in purchasing power caused by price inflation.

When you are considering investments, it is often more useful to look at the rate of return earned on an investment after taking into account the erosion caused by inflation.
This is done by looking at the real rate of interest (or real rate of return).

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The idea of a real rate of interest, as distinct from a money rate of interest, was introduced in Chapter 4. Ways of calculating real rates of interest will now be examined. The real rate of interest of a transaction is the rate of interest after allowing for the effect of inflation on a payment series.

3.1

Inflation adjusted cashflows
The effect of inflation means that a unit of money at, say, time 0 has different purchasing power than a unit of money at any other time. We find the real rate of interest by first adjusting all payment amounts for inflation, so that they are all expressed in units of purchasing power at the same date.
As a simple example, consider a transaction represented by the following payment line:
Time:
Payment:

0

1

−100

120

That is, for an investment of 100 at time 0 an investor receives 120 at time 1.
The effective rate of interest on this transaction is clearly 20% per annum. The real rate of interest is found by first expressing both payments in units of the same purchasing power. Suppose that inflation over this one year period is 5% per annum. This means that 120 at time 1 has a value of 120 1.05 = 114.286 in terms of time 0 money units.
So, in “real” terms, that is, after adjusting for the rate of inflation, the transaction is represented as:
Time:
Payment:

0

−100

1
114.286

Hence, the real rate of interest is 14.286%.

Question 12.6
If inflation had been 5% pa for the first nine months of the year but only 3% pa for the remaining three months, what is the real rate of return?

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3.2

CT1-12: Elementary compound interest problems

Calculating real yields using an inflation index
Where the rates of inflation are known (that is, we are looking back in time at a transaction that is complete) we may adjust payments for the rate of inflation by reference to a relevant inflation index.
For example, assume we have an inflation index, Q (t k ) at time t k , and a payment series as follows:
Time, t:
Payment:
Q (t )

0

1

2

3

−100
150

8
156

8
166

108
175

Clearly the rate of interest on this transaction is 8%.

This is because for an initial investment of 100 (working in years), we receive interest payments of 8 at the end of each year plus a return of capital at the end of three years.
If you are still not sure then check that 8% solves the equation of value:
100 = 8a3| + 100v 3
Now we can change all these amounts into time 0 money values by dividing the payment at time t by the proportional increase in the inflation index from 0 to t.
For example the inflation-adjusted value of the payment of 8 at time 1 is
8 ÷ (Q(1) / Q(0)) . The series of payments in time 0 money values is then as follows: Time, t:
Payment:

0

1

2

3

−100

7.6923

7.2289

92.5714

This gives a yield equation for the real yield:

−100 + 7.6923v i ′ + 7.2289v i2 + 92.5714v i3 = 0

′ where i ′ is the real rate of interest, which can be solved using numerical methods to give i ′ = 2.63% .

Question 12.7
Check that i ′ = 2.63% solves the equation of value.

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In general, the real yield equation for a series of cashflows {Ct1 ,Ct 2 , … ,Ct n } , given associated inflation index values {Q (0), Q (t1), Q (t2 ),… , Q (t n )} is, using time 0 money units: n ∑

Ct k

k =1

Q(0) t k v =0
Q (t k ) i ′



n

Ct k

∑ Q(t k ) v it′

k

=0

k =1

Until now we have been expressing all payments in terms of time 0 money units.
However, this choice was arbitrary, as we could have chosen any date on which to value all the payments.
The second equation here, in which all terms are divided by Q (0) , demonstrates that the solution of the yield equation is independent of the date the payment units are adjusted to.

3.3

Calculating real yields given constant inflation assumptions
If we are considering future cashflows, the actual inflation experience will not be known, and some assumption about future inflation will be required. For example, if it is assumed that a constant rate of inflation of j per annum will be experienced, then a cashflow of, say, 100 due at t has value 100(1 + j )-t in time
0 money values.
So, for a fixed net cashflow series {Ct k } , k = 1,2,… , n , assuming a rate of inflation of j per annum, the real, effective rate of interest, i ′ , is the solution of the real yield equation: n ∑ Ct v tj v it′ k k =1

k

k

=0

We also know that the effective rate of interest with no inflation adjustment which may be called the “money yield” to distinguish from the real yield, is i where: n

∑ Ct v it

k =1

k

k

=0

So the relationship between the real yield i ′ , the rate of inflation j and the money yield i is v i = v j v i ′ .

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Therefore 1 + i ′ =

1+ i
1+ j

⇒ i′ =

i− j
1+ j

This is sometimes approximated to i ¢ i - j if only estimates are required. However, you should only use this approximation if the question specifically asks for an estimate or an approximation since the “exact” method isn’t much more difficult.

Example
Ten years ago, a saver invested £5,000 in an investment fund operated by an insurance company. Over this period the rate of return earned by the fund has averaged 12% per annum. If prices have increased by 80% over this period, calculate the average annual real rate of interest earned by the fund over this period.

Solution
The average annual rate of inflation j over the 10 year period can be found from:
(1 + j )10 = 18
.

⇒ j = 181/10 − 1 = 0.0605
.

ie 6.05%

The average annual rate of interest is i = 012 . So the average annual real rate is:
.

i′ =

012 − 0.0605
.
= 0.0561
10605
.

ie 5.61%

The graph below shows the real growth over time of a single investment when the rate of inflation is (a) higher than, (b) equal to and (c) less than the rate of interest (taken to be
8% pa).
Real growth j=4% j=8%

j=12%

Time

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If an investment is achieving a positive real rate of return ( i > j ), then it is outstripping inflation. If it is achieving a negative real rate of return (i < j ), then it is falling behind inflation (which means that you would be better spending the money now, rather than
“investing” it).

Question 12.8
You are the investment manager for a pension scheme. The trustees have paid over a sum of £250,000, instructing you to invest it so as to obtain the highest real rate of return over the next 5 years. Your researchers have advised you that suitable investments are available in the following countries:
Expected rate of return

Expected inflation rate

Japan:

3% pa

1% pa

United Kingdom:

7% pa

3% pa

Malaysia:

15% pa

10% pa

Mexico:

25% pa

40% pa

Which country would you select? (Assume that the investment opportunities in each country are otherwise identical and ignore currency risks and all other factors that might influence your decision.)
In some cases a combination of known inflation index values and an assumed future inflation rate may be used to find the real rate of interest.

Question 12.9
For the last 10 years a man has paid £50 at the start of each month into a savings account that has achieved a real rate of interest of 3% per annum over this period. If inflation has been at a constant rate of 5% per annum, calculate the balance of the man’s account today.
Note that the calculations for real rates of interest are very similar to the calculations used for valuing compound increasing annuities (and, in fact, the formulae are identical). Formulae of this type, which involve applying an adjustment to the interest rate used in the calculations, arise frequently in actuarial work in situations where there is a second factor, apart from interest, influencing the growth of a fund.

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Question 12.10
To provide for her retirement a woman has decided to pay 5% of her annual salary, currently £30,000 per annum, into a special savings account at the start of each of the next 15 years. If the fund is expected to earn 8% interest each year and her salary is expected to increase by 6% each year, calculate the approximate amount of the fund at the end of the 15 years.

3.4

Payments related to the rate of inflation
Some contracts specify that the cashflows will be adjusted to allow for future inflation, usually in terms of a given inflation index.
The index-linked government security is an example. The actual cashflows will be unknown until the inflation index at the relevant dates are known. The contract cashflows will be specified in terms of some nominal amount to be paid at time t, say ct . If the inflation index at the base date is Q (0) and the relevant value for the time t payment is Q (t ) then the actual cashflow is:

Ct = c t

Q(t )
Q(0)

It is easy to show that if the real yield i ′ is calculated by reference to the same inflation index as is used to inflate the cashflows, then i ′ is the solution of the real yield equation: n ∑

Ct k

k =1

Q(0) t k v =0
Q (t k ) i ′



n

∑ ct v it′

k

k =1

k

=0

In other words we can solve the yield equation using the nominal amounts.
However, it is not always the case that the index used to inflate the cashflows is the same as that used to calculate the real yield. For example the index-linked
UK government security has coupons inflated by reference to the inflation index value 3 months before the payment is made. The real yield, however, is calculated using the inflation index at the actual payment dates.

As mentioned in Chapter 1, borrowers and investors may need to know the amount of individual payments in advance. The lag of 3 months in the UK means that the amount of the next coupon can be determined before it is received. Consider coupons arriving in January and July. The July coupon will use the index from April and the January coupon will use the index from the preceding October.

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This makes things a little more complicated and is illustrated in the example below.

Worked Example
A three-year index-linked security is issued at time 0. The security pays nominal coupons of 4% annually in arrear and is redeemed at par. The coupons and capital payment are inflated by reference to the inflation index value 3 months before the payment is made. The inflation index value 3 months before issue was 110. The table below shows the index values at other times.
Time

0

9
12

1

9
1 12

2

9
2 12

3

Index

112

115

116

118

119

122

125

Calculate the real yield if the price of the stock is £100.

Solution
First we must calculate the monetary amount of each payment using the inflation index values 3 months before the payment date. We will then express these amounts in terms of time 0 money units. The results are shown in the following table.
Time
Nominal payment (ie cash actually received) Payment in time 0 units

1



115
= 4.1818
110

4.18 ¥

112
= 4.0376
116

2



118
= 4.2909
110

4.29 ¥

112
= 4.0385
119

3

104 ×

122
= 115.35
110

115.35 ×

112
= 103.35
125

The real yield is then found by solving for i the equation of value:
100 = 4.0376v + 4.0385v 2 + 103.35v3 which gives, by trial and error or other method, a real yield of 3.8%.
We will look further at index-linked bonds in Section 4.

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3.5

CT1-12: Elementary compound interest problems

The effects of inflation
Consider the simplest situation, in which an investor can lend and borrow money at the same rate of interest i1 . In certain economic conditions the investor may assume that some or all elements of the future cashflows should incorporate allowances for inflation (ie increases in prices and wages). The extent to which the various items in the cashflow are subject to inflation may differ. For example, wages may increase more rapidly than the prices of certain goods, or vice versa, and some items (such as the income from rent-controlled property) may not rise at all, even in highly inflationary conditions.
The case when all items of cashflow are subject to the same rate of escalation j per time unit is of special interest. In this case we find or estimate ctj and ρ j (t ) , the net cashflow and the net rate of cashflow allowing for escalation at rate j per unit time, by the formulae:

ctj = (1 + j )t ct

ρ j (t ) = (1 + j )t ρ (t ) where ct and ρ (t ) are estimates of the net cashflow and the net rate of cashflow respectively at time t without any allowance for inflation. It follows that, with allowance for inflation at rate j per unit time, the net present value of an investment or business project at rate of interest i is:

NPV j (i ) =
=

where 1 + i0 =



∑ ct (1 + j )t (1 + i )−t + ∫0


∑ ct (1 + i0 )−t + ∫0

ρ (t )(1 + j )t (1 + i )−t dt

ρ (t )(1 + i0 )−t dt

1+ i i- j
, or i0 =
.
1+ j
1+ j

(3.1)

(3.2)

Remember that j is the rate of inflation, i0 is the real yield. If j is not too large, so that

1 + j is close to 1, we could use the approximation: i0 ª i - j

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Example
Consider a property developer who buys a property at time 0 for £90,000. He also spends £10,000 at time 0 to buy some materials he will use to develop the property.
Ignoring inflation, the investor thinks that the property will be worth £110,000 today if some improvements to the property are made.
The developer expects to make the improvements over the next year and sell the property at time 1. Inflation is 5% pa.
In one year’s time the property will be worth 1.05 ¥ 110, 000 .
The nominal yield of the project is such that:

90, 000 + 10, 000 =

1.05 ¥ 110, 000 fi i = 15.5%
1+ i

The real yield of the project is such that:

100, 000 =

110, 000 ¥ 1.05 fi i0 = 10%
1.05(1 + i0 )

These results are of considerable practical importance, because projects which are apparently unprofitable when rates of interest are high may become highly profitable when even a modest allowance is made for inflation.
It is, however, true that in many ventures the positive cashflow generated in the early years of the venture is insufficient to pay bank interest, so recourse must be had to further borrowing (unless the investor has adequate funds of their own). This in itself does not undermine the profitability of the project, but the investor would require the agreement of his lending institution before further loans could be obtained and this might cause difficulties in practice.

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Example
An entrepreneur is considering a business project that will be financed by an unrestricted loan (effectively an overdraft). The only outlay required is an initial cost of
£80,000. The income is received annually in arrears. At the end of the first year the income is expected to be £8,800 and inflation is expected to increase this each year thereafter by 10% pa compound. The entrepreneur may borrow and invest money at
12% pa interest.
If the project is expected to last for twelve years, calculate:
(i)

the largest overdraft held during the term of the project, and

(ii)

the net present value of the project at 12% pa effective.

Solution
Part (i)
Year 1
The interest due at the end of the first year is 012 × 80,000 = £9,600 . The income is
.
only £8,800 and so the overdraft increases to:
80,000 + 9,600 − 8,800 = £80,800

Year 2
The interest due at the end of the second year is 012 × 80,800 = £9,696 . The income is
.
only £9,680 and so the overdraft increases to:
80,800 + 9,696 − 9,680 = £80,816

Year 3
The interest due at the end of the third year is 012 × 80,816 = £9,697.92 whereas the
.
income is now £10,648 and so the overdraft will decrease.
The maximum overdraft is therefore £80,816.

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Part (ii)
The net present value equals:

NPV = -80, 000 + 8,800v + 8,800 ¥ 1.1v 2 +
= -80, 000 +

+ 8,800 ¥ 1.111 v12

8,800v(1 - 1.112 v12 )
= £5,555
1 - 1.1v

Question 12.11
What is the accumulated profit at the end of the twelve years?

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4

CT1-12: Elementary compound interest problems

Index-linked bonds
Index-linked bond cashflows are described in Chapters 1 and 11. The coupon and redemption payments are increased according to an index of inflation.

We have also already looked at an example involving index-linked securities in the previous section of this chapter.
Given simple assumptions about the rate of future inflation it is possible to estimate the future payments. Given these assumptions we may calculate the price or yield by solving the equation of value using the estimated cashflows.
For example, let the nominal annual coupon rate for an n-year index-linked bond be D per £1 nominal face value with coupons payable half-yearly, and let the nominal redemption price be R per £1 nominal face value. We assume that payments are inflated by reference to an index with base value Q (0) , such that the coupon due at time t years is:

D Q(t )
2 Q(0)
Then the equation of value, given an effective (money) yield of i per annum, and a present value or price A per £1 nominal at issue or immediately following a coupon payment, is:

A=

2n

D Q( k 2) k / 2
Q(n ) n vi + R vi Q(0)
Q(0)
k =1

∑2

We estimate the unknown value of Q (t ) using some assumption about future inflation and using the latest known value – which may be Q (0) . For example, assume inflation increases at rate jt per annum in the year t - 1 to t, then we have: Q(½) = Q(0)(1 + j1)½
Q(1) = Q(0)(1 + j1 )

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Example
An index-linked bond pays half-yearly coupons and is redeemable at par on
28 February 2023. The coupon paid on 28 February 2007 was £2.10. A non-taxpayer buys £100 nominal of the bond on 29 February 2007. Assuming that future inflation is
5.25% pa, how much should this investor pay in order to obtain a money rate of return of 10% pa?

Solution
The value of the payments can be obtained as follows:
P = 2.10 ¥ 1.05250.5 v 0.5 + 2.10 ¥ 1.0525v + 2.10 ¥ 1.05251.5 v1.5 +
+ 2.10 ¥ 1.052516 v16 + 100 ¥ 1.052516 v16
= 2.10 ¥ 1.05250.5 v 0.5

(1 - (1.05250.5 v 0.5 )32 )
0.5 0.5

1 - 1.0525

v

+ 100 ¥ 1.052516 v16

= 47.6642 + 49.3481 = £97.01
Alternatively, this could also be evaluated using annuities:
(2)
P = 4.20a16| ¢ + 100v ¢16

where the functions are calculated at i ¢ =

i - j 0.10 - 0.0525
=
= 4.51% .
1+ j
1 + 0.0525

Evaluating this we get:
(2)
P = 4.20a16| ¢ + 100v ¢16 = 4.20 ¥ 11.3486 + 100 ¥ 0.49348 = £97.01

As we have already mentioned earlier in this chapter…
It is important to bear in mind that the index used may not be the same as the actual inflation index value at time t that one would use, for example, to calculate the real (inflation-adjusted) yield. In the case of UK index-linked bonds, the payments are increased using the index values from 3 months before the payment date. Real yields would be calculated using the inflation index values at the payment date.

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Like equities, index-linked bonds (and fixed interest bonds) may be offered for sale “ex-dividend”. No new principles are involved in the valuation of exdividend index-linked bonds.

Question 12.12
What does it mean if a bond is offered for sale ex-dividend?

Question 12.13
An index takes the following values:
1/1/05:
1/4/05
1/7/05
1/10/05

121.2
122.8
123.1
123.6

1/1/06
1/4/06
1/7/06
1/10/06

123.9
124.2
124.4
124.9

1/1/07
1/4/07

125.2
126.0

An index-linked bond is purchased on 1/4/05 (when the remaining term is two years) for a price of £101 per £100 nominal. The bond is due to be redeemed at par. All coupon and redemption payments are linked to the inflation index three months prior to the payment date. The coupons on the bond are of nominal amount 4% pa payable halfyearly in arrears on 1st April and October every year.
Calculate the real yield obtained by the investor. You may ignore tax.

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Page 33

Capital gains tax
Capital gains tax is a tax levied on the difference between the sale or redemption price of a stock (or other asset) and the purchase price, if lower. In contrast to income tax, this tax is normally payable once only in respect of each disposal, at the date of sale or redemption.
This section largely relates to the effect of capital gains tax on the prices and yields of fixed interest securities. The principles can however be applied to other investments that are subject to such tax. In general we shall assume that the stock in question will be held to redemption. If a stock is sold before the final maturity date, the capital gains tax liability will in general be different, since it will be calculated with reference to the sale proceeds rather than the corresponding redemption money.

5.1

Valuing a loan with allowance for capital gains tax
Consider a loan that pays coupons at rate D per annum, payable 1/pthly in arrear, and with redemption amount R payable at time n .
At an effective rate of interest i per annum, the value of this loan with no allowance for tax is clearly:

A = Da( p ) + Rv in
|
n

at rate i per annum

The rate of interest on the transaction with no allowance for taxation is the gross rate of interest.

Question 12.14
What is another name for the gross rate of interest of a fixed interest security?
It is convenient to split the equation of value into the two parts:

A=I+K where I = Da ( p ) and K = Rv in . K is the value of the redemption payment and I
|
n

is the value of the coupon (or ‘interest’) payments.

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If the investor is liable to income tax at rate t1 , but is not liable to capital gains tax, the price to be paid for a net effective rate of interest of i per annum is A′ , say, where:

A ¢ = D(1 - t1)a ( p ) + Rv in = (1 - t1)I + K
|
n

If the investor is also subject to tax at rate t2 ( 0 < t2 < 1 ) on the capital gains, then let the new price payable be A′′ .
If there is no capital gain (ie if the redemption price R is less than the purchase price) then there is no additional tax liability, and the price paid is simply A′. (We are assuming that it is not permissible to offset the capital loss against any other capital gain: see example 5.3).
If there is a capital gain, then at the redemption date of the contract there is an additional liability of t2 (R − A′′) .
We already know how to test for a capital gain in the case where there is no capital gains tax liability, that is, we can test whether R > A′ by comparing

g (1 − t1) with i ( p ) .
If i ( p ) ≤ g (1 − t1) then A′ ≥ R and there is no capital gain and no capital gains tax liability due at redemption. Hence A′′ = A′ = (1 − t1)I + K .

i ( p ) > g (1 − t1) then A′ < R and there is a capital gain.
As t2 < 1,
A′ < R ⇒ A′′ < R , so there is a capital gains tax liability at redemption of t2 (R − A′′) .

If

In this case:

A′′ = D (1 − t1)a( p ) + Rv n − t2 (R − A′′)v n = (1 − t1)I + (1 − t2 )K + t2 A′′v n n =

(1 − t1)I + (1 − t2 )K
1 − t2v n

So, in general:

if i ( p ) ≤ g (1 − t1)
(1 − t1)I + K


A′′ = ⎨ (1 − t1)I + (1 − t2 )K if i ( p ) > g (1 − t1)

1 − t2v n


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So we carry out a capital gains test to establish if there is a capital gain before we can set up the equation of value.

Example
On 15 July 2007 an investor purchases a six-year stock with half-yearly coupon payments of 13% pa. The investor pays 33% tax on income and capital gains. What price should the investor pay for £100 nominal of the stock if she requires a yield of
10% per annum?

Solution
Carrying out the capital gains test:
(1 - t1 ) g = 0.67 ¥ 0.13 = 0.0871

i (2) @10% = 9.76%

Since i (2) > (1 - t1 ) g , there is a capital gain.
So the price P is the PV of the net proceeds (valued at 10% interest):
(2)
P = 0.67 ¥ 13a6| + 100v 6 - 0.33(100 - P)v 6

= 38.860 + 37.820 + 0.33v 6 P
Rearranging to find P:

P=

76.680
1 - 0.33v 6

= £94.23

Note that this price is lower than the price calculated in the example on page 5, because the investor in this example paid tax on capital gains as well as on income. Also note that the price paid is lower than the redemption value so there is a capital gain as we expected. The Actuarial Education Company

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Question 12.15
What price would the investor in the previous example have to pay to obtain a yield of
5% per annum?

5.2

Finding the yield when there is capital gains tax
An investor who is liable for capital gains tax may wish to determine the net yield on a particular transaction in which he has purchased a loan at a given price.
One possible approach is to determine the price on two different net yield bases and then estimate the actual yield by interpolation. This approach is not always the quickest method. Since the purchase price is known, so too is the amount of the capital gains tax, and the net receipts for the investment are thus known. In this situation one may more easily write down an equation of value which will provide a simpler basis for interpolation, as illustrated by the next example.

Example 5.1
A loan of £1,000 bears interest of 6% per annum payable yearly and will be redeemed at par after ten years. An investor, liable to income tax and capital gains tax at the rates of 40% and 30% respectively, buys the loan for £800. What is his net effective annual yield?

Solution
Note that the net income each year of £36 is 4.5% of the purchase price. Since there is a gain on redemption, the net yield is clearly greater than 4.5%.
The gain on redemption is £200, so that the capital gains tax payable will be £60 and the net redemption proceeds will be £940. The net effective yield pa is thus that value of i for which:
800 = 36a10| + 940v 10
If the net gain on redemption (ie £140) were to be paid in equal instalments over the ten-year duration of the loan rather than as a lump sum, the net receipts each year would be £50 (ie £36 + £14 ). Since £50 is 6.25% of £800, the net yield actually achieved is less than 6.25%. When i = 0.055 , the right-hand side of the above equation takes the value 821.66, and when i = 0.06 the value is 789.85.

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By interpolation, we estimate the net yield as:

i = 0.055 +

821.66 − 800
0.005 = 0.0584
82166 − 789.85
.

The net yield is thus 5.84% per annum.
Alternatively, we may find the prices to give net yields of 5.5% and 6% per annum. These prices are £826.27 and £787.81, respectively. The yield may then be obtained by interpolation. However, this alternative approach is somewhat longer than the first method.

Question 12.16
Explain why the price that gives a net yield of 5.5% isn’t just the RHS (@5.5%) of the equation of value given above which equals 821.66.
Check that the price does in fact equal £826.27.

5.3

Optional redemption dates
In Section 1.5 we considered the consequences for a lender, subject only to income tax, of the borrower having a choice of redemption dates. In fact even with the additional complication of capital gains tax the strategy adopted should be the same.
The intuition for this is simple. We ensure a certain rate of return by assuming a worst case result for the investor – and the investor then should pay no more than this worst case value.
The worst case differs according to if:
(i)

i ( p ) ≤ g (1 − t1) then there is a capital loss and the worst case is that the redemption money is paid at the earliest possible date.

(ii)

i ( p ) > g (1 - t1) then there is a capital gain and the worst case is that the redemption money is paid as late as possible.

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A capital gains tax liability does not change any of this – an investment which has a capital gain before allowing for capital gains tax must still have a net capital gain after allowing for the capital gains tax liability, so that the “worst case” for the investor is still the latest redemption. If there is a capital loss then the situation is exactly as in Section 1.5.
In every case, with or without capital gains tax, the “worst case” method for valuing optional redemption date bonds is to calculate the minimum possible value for a given yield, and that is the maximum price that should be paid to ensure that the yield earned is at least the required yield. The problem then becomes that of determining the “worst case” conditions.

Question 12.17
A fixed interest stock with a coupon of 8% per annum payable half yearly in arrears can be redeemed at par at the option of the lender at any time between 10 and 15 years from the date of issue.
What price should an investor subject to tax at 25% on income and capital gains, who wishes to obtain a net yield of at least 7% per annum, pay for £100 nominal of this stock? In some cases, for example if the redemption price varies, the simple strategy described above will not be adequate, and several values may need to be calculated to determine which is lowest.

5.4

Offsetting capital losses against capital gains
Until now we have considered the effects of capital gains tax on the basis that it is not permitted to offset capital gains by capital losses. In some situations, however, it may be permitted to do so. This may mean that an investor, when calculating his liability for capital gains tax in any year, is allowed to deduct from his total capital gains for the year the total of his capital losses (if any). If the total capital losses exceed the total capital gains, no “credit” will generally be given for the overall net loss, but no capital gains tax will be payable.

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Example 5.2
Suppose that in a particular tax year an investor sold the following two assets, both of which were purchased some years ago.
Asset A

Sold for £1,865 (Purchase price £1,300)

Asset B

Sold for £500 (Purchase price £900)

The sale of asset A produces a capital gain of £565 while for asset B there is a capital loss of £400.
In the absence of both the right to offset losses against gains and of indexation the sales of these two assets lead to an overall capital gain of £565. (The loss of
£400 is simply regarded as a zero capital gain.)

The indexation of capital gains is covered in the next section.
If the offsetting of losses against gains is allowed, the sales of these two assets lead to an overall capital gain of £165 (ie £565 - £400 ).

In some countries investors may also have a capital gains allowance. Investors do not have to pay CGT on their allowance and so it acts to reduce the tax bill.
For example if the investor in the above example had an allowance of £150 and offsetting is allowed then the overall capital gain would only have been £15.

5.5

The indexation of capital gains
When an asset is sold at a profit, it may be permitted to reduce the capital gain for taxation purposes by determining the amount of the capital gain not by reference to the actual purchase price but by reference to a (greater) “notional” purchase price. The notional purchase price is the actual purchase price increased in line with an approved index. This practice is generally referred to as the “indexation of gains”.

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CT1-12: Elementary compound interest problems

Example 5.3
In Example 5.2, suppose that indexation is also allowed and that over the period during which the investor owned asset A the approved index increased by 18%.
In this case the “notional” purchase price of asset A is £1,534 (ie £1,300 ¥ 1.18 ) and the capital gain arising on the sale of this asset is reduced to £331 (ie
£1,865 - £1,534 ). If no offsetting of losses is allowed, this last amount will be the total capital gain arising from the sale of the two assets. If, however, offsetting is permitted, there is zero overall capital gain – since the loss of £400 on the sale of asset B exceeds the (reduced) gain arising from the sale of asset A.
It is perhaps worth pointing out that indexation of losses is not usually permitted. Thus, for example, if over the period during which the investor owned asset B the approved index increased by 12%, the loss on the sale of this asset is still considered to be £400. It is not taken as £508 (ie £900 × 1.12 − £500 ).

Therefore indexation can often be used to reduce capital gains but it can not usually be used to increase capital losses.

Question 12.18
In 2007, an investor with a capital gains allowance of 530 sold two assets.
The first asset was purchased on 1/7/1988 for 3,000 and sold on 1/1/2007 for 20,000.
The second asset was purchased on 31/12/2006 for 1,450 and sold on 31/12/2007 for
970.
If indexation of gains by inflation and offsetting capital losses are allowed and inflation has been a constant 3% pa for the last twenty years, calculate the taxable capital gain in
2007.

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Chapter 12 Summary
The price for a fixed interest security can be calculated as the present value of the interest and redemption payments. When the proceeds are subject to income tax or capital gains tax, the net payments must be used. The formulae used for calculating the price are:
(
Ignoring tax: A = Dan2 ) + Rv n
|

Allowing for income tax:

(
A′ = D(1 − t1 )an2) + Rv n
|

(
Allowing for income and capital gains tax: A′′ = D(1 − t1 )an2 ) + Rv n − t2 ( R − A′′)v n
|

Based on a given purchase price, the running yield and the redemption yield can be calculated, on either a gross or a net basis.
The running yield is the coupon divided by the price. The redemption yield is the rate of interest that equates the price with the present value of the interest and redemption payments. If a security has optional redemption dates and an investor wishes to achieve a yield of at least i, then it should value the security on the assumption that will give the lowest yield, ie the worst case. The worst case is the latest possible date for a gain and the earliest possible date for a loss.
Equities are usually valued by assuming that dividends increase at a constant rate and continue in perpetuity. If A is the value of an equity paying annual dividends just after a dividend payment, then:
A=

D(1 + g ) where D is the amount of the dividend just paid i−g Property is valued in a similar way although rents are generally fixed for a number of years at a time and some contracts are for a fixed term. The equation of value for property is:


1
A = ∑ m Dk / m v k / m where k =1

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The real rate of interest of a transaction is the rate of interest after allowing for the effect of inflation on a payment series:

i¢ =

i- j
1+ j

i- j

Index linked bonds can be valued by allowing for the increases in the coupons and the redemption payment. The equation of value for an index linked bond is:
2n

D Q( k 2) k / 2
Q( n) n vi + R vi where Q(t ) is the index value at time t
Q(0)
k =1 2 Q( 0)

A=∑

Capital gains tax is a tax levied on the difference between the sale or redemption price of a stock (or other asset) and the purchase price, if lower.
Sometimes it is possible to offset capital losses against capital gains or to reduce the capital gain by the inflation over the period of the gain. This latter practice is generally referred to as the “indexation of gains”.

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Chapter 12 Solutions
Solution 12.1 i ′ = 11½ − 1 = 4.8809%
.

A = 6.5a12| + 100v12

@ i ¢%

1 - 1.048809-12 fi A = 6.5 ¥
+ 100 ¥ 1.048809 -12 = 114.45
0.048809

Solution 12.2
(i)

This investor has paid £120 for a £100 nominal holding of the stock which will pay a total dividend of £13 during the coming year. So the running yield is:

13
× 100% = 10.8%
120
(ii)

We also require the net redemption yield. This will be lower than the gross redemption yield. The new equation of value is:
(2)
(2)
120 = 13 ¥ 0.75a6| + 100v6 = 9.75a6| + 100v6
(2)
At 6%: 9.75a6| + 100v 6 = 9.75 ¥ 4.9900 + 100 ¥ 0.70496 = 119.15
(2)
At 5%: 9.75a6| + 100v 6 = 9.75 ¥ 5.1384 + 100 ¥ 0.74622 = 124.72

We can approximate i by interpolating (linearly) using these two values:

i -5
120 - 124.72
ª
fi i
6 - 5 119.15 - 124.72

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5.8%

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CT1-12: Elementary compound interest problems

Solution 12.3
(i)

If the purchase price is £80 we must find i that solves the equation:
80 = 100v n

(a)

If n = 5 , i = 4.6% pa.

(b)

If n = 10 , i = 2.3% pa.

This implies that if there is a capital gain then the sooner the bond is redeemed, the higher the yield.
(ii)

If the purchase price is £120 we must find i that solves the equation:
120 = 100v n

(a)

If n = 5 , i = −3.6% pa.

(b)

If n = 10 , i = −1.8% pa.

This implies that if there is a capital loss then the later the bond is redeemed, the higher the yield.

Solution 12.4
Here there will be a loss on redemption. So the earliest redemption date is the worst case scenario and will give the minimum yield.
So:
110 = 0.75 × 8a ( 2|) + 100v 10
10

i = 5% gives 108.29 and i = 4½% gives 112.40. Interpolating gives:

i − 4.5
110 − 112.40
=
⇒ i = 4.8% .
5 − 4.5 108.29 − 112.40

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Solution 12.5
The yield can be found from the equation:
1

120 = v 12

8 ¥ 1.042 ¥ 0.65v 8.32 ¥ 0.65 ¥ 1.04
=
1 - 1.04v
(1 + i )1/12 (i - 0.04)

Using trial and error this gives i = 8.7%

Solution 12.6
120 at time 1 has a value of

120
9 /12

105
.

1033/12
.

= 114.84 in terms of time 0 money units.

Therefore the real rate of return is 14.84%.

Solution 12.7
No solution necessary.
This question may have been very easy but the more you use your calculator for actuarial calculations, the quicker you will get. This will then leave more time in the exam for thinking.

Solution 12.8
Country
Japan:
UK:
Malaysia:
Mexico:

Expected rate of return
3% pa
7% pa
15% pa
25% pa

Expected inflation rate Real rate of return
1% pa
1.98% pa
3% pa
3.88% pa
10% pa
4.55% pa
40% pa
–10.71% pa

So Malaysia is expected to provide the highest real rate of return.

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CT1-12: Elementary compound interest problems

Solution 12.9
The real rate of interest was 3% and inflation has been 5%. Hence if the actual rate of i - 0.05 interest was i , then 0.03 = fi i = 8.15% . So the accumulated value of the
1.05
man’s account today will be:
(12
600s10| ) = 600 ×

.
1081510 − 1
= 600 × 15.2265 = £9,136
0.07809

Solution 12.10
Consider the present value at the outset of the payment that will be made at time t .
This will be:
0.05 × 30,000 ×

106 t
.
108 t
.

This expression contains a 6% factor on the top and an 8% factor on the bottom ie the payments are being discounted at a net rate of approximately 2% per annum. So we can find the approximate total present value of all the payments by using an annuity factor based on 2% interest:
@2%
15|

PV = 1,500a

The accumulated value at the end of the 15 years can then be calculated as:
@
Acc.V = 1,500a15| 2% ¥ 1.0815 = 1,500 ¥ 13.1062 ¥ 3.1722 = £62, 400

In this example, the other factor involved was salary inflation, rather than price inflation. (If you’ve used the “exact” formula you should get £62,824 – but the question did ask you to find the approximate amount.)

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Solution 12.11
The accumulated profit is easily found from the net present value:

AP = 5,556 ¥ 1.1212 = £21, 643

Solution 12.12
If a bond is offered for sale ex-dividend then the purchaser will not receive the next coupon. Solution 12.13
We will work with £100 nominal. The coupons are £4 pa payable half-yearly, which means that each payment is £2. We first need to calculate the actual coupon and redemption payments received:
Coupon 1/10/05:



index1/ 7 / 05
123.1
= 2¥
= 2.031353
121.2
index1/1/ 05

Coupon 1/4/06:



index1/1/ 06
123.9
= 2¥
= 2.044554 index1/1/ 05
121.2

Coupon 1/10/06:



index1/ 7 / 06
124.4
= 2¥
= 2.052805 index1/1/ 05
121.2

Coupon 1/4/07:



index1/1/ 07
125.2
= 2¥
= 2.066007
121.2
index1/1/ 05

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Redemption 1/4/07:

100 ¥

index1/1/ 07
125.2
= 100 ¥
= 103.30033 index1/1/ 05
121.2

Alternatively, you could have combined these last two payments since they are made at the same time.
We can then set up the equation of value using the actual coupon and redemption payments, but taking into account inflation. This time there is no lag in the inflation index: 1

101 = 2.031353v 2 ¥

122.8
122.8
11 122.8
+ 2.044554v ¥
+ 2.052805v 2 ¥
123.6
124.2
124.9

+ 105.366337v 2 ¥
1

122.8
126.0
11

= 2.018205v 2 + 2.021508v + 2.018291v 2 + 102.690366v 2
As a first guess, we can try the nominal coupon rate 4%.
At 4%, the right hand side is 100.7688.
At 3%, the right hand side is 102.6775.
Interpolating:

i -3
101 - 102.6775
=
fi i = 3.9%
4 - 3 100.7688 - 102.6775
So the real yield is approximately 3.9%.

Solution 12.14
Gross redemption yield.

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Solution 12.15
Carrying out the capital gains test: i (2) = 4.939%

(1 - t1 ) g = 0.13 ¥ 0.67 = 8.71%

Since i (2) < (1 - t1 ) g , there is no capital gain.
So the price P is the present value of the net proceeds (valued at 5% interest):
(2)
P = 0.67 ¥ 13a6| + 100v 6

= 0.67 ¥ 13 ¥ 5.1384 + 100 ¥ 0.74622
= £119.38
Notice that P is greater than £100 and hence no CGT is payable which confirms the findings of the test.

Solution 12.16
The RHS of the equation of value, 36a10| + 940v10 , is the present value of the benefits if the price was £800. If the price changes, the capital gains tax will also change and so the net redemption payment is not £940.
The price is found from the equation:
P = 36a10| + [1, 000 - 0.3(1, 000 - P)]v10 fiP= 36 ¥ 7.5376 + 700 ¥ 0.58543
= £826.27
1 - 0.3 ¥ 0.58543

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Solution 12.17
Carrying out the capital gains test: i (2) = 6.88%

(1 - t1 ) g = 6%

Since i (2) > (1 - t1 ) g , there is a capital gain. However the worst case for the investor is the latest redemption date.
In this question it doesn’t matter who has the option to redeem. If the borrower has the option, because the question is asking for the price that gives the investor a yield of at least 7%, then the worst case for the investor would still have been the latest redemption date. So:
(2)
P = 0.75 ¥ 8a15| + 100v15 - 0.25(100 - P)v15 @ 7%
(2)
= 6a15| + 75v15 + 0.25 Pv15

=

55.59 + 27.18
= £91.02
0.9094

Solution 12.18
The notional purchase price of the first asset is 3,000 × 10318.5 = 5,183 .
.
The capital gain is therefore 20,000 − 5,183 = 14,817 .
There is a capital loss for the second asset of 480.
Therefore the taxable gain is 14,817 − 480 − 530 = 13,807 .

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CT1-13: Arbitrage and forward contracts

Page 1

Chapter 13
Arbitrage and forward contracts
Syllabus objective
(xii)

Calculate the delivery price and the value of a forward contract using arbitrage free pricing methods.
1.

Define “arbitrage” and explain why arbitrage may be considered impossible in many markets.

2.

Calculate the price of a forward contract in the absence of arbitrage assuming: •

no income or expenditure associated with the underlying asset during the term of the contract



a fixed income from the asset during the term



a fixed dividend yield from the asset during the term

3.
4.

0

Explain what is meant by “hedging” in the case of a forward contract.
Calculate the value of a forward contract at any time during the term of the contract in the absence of arbitrage, in the situations listed in 2 above. Introduction
Arbitrage is the simultaneous buying and selling of two economically equivalent but differentially priced portfolios (collections of assets) so as to make a risk-free profit.
You may think that this sounds too good to be true which is why an assumption is often made that arbitrage opportunities do not exist.

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This chapter shows how this assumption can be used to value forward contracts and to calculate the price at which securities will be traded under the terms of a forward contract. © IFE: 2009 Examinations

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CT1-13: Arbitrage and forward contracts

1

The “No Arbitrage” assumption

1.1

Page 3

Introduction
Arbitrage in financial mathematics is generally described as a risk-free trading profit. More accurately, an arbitrage opportunity exists if either:
(a)

an investor can make a deal that would give her or him an immediate profit, with no risk of future loss; or: (b)

an investor can make a deal that has zero initial cost, no risk of future loss, and a non-zero probability of a future profit.

Question 13.1
For each of the following, state whether or not there is an arbitrage opportunity.
(i)

An individual wishing to make a telephone call is desperate for some small change and so offers you a five pound note in return for four pounds of small change. (ii)

USA are playing Luxembourg in a basketball match. A friend is so confident in his team that he says that he will pay you £400 if USA lose.

(iii)

Coventry City football club has just paid £800,000 for a Croatian footballer.
Real Madrid of Spain is happy to pay £4m for the same player.

(iv)

A friend offers to pay you £100 if one roll of a fair dice is a six. If it is a one, two, three, four or five you will have to pay her £1.

(v)

As (iv) but another friend is also prepared to pay you, based on a subsequent roll of the same dice, £1 for a one, £2 for a two, £3 for a three, £4 for a four, £5 for a five and you must pay him £12 for a six.

(vi)

As (v) but the friends are prepared to make the above payments on the same roll of a dice.

(vii)

Lend £1 for six months, receive £1.05 back with no risk of default and simultaneously borrow £1 for six months, paying £1.04 back.

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CT1-13: Arbitrage and forward contracts

The concept of arbitrage is very important because we generally assume that in modern developed financial markets arbitrage opportunities do not exist. This assumption is referred to as the “No Arbitrage” assumption, and is fundamental to modern financial mathematics.
If we assume that there are no arbitrage opportunities in a market, then it follows that any two securities or combinations of securities that give exactly the same payments must have the same price. This is sometimes called the “Law of One
Price”.

If two securities with identical payments have different prices then an arbitrage opportunity exists. Arbitrageurs would spot this opportunity and buy the cheaper security and sell the dearer security so as to make an immediate, risk-free profit. The arbitrage position would soon disappear however because of the increased demand for the cheaper security and the lack of demand for the dearer security. This would force the security prices back into line.
The ideas are demonstrated in the following example.

Example 1
Consider a very simple securities market, consisting of two securities, A and B.
A
B
At time t = 0 the prices of the securities are P0 and P0 respectively. The term

of both the securities is 1 year. At t = 1 there are two possible outcomes. Either
B
the “market” goes up, in which case security A pays P1A (u ) and B pays P1 (u ) , or
B
it goes down, with payments P1A (d ) and P1 (d ) respectively.

Investors can buy securities, in which case they pay the time 0 price and receive the time 1 income, or they can sell securities, in which case they receive the time
0 price and must pay the time 1 outgo.

If you sell something then, compared with the alternative of “not selling”, you are better off (at time 0) by the amount of the sale price and worse off (at time 1) by the value of the future proceeds from the asset. The cashflows associated with selling an asset are the opposite of those associated with buying an asset.
Now, assume first that we have the following payment table:

Security:

Time 0 price P0

A
B

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£
6
11

Market goes up
P1(u )

Market goes down
P1(d )

£
7
14

£
5
10

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There is an arbitrage opportunity here. An investor could buy one unit of security B and sell two units of security A. This would give income at time 0 of
£12 from the sale of security A and an outgo of £11 from the purchase of security
B – which gives a net income at time 0 of £1. At time 1 the outgo due on the portfolio of 2 units of security A exactly matches the income due from security B, whether the market moves up or down. Thus, the investor makes a profit at time
0, with no risk of a future loss.
It is clear that investment A is unattractive compared with investment B. This will cause pressure to reduce the price of A and to increase the price of B, as there will be no demand for A and an excessive demand for B. Ultimately we
A
B would achieve balance, when P0 = P0 / 2 , when the arbitrage opportunity is eliminated, and the prices are consistent.

The situation where P0A = P0B / 2 satisfies the “law of one price” explained earlier.

Example 2
Another example is given in the following table:

Security:
A
B

Time 0 price P0
£
6
6

Market goes up
P1(u )

Market goes down P1(d )

£
7
7

£
5
4

An arbitrage opportunity exists, as an investor could buy one unit of A and sell one unit of B. The net income at time 0 is £0, as the income from the sale of B matches the outgo on the purchase of A. At time 1 the net income is £0 if the market goes up, and £1 if the market goes down. So, for a zero investment, the investor has a possibility of making a profit (assuming the probability that the market goes down is not zero) and no possibility of making a loss.
With these prices, investors will naturally choose to buy investment A and will want to sell investment B. This will put pressure on the price of A to increase, and on the price of B to decrease. The arbitrage opportunity is eliminated when
A
B
P0 > P0 .

The law of one price is not directly relevant to this example because no two combinations of A and B give the same payouts.

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1.2

CT1-13: Arbitrage and forward contracts

Why do we assume “No Arbitrage”?
The “No Arbitrage” assumption is very simple and very powerful. It enables us to find the price of complex instruments by “replicating” the payoffs. This means that if we can construct a portfolio of assets with exactly the same payments as the investment that we are interested in, then the price of the investment must be the same as the price of the “replicating portfolio”.
In practice, in the major developed securities markets arbitrage opportunities, when they do arise are very quickly eliminated as investors spot them and trade on them. Such opportunities are so fleeting in nature, according to the empirical evidence, that it is sensible, realistic and prudent to assume that they do not exist. We also assume here that there are no transaction costs or taxes associated with buying, selling or holding assets. These are also idealised assumptions, but they enable us to develop a methodology that may be adapted to deal with these institutional features if necessary.
The “No Arbitrage” assumption will be used extensively in Subject CT8,
Financial Economics. In this subject we introduce the ideas in the context of
“Forward Contracts”.

Question 13.2
State the two alternative conditions required for the existence of an arbitrage opportunity. Question 13.3
If the law of one price is satisfied, do all investments give the same expected returns?

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2

Forward contracts

2.1

Page 7

Introduction
A forward contract is an agreement made at some time t = 0 , say, between two parties under which one agrees to buy from the other a specified amount of an asset (denoted S) at a specified price on a specified future date. The investor agreeing to sell the asset is said to hold a short forward position in the asset, and the buyer is said to hold a long forward position.

Question 13.4
Forward contracts were mentioned earlier in the course when we introduced derivatives.
Define a derivative and name three other types of derivatives.
Let Sr be the price of the underlying asset (for example, a unit of equity stock) at time r. The price will not generally be predictable – for example, we may contract to buy shares in a company in 6 months time. We know what the current price of the shares is, but the price will vary more or less continuously, so we do not know with certainty what the share price will be at any future date.
Let K be the price agreed at time t = 0 to be paid at time t = T , called the
“forward price”. t = 0 is the time the forward contract is agreed, T is the time the contract matures (that is, when the sale actually happens). We also assume there is a known force of interest d that is available on a risk-free investment over the term of the contract. This is known as the “risk-free” force of interest.

The risk-free force of interest is likely to be based on the yield obtainable by borrowing or lending a fixed-interest security issued by a major, developed country’s government for the same term as the forward contract.
At time 0 when the agreement is made no money changes hands (except possibly a “good faith” deposit – we will ignore this here). The price K agreed at time t = 0 is determined such that the value of the forward contract at time t = 0 is zero.

Methods for calculating the forward price are discussed in the next and subsequent sections of this chapter.
The forward contract will generally have non-zero value at time T; if K > ST then the seller receives K for an asset worth (at that time) ST , and has made a profit at time T of K − ST . Similarly, if K < ST then the buyer has paid K for an asset worth ST , giving the buyer a profit at time T of ST − K .

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CT1-13: Arbitrage and forward contracts

Example
Investor I agrees to sell 1,500 Coca-Cola shares in six months to Investor J at a price of
£1.40 per share. The current price is £1.33 per share.
£1.40 is the forward price.
Six months later the share price is in fact £1.35.
Investor I has to sell 1,500 shares, which are currently worth £2,025, for £2,100. He therefore makes a profit from the futures contract of £75.
Similarly, Investor J has to buy the shares for a loss of £75.

2.2

Calculating the forward price for a security with no income
One important question is how to determine the forward price, K. This is the price agreed at time t = 0 but not actually paid until the contract ends, at t = T .
The price Sr is uncertain for all r > 0 .

However, using the no arbitrage

assumption we can find the forward price without having to make any assumption about the statistical properties of the process Sr . Instead, we can use a replication argument.
We assume at this stage that there are no payments or costs associated with holding the stock (eg dividends, custody fees).
Consider the following two investment portfolios:
Portfolio A:

Enter a forward contract to buy one unit of an asset S, with forward price K, maturing at time T; simultaneously invest an amount
Ke −δ T in the risk-free investment.

Portfolio B:

Buy one unit of the asset, at the current price S0 .

At time t = 0 the price of Portfolio A is Ke −δ T for the risk-free investment; recall that the price of a forward contract is zero.
The price of Portfolio B is S0 .

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At time t = T the cashflows for Portfolio A are:
An amount K is received from the risk-free investment ( Ke −δ T invested at force of interest δ for T years gives an accumulated value of K). The same amount K is paid on the forward contract. Receive 1 unit of asset S.
The payout from Portfolio B is one unit of asset S.
Now, the future cashflows of portfolio A are identical to those of portfolio B – both give a net portfolio of one unit of the underlying asset S. The no arbitrage assumption states that when the future cashflows of two portfolios are identical, the price must also be the same – that is:

Ke − δT = S0



K = S0e δT

So the forward price for a security with no income is the accumulated value at time T of the price of the underlying security when the contract was agreed. Note that the accumulation is at the risk-free interest rate, not the rate of return we expect to get on the assets.
The no arbitrage assumption gives the price for the forward contract with no need for any model of how the asset price St will actually move over the term of the contract.

Question 13.5
A three-month forward contract exists in a zero-coupon corporate bond with a current price per £100 nominal of £42.60. The yield available on three-month government securities is 6% pa effective. Calculate the forward price.

2.3

Calculating the forward price for a security with fixed cash income Assume now that at some time t1 , 0 < t1 < T , the security underlying the forward contract provides a fixed amount c to the holder. For example, if the security is a government bond, there will be fixed coupon payments due every six months.
Now consider the following two portfolios:
Portfolio A:

Enter a forward contract to buy one unit of an asset S, with forward price K, maturing at time T; simultaneously invest an amount

Ke − δT + ce − δt1 in the risk-free investment.

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CT1-13: Arbitrage and forward contracts

We are now assuming that the risk-free force of interest is constant throughout the term of the contract so that δ is appropriate for a term of T and a term of t1 . If this was not the case, we would simply use two different forces of interest corresponding to the appropriate time periods.
Portfolio B:

Buy one unit of the asset, at the current price S0 . At time t1 invest the income of c in the risk-free investment.

At time t = 0 the price of Portfolio A is Ke − δT + ce − δt1 for the risk-free investment, and zero for the forward contract.
The price of Portfolio B is S0 .
At time t = T the payout from Portfolio A is: Income of K + ce δ (T − t1 ) from the risk-free investment; Outgo of K on the forward contract. Receive 1 unit of asset, value ST . The net portfolio at T is one unit of the asset S plus ce δ (T − t1 ) units of the risk-free security.
The payout from Portfolio B is one unit of the asset, value ST , plus ce δ (T − t1 ) units of the risk-free security, from the invested coupon payment.
The net cashflows of portfolio A at time T are identical to those of portfolio B – both give a net portfolio of one unit of the underlying asset S plus ce δ (T − t1 ) units of the risk-free security. Using the no arbitrage assumption the prices must also be the same – that is:

Ke −δ T + ce −δ t1 = S0



K = S0eδ T − ceδ (T − t1 )

So the forward price is equal to the accumulated value at time T of the current price, less the accumulated value at time T of the income payment, which will not be received by the buyer of the forward contract.

Example
A fixed interest security pays coupons of 8% pa half-yearly in arrear and is redeemable at 110%. Two months before the next coupon is due, an investor negotiates a forward contract in which he agrees to buy £60,000 nominal of the security in six months’ time.
The current price of the stock is £80.40 per £100 nominal and the risk-free force of interest is 5% pa.
Calculate the forward price.

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Solution
Using the formula given above, the forward price is equal to:

K = 80.4 ×

60,000 0.05×½ e − ½ × 0.08 × 60,000e0.05×( 6/12 −2 /12) = £47,021
100

The redemption rate (110%) is not required, only the price.
For a long forward contract on a fixed interest security there may be more than one coupon payment. It is easy to adapt the above method to allow for this. If we let I denote the present value at time t = 0 of the fixed income payments due during the term of the forward contract, then the forward price at time t = 0 per unit of security S is

K = (S0 − I )e δT

The word ‘long’ here is being used to say that the term is long. The word ‘long’ has a different meaning later on in this chapter.
So the forward price is equal to the current price, less the present value at time 0 of the income payments during the term of the contract, accumulated to time T .

Question 13.6
Consider the security in the last example. On the same day, a different investor negotiates a forward contract to purchase £50,000 nominal of the stock in ten months’ time. Calculate the forward price of this contract.

2.4

Calculating the forward price for a security with known dividend yield
We will now consider equities. The dividend yield for an equity is defined to be:

Dividend yield =

Dividend per share
Price per share

Let D be the known dividend yield per annum. We assume that dividends are received continuously, and are immediately reinvested in the security of S.

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CT1-13: Arbitrage and forward contracts

This has the effect that the number of shares will increase by a constant force, D. This rate of increase will not be affected by changes in the share price. If the share price increases then the value of dividends will increase by the same proportion.
If we start with one unit of the security at time t = 0 , the accumulated holding at time T would be e DT units of the security. This is because the number of units owned is continuously compounding at rate D per annum for T years. If instead of 1 unit at time t = 0 we hold e -DT units, reinvesting the dividend income, at time T we would hold e DT e - DT = 1 unit of the security.
Now consider the following two portfolios:
Portfolio A:

Enter a forward contract to buy one unit of an asset S, with forward price K, maturing at time T; simultaneously invest an amount

Ke -d T in the risk-free investment.
Portfolio B:

Buy e -DT units of the asset S, at the current price S0 . Reinvest dividend income in the security S immediately it is received.

At time t = 0 the price of Portfolio A is Ke -d T for the risk-free investment, and zero for the forward contract.
The price of Portfolio B is e -DT S0 .
At time t = T the cashflows of Portfolio A are: an amount K is received from the risk-free investment. Outgo K is paid on the forward contract. Receive 1 unit of asset S. The net portfolio at T is one unit of the asset S.
The payout from Portfolio B is one unit of the asset S.
The net cashflows of portfolio A at time T are identical to those of portfolio B – both give a net portfolio of one unit of the underlying asset S. Using the no arbitrage assumption the prices must also be the same – that is:

Ke −δ T = S0e − DT

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K = S0e (δ − D )T

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Example
The dividend yield of a portfolio of shares with current price of £673,000 is 2.8% pa payable continuously. Calculate the forward price of a one-year forward contract, based on the portfolio, if we assume dividends are received continuously and the risk-free rate of interest is 4.6028% pa effective.

Solution
4.6028% effective is equivalent to a force of interest of 4.5%.
Therefore K = 673,000e 0.045−0.028 = £684,539 .

Question 13.7
The current share price of a stock is £200. Dividends are paid continuously and the current dividend is £5 pa. Calculate the forward price of a five-year contract on the asset if the risk-free force of interest is 5% pa.
It is simple to adjust the portfolios to get the forward price if the dividends are paid discretely.

Question 13.8
Deduce a formula for the forward price, K, for an equity forward contract. Assume a constant dividend yield D, and that dividends are received at the end of each year and are immediately reinvested. The term of the forward contract T is a whole number of years. The important principle for this case and the known income case is that, if the income is proportional to the underlying security, S, we assume the income is reinvested in the security. If the income is a fixed amount regardless of the price of the security at the payment date, then we assume it is invested in the risk-free security. This is because when the payment is proportional to the stock price
(eg dividends) we know how many units of stock they will purchase, but we do not know how much cash is paid (as the stock price is unknown). So we can predict the amount of stock held at the end if we assume reinvestment in the stock. The Actuarial Education Company

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CT1-13: Arbitrage and forward contracts

With a cash payment on the other hand, we would not know how much stock could be bought, but we do know how much the cash would accumulate to at the risk-free force of interest. Assuming dividends are reinvested in the security, but cash is invested at the risk-free (and known) force of interest enables us to predict the final portfolio without requiring any information about the price of the asset S during the course of the contract.

2.5

Hedging
Hedging is a general term which describes the use of financial instruments
(including stocks, bonds, forward contracts and more complex financial contracts such as options) to reduce or eliminate a future risk of loss.
An investor who agrees to sell an asset at a given price in a forward contract need not hold the asset at the start of the contract. However, by the end of the contract he or she must own the asset ready to sell under the terms of the forward contract. If the investor waits until the end of the contract to buy the asset S the risk exists that the price will rise above the forward price, and they will have to pay more than the forward price K that they receive for the asset. On the other hand, if they buy the asset at the start of the forward contract, and hold it until the contract matures, there is a risk that the price will have fallen, and they have paid more than they needed to.

We are assuming here that there is no interest or dividend income associated with the asset S.

Question 13.9
What does the forward price equal in terms of the price of the asset at the start of the contract, S0 ?
In fact the price does not have to fall for the investor to lose out. If the price rises at a lower rate than the risk-free force of interest then the investor would have been better off waiting, and buying the asset at a later date.

Example
An investor agrees to sell an asset with no income in five years’ time. The current price of the asset is 100 and the risk-free force of interest is 5%. We assume that the investor borrows and invests money at the risk-free rate.

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The forward price must be 100e5×0.05 = 128.40 . This is the amount that the investor will receive at time 5 for selling the asset.
In order to sell the asset, the investor first needs to buy the asset.
Suppose the investor chooses to buy the asset now and so the accumulated profit at time
5 is 0.
If the price of the asset at time 5 turns out to be 125.80, then the investor would have made a profit if she had waited and purchased the asset at time 5. The accumulated profit would have been 128.40 − 12580 = 2.60 .
.

Question 13.10
Suppose that the price of the asset in the above example turned out to be 105.00 at time 2 and 128.40 at time 5. Was the investor wise to buy the asset at time 0?
To hedge the risk the investor could borrow an amount Ke -d T at the risk-free force of interest, buy the asset S at the start of the contract, at the price S0 , and hold it until it is to be handed over at time T. The price of this “hedge portfolio” is -Ke -d T + S0 = 0 .
At time T the investor owes K that is exactly covered by the forward price received at T. He or she also owes one unit of asset S under the forward contract, which is also paid from the hedge portfolio.
This way, if the investor holds the hedge portfolio he or she is certain not to make a loss on the forward contract. There is also no chance of making a profit.
This is called a “static hedge” since the hedge portfolio, which consists of the asset to be sold plus the borrowed risk-free investment, does not change over the term of the contract. For more complex financial instruments, such as options, the hedge portfolio is more complex, and requires (in principle) continuous rebalancing to maintain. This is called a “dynamic hedge”.

Hedging is discussed further in the Financial Economics Subject.

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2.6

CT1-13: Arbitrage and forward contracts

The value of a forward contract
No interest or dividend income
Consider a forward contract agreed at time t = 0 , with a forward price K0 , for one unit of security S. The maturity date of the contract is time T.
At the start of the contract the value, to buyer and seller of the asset S, is 0. At the maturity date the value of the contract to the seller of the asset is K0 - ST and to the buyer is -(K0 - ST ) .
It is of interest to find the value of the contract at intermediate times.

These values can be found using the “no arbitrage” assumption and similar techniques to those used to find the forward price.
Suppose at time r > 0 an investor holds a long forward contract – that is, holds a contract agreeing to buy an asset S at T > r at a price agreed at time t = 0 of K0 .
The investor wants to know the value of this contract during the term at time r .

We will use the notation Vl and Vs to denote the value of long and short positions in the forward contract at time r .
Consider the following two portfolios purchased at time r :
Portfolio A:

Consists of the existing long forward contract (bought at time 0) with current value Vl . Invest K0e −δ (T − r ) at time r in the risk-free investment for T - r years.

Portfolio B:

Buy a new long forward contract at time r , with maturity at T , forward price Kr = Sr e δ (T − r ) . The price of a forward contract at issue is zero. Also, invest Kr e −δ (T − r ) in the risk-free investment for T - r years.

The price of portfolio A at r is Vl + K0e −δ (T − r ) .
The price of portfolio B at r is Kr e −δ (T − r ) .
The payout from portfolio A at T is one unit of the asset S; the investment of

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The payout from portfolio B at T is also one unit of the asset S; again, the riskfree investment accumulates to Kr , which meets the forward price required.
By the no arbitrage assumption we have:

Vl + K0e −δ (T − r ) = K r e −δ (T − r )



Vl = (K r − K0 )e −δ (T − r )

This is a general equation that actually works for all three cases (no income, income and dividend yield).
The value of a short forward contract may be determined by similar arguments, to be
Vs = ( K 0 - K r )e -d (T - r ) . Again, this formula works in all three cases.

Special case – no income
We may substitute K r and K0 to get the value of the forward contract in terms of the asset price:

Vl = Sr − S0e δr which gives the value of the long forward contract at time r , but only for an asset

which provides no income or for the very special case where there is fixed income after time t1 (r < t1 ) (see the example on page 19).
The value of a short forward contract may be determined by similar arguments, to be Vs = S0eδ r − Sr , that is Vs = −Vl .
The value of forward contracts where there is some interest or dividend income associated with the underlying asset may be determined easily using similar arguments. Question 13.11
Which of the following describes the long forward contract referred to above?
A:

A forward contract under which the investor concerned agrees to sell an asset in the future.

B:

A forward contract under which the investor concerned agrees to buy an asset in the future.

C:

A forward contract with a very long term.

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CT1-13: Arbitrage and forward contracts

Let us now consider some examples of various cases.

Example
On 1 January 2006 an investor agrees to pay the fair price of £3,000 in four years’ time for a security. The security pays no interest and the price of the security at the time of the agreement was £2,680. On 1 July 2007 the price of the security is £2,800.
Calculate the value of the forward contract on 1 July 2007, assuming no arbitrage and a constant risk-free force of interest over the four years.

Solution
The risk-free force of interest can be found from the equation:
K = Se 4δ

where K is the forward price (£3,000) and S is the price of the security at issue (£2,680).

δ = ¼ log

3,000
= 2.82%
2,680

Let K1.5 be the forward price of a contract set up on 1 July 2007 with a term of 2.5 years. Then:
K1.5 = 2,800e2.5¥0.0282 = 3, 004.516
So the value of the long forward contract is:
(3, 004.516 - 3, 000)e -2.5¥0.0282 = £4.21

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Example
Assume now that at some time t1 , 0 ≤ t1 < T , the security underlying the forward contract provides a fixed amount c to the holder. Deduce a formula for the value of a long forward contract at time r where 0 < r < t1 .

Solution
Consider the following two portfolios set up at time r :
Portfolio A:

Consists of the long forward contract with current value Vl .
Simultaneously invest an amount of cash K 0e -d (T - r ) in the risk-free investment. Portfolio B:

Buy one unit of the asset, at the current price Sr . Borrow ce -d (t1 - r ) at the risk-free rate of interest.

At time t = r the value of Portfolio A is K 0e −δ (T −r ) + Vl . The value of portfolio B is
S r - ce -d (t1 - r ) .
At time T the payout from Portfolio A is income of K 0 from the risk-free investment, which enables you to purchase the asset. This is worth ST . The payout from Portfolio
B is one unit of asset, value ST . Note that the income from the asset will have paid off the loan.
The net cashflows of portfolio A at time T are identical to those of portfolio B. Using the no arbitrage assumption the prices must also be the same – that is:
Vl + K 0 e -d (T - r ) = Sr - ce -d (t1 - r )

fi Vl = Sr - S0 ed r

since K0 = S0eδT − ceδ ( T −t1 ) .
This is the formula that Core Reading derived from the general equation
Vl = ( K r - K 0 )e -d (T - r ) in the case where there was no income.

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CT1-13: Arbitrage and forward contracts

Question 13.12
Two months ago an investor agreed to sell a zero-coupon bond four months after the start date of the contract. The price of the bond at the start of the contract was £86.90.
The current price is £90.10 and the risk-free force of interest is 4% pa.
Calculate the forward price and the current value of the contract.

Question 13.13
The current price of a stock is £200. Dividends are paid continuously and the current dividend yield is 4% pa. Calculate the value of the long position of the contract 2 years into the five-year term if the risk-free force of interest is 5% pa and the stock price has risen to £205 at that stage.

2.7

Note
We have simplified the calculations by assuming that the risk-free force of interest is independent of the time or duration of the investment. In fact, as shown in the next chapter there is a term structure to interest rates – that is, the interest rate earned on an investment depends on both the time a sum is invested and on the length of time for which it is invested. The results above may be adjusted to allow for this, replacing δ with the appropriate spot or forward force of interest.

Question 13.14
An 18-month forward contract is set up on a bond bearing no income that has a current price of £87.20 per £100 nominal. Assuming the risk-free rate of interest is 6% pa effective for the first year and 5% pa effective for the second, calculate the forward price. © IFE: 2009 Examinations

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3

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Exam-style question
Have a go at the following exam-style question before turning the page and checking the solution.

Question
(i)

Define arbitrage and explain why arbitrage may be considered impossible in many markets.
[4]

An investor wishes to enter a forward contract to buy some shares in Company X, maturing in 10 years’ time. The current share price is £2.50 and the current instantaneous dividend yield is 8p per annum. The risk-free rate of interest is assumed to be 4% pa for the next five years and 5% pa for the following five years.
(ii)

Calculate the forward price of the contract assuming a constant dividend yield and assuming that dividends are paid continuously.
[5]

(iii)

Determine the value of the forward contract after 6 years, when the share price is
£2.90, assuming that the risk-free rate of interest for the remaining term is still
5% pa.
[6]
[Total 15]

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CT1-13: Arbitrage and forward contracts

Solution
(i)

Arbitrage in financial mathematics is generally described as a risk-free trading profit. More accurately, an arbitrage opportunity exists if either:
(a)

an investor can make a deal that would give her or him an immediate profit, with no risk of future loss;

or:
(b)

an investor can make a deal that has zero initial cost, no risk of future loss, and a non-zero probability of a future profit.

In the major developed securities markets arbitrage opportunities, when they do arise are very quickly eliminated as investors spot them and trade on them. Such opportunities are so fleeting in nature, according to the empirical evidence, that it is sensible and realistic to assume that they do not exist.
(ii)

The dividend yield is D = 8 / 250 = 0.032 .

Ïlog1.04 = 3.922%
Ô
The force of interest is d = Ì
Ôlog1.05 = 4.879%
Ó

for five years for next five years

The forward price K can be found from the formula:
K = S 0 e (δ − D ) T

where S0 is the share price (£2.50) and T is the term of the contract (10 years).
Therefore:
K = 2.5e(0.03922-0.032)¥5e(0.04879- 0.032)¥5 = £2.8189

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(iii)

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We can use the general formula:
Vl = ( K r - K 0 )e -d (T - r ) with r = 6 and T = 10

We calculated K 0 in part (i) to be £2.82.
K 6 = S6e 4(δ − D ) = 2.90e 4(0.04879−0.032) = 3.1015

So:
Vl = (3.1015 - 2.8189)e -0.04879¥ 4 = £0.23 per share

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CT1-13: Arbitrage and forward contracts

End of Part 3
You have now completed Part 3 of the Subject CT1 Notes.

Review
Before looking at the Question and Answer Bank we recommend that you briefly review the key areas of Part 3, or maybe re-read the summaries at the end of Chapters
11 to 13.

Question and Answer Bank
You should now be able to answer the questions in Part 3 of the Question and Answer
Bank. We recommend that you work through several of these questions now and save the remainder for use as part of your revision.

Assignments
On completing this part, you should be able to attempt the questions in Assignment X3.

Reminder
If you have not booked a tutorial, then maybe now is the time to do so.

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Chapter 13 Summary
Arbitrage refers to a risk-free trading profit. An arbitrage opportunity exists if either


an investor can make a deal that would give her or him an immediate profit, with no risk of future loss, or



an investor can make a deal that has zero initial cost, no risk of future loss, and a non-zero probability of a future profit.

The “Law of One Price” states that if we assume that there are no arbitrage opportunities in a market, then it follows that any two securities or combinations of securities that give exactly the same payments must have the same price.
A forward contract is an agreement made between two parties under which one agrees to buy from the other a specified amount of an asset at a specified price, called the forward price, on a specified future date.
The investor agreeing to sell the asset is said to hold a short forward position in the asset, and the buyer is said to hold a long forward position.
The forward price and the value of a forward contract can be found using the no arbitrage assumption and a risk-free investment rate:


The forward price for a security with no income is:
K = S0eδT



The forward price for a security with fixed cash income is:
K = ( S0 − I )eδT

where I is the present value of income received during the term of the contract.
The forward price for a security with known dividend yield, D, where dividends are payable continuously is:
K = S 0 e (δ − D ) T

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CT1-13: Arbitrage and forward contracts

The value of a forward contract is:
Vl = ( K r - K 0 )e -d (T - r )

Vs = −Vl

where K r is the forward price for a contract set up at time r with a term of T - r .
If the income is proportional to the underlying security, S, we can carry out calculations using the no arbitrage assumption if we assume the income is reinvested in the security.
If the income is a fixed amount regardless of the price of the security at the payment date, then we assume it is invested in the risk-free security.
Hedging is a general term which describes the use of financial instruments to reduce or eliminate a future risk of loss.

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Chapter 13 Solutions
Solution 13.1
(i)

Yes. You can make an immediate £1 profit with no risk.

(ii)

Yes. There are no initial costs and there is a chance, albeit small, that you will get £400.

(iii)

Yes. Coventry City make an immediate £3.2m profit.

(iv)

No. There is a risk of a loss and no arbitrage opportunity.

(v)

No. Again there is a risk of a loss, ie the first roll is 1–5 and the second roll a six. (vi)

Yes. If you take both bets simultaneously you never make a loss.

(vii)

Yes. This gives a risk-free profit with no initial outlay.

Solution 13.2
An arbitrage opportunity exists if:


an investor can make a deal that would give her or him an immediate profit, with no risk of future loss.



an investor can make a deal that has zero initial cost, no risk of future loss, and a non-zero probability of a future profit.

Solution 13.3
No! Investments with the same payouts (and therefore prices) as each other will give the same expected returns but different assets will have different payouts and therefore different prices and expected returns.

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CT1-13: Arbitrage and forward contracts

Solution 13.4
A derivative is a financial instrument with a value dependent on the value of some other, underlying asset.
Three types are futures, options and swaps.

Solution 13.5
3d

K = 42.6e12

3

= 42.6(1 + i ) 12
3

= 42.6 ¥ 1.0612 = £43.23

Solution 13.6
The amount of each coupon is ½ × 0.08 × 50,000 = £2,000 .
Using the formula in the Core Reading:

K = (80.4 ¥ 500 - 2, 000(e -0.05¥ 2 /12 + e -0.05¥8 /12 ))e0.05¥10 /12
= £37,826
Alternatively you can think of the forward price as the accumulated value of the current price less the accumulated value of the coupon payments due in two months and eight months, giving the equivalent formula:
K = 80.4 ¥ 500e0.05¥10 /12 - 2, 000(e0.05¥8 /12 + e0.05¥ 2 /12 )
= £37,826

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Solution 13.7
The current dividend yield is

5
= 2.5% .
200

The forward price is therefore:
K = 200e ( 0.05−0.025) ×5 = £226.63 per share

Solution 13.8
If we start with one unit of the security at time t = 0, the accumulated holding at time T would be (1 + D) T units of the security. If instead of 1 unit at time t = 0 we hold
(1 + D) − T units, reinvesting the dividend income, then at time T we would hold 1 unit of the security.
Now consider the following two portfolios:
Portfolio A:

Enter a forward contract to buy one unit of an asset S, with forward price
K, maturing at time T; simultaneously invest an amount Ke -d T in the risk-free investment.

Portfolio B:

Buy (1 + D) − T units of the asset S, at the current price S0 . Reinvest dividend income in the security S immediately it is received.

At time t = 0 the price of Portfolio A is Ke -d T for the risk-free investment, and zero for the forward contract.
The price of Portfolio B is S0 (1 + D) − T
The net cashflows of portfolio A at time T are identical to those of portfolio B – both give a net portfolio of one unit of the underlying asset S. Using the no arbitrage assumption the prices must also be the same – that is:
Ke −δT = S0 (1 + D) − T ⇒ K = S0eδT (1 + D) − T

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Solution 13.9
K = S0eδT

where δ is the risk-free force of interest.

Solution 13.10
The investor would have been better off buying the asset at time 2. The investor would then have made a profit at time 5 of:
128.40 − 105e 3×0.05 = 6.41

Solution 13.11
B:

A forward contract under which the investor concerned agrees to buy an asset in the future.

Solution 13.12
Forward price
K0 = 86.90e

4
0.04 × 12

= £88.07

Current value
Firstly, we will calculate the forward price for a contract set up now for a two-month period: K 2 = 90.10e

2
0.04¥ 12

= £90.70

So the value of the long forward contract is:
Vl = ( K 2 - K 0 )e

2
-0.04¥ 12

= £2.62

Alternatively, we could use the special formula developed for this situation:

Vl = 9010 − 86.90e
.

2
0.04 × 12

= £2.62

However the question refers to selling the bond, so we require Vs = -£2.62 .

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Solution 13.13
Using the formula Vl = ( Kt - K 0 )e -d (T -t ) to calculate the value, we need to calculate the forward prices:
K 0 = 200e5(0.05−0.04) = 210.254
K 2 = 205e3(0.05−0.04) = 211.243
So the value is:
Vl = (211.243 - 210.254)e -3¥0.05 = 0.85

Solution 13.14
If we have a risk-free rate of interest i, the forward price for a T year contract, K is given by: K = S0 (1 + i ) T
However here we have a change of interest rate. We adapt the formula as follows:
K = S0 (1 + i1 )t1 (1 + i2 )t2
1

= 87.20 ¥ 1.06 ¥ 1.05 2
= £94.71
So the forward price is £94.71 per £100 nominal.

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Chapter 14
Term structure of interest rates
Syllabus objective
(xiii)

Show an understanding of the term structure of interest rates.
1.

Describe the main factors influencing the term structure of interest rates.

2.

Explain what is meant by the par yield and yield to maturity.

3.

Explain what is meant by, derive the relationships between and evaluate:


discrete spot rates and forward rates



continuous spot rates and forward rates

4.

5.

Evaluate the duration and convexity of a cashflow sequence.

6.

0

Define the duration and convexity of a cashflow sequence, and illustrate how these may be used to estimate the sensitivity of the value of the cashflow sequence to a shift in interest rates.

Explain how duration and convexity are used in the (Redington) immunisation of a portfolio of liabilities.

Introduction
So far in this course it has generally been assumed that the interest rate i or force of interest δ earned on an investment are independent of the term of that investment. In practice the interest rate offered on investments does usually vary according to the term of the investment. It is often important to take this variation into consideration.
In investigating this variation we make use of unit zero coupon bond prices. A unit zero coupon bond of term n, say, is an agreement to pay £1 at the end of n years. No coupon payments are paid. It is also called a pure discount bond.

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CT1-14: Term structure of interest rates

We denote the price at issue of a unit zero coupon bond maturing in n years by
Pn .

Sections 1 and 2 look at ways of expressing interest rates that vary by term and Section
3 gives the reasons why interest rates vary by term. The final section looks at some numerical measures that allow us to quantify the effects of changes in interest rates on cashflow series, and techniques used to minimise the risk of interest rate changes.

Question 14.1
Calculate the yield achieved by investing in a 15-year pure discount bond if P = 0.54 .
15

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1

Discrete time

1.1

Page 3

Discrete time spot rates
The yield on a unit zero coupon bond with term n years, y n , is called the
“n-year spot rate of interest”.

The n-year spot rate is a measure of the average interest rate over the period from now until n years’ time.
Using the equation of value for the zero coupon bond we find the yield on the bond, y n , from:
Pn =

1
(1 + y n )n

1
−n

⇒ (1 + y n ) = Pn

Example
The prices for zero coupon bonds of various terms are as follows:
1 year = £94%

5 years = £70%

10 years = £47%

15 years = £30%

£x% means £x per £100 nominal.
Calculate the spot rates for these terms and sketch a graph of these rates as a function of the term.
Solution
The spot rates for the various terms can be found from the equations of value:
1 year:

100(1 + y1 ) −1 = 94



y1 = 6.4%

5 years:

100(1 + y5 ) −5 = 70



y5 = 7.4%

10 years:

100(1 + y10 ) −10 = 47 ⇒

y10 = 7.8%

15 years:

100(1 + y15 ) −15 = 30 ⇒

y15 = 8.4%

The graph of the spot rates is given on the next page:

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CT1-14: Term structure of interest rates

Since rates of interest differ according to the term of the investment, in general y s π y t for s π t . Every fixed-interest investment may be regarded as a combination of (perhaps notional) zero coupon bonds. For example, a bond paying coupons of D every year for n years, with a final redemption payment of R at time n may be regarded as a combined investment of n zero coupon bonds with maturity value D, with terms of 1 year, 2 years ..., n years, plus a zero coupon bond of nominal value R with term n years.
Defining v yt = (1 + y t ) −1 , the price of the bond is:

A = D(P1 + P2 +
2
= D(v y 1 + v y +
2

+ Pn ) + RPn n n
+ v y ) + Rv y n n

This is actually a consequence of “no arbitrage”; the portfolio of zero coupon bonds has the same payouts as the fixed-interest bond, and the prices must therefore be the same.

Question 14.2
What would happen if the price of a ten-year fixed-interest security was greater than the price given by the above formula?
The variation by term of interest rates is often referred to as the term structure of interest rates. The curve of spot rates { y t } is an example of a yield curve.

The graph above is an example of a yield curve. It isn’t always spot rates that are plotted in a yield curve. It might instead be redemption yields or forward rates (which you will meet shortly).

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Example
Calculate the price of a five-year fixed-interest security, redeemable at par, with 6% annual coupons if the annual term structure of interest rates is:
(7%, 7¼%, 7½%, 7¾%, 8%, …)

Solution
The price per £100 nominal is given by the equation:
2
3
4
5
5
P = 6(v7% + v7.25% + v7.5% + v7.75% + v8% ) + 100v8%

fi P = 6 ¥ 4.0314 + 100 ¥ 0.6806 = £92.25

Question 14.3
Calculate the gross redemption yield of this fixed-interest security.

1.2

Discrete time forward rates
So far we have looked at spot rates, which tell us about interest rates over a period that starts now. We can also look at forward rates, which tell us about interest rates over future periods that may start at a future time.
The discrete time forward rate, ft ,r , is the annual interest rate agreed at time 0 for an investment made at time t > 0 for a period of r years.
That is, if an investor agrees at time 0 to invest £100 at time t for r years, the accumulated investment at time t + r is:

100(1 + ft ,r )r

The forward rate, f t ,r , is a measure of the average interest rate between times t and t +r.

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Forward rates, spot rates and zero-coupon bond prices are all connected. The accumulation at time t of an investment of 1 at time 0 is (1 + y t )t . If we agree at time 0 to invest the amount (1 + y t )t at time t for r years, we will earn an annual rate of ft ,r . So we know that £1 invested for t + r years will accumulate to

(1 + y t )t (1 + ft ,r )r . But we also know from the (t + r ) spot rates that £1 invested for t + r years accumulates to (1 + y t + r )t + r , and we also know from the zero coupon bond prices that £1 invested for t + r years accumulates to Pt−1r . Hence
+
we know that

(1 + y t )t (1 + ft ,r )r = (1 + y t + r )t + r = Pt-1
+r
from which we find that

(1 + ft ,r )r =

(1 + y t + r )t + r t (1 + y t )

=

Pt
Pt + r

so that the full term structure may be determined given the spot rates, the forward rates or the zero coupon bond prices.

The connection between the spot rates and the forward rates can be represented on a time line.
(1 + yt ) t
(1+ f t ,r ) r t +r

t

0

(1 + yt + r ) t +r
Accumulating payments from time 0 to time t + r using the spot rate yt + r is equivalent to first accumulating to time t using yt , and then accumulating from time t to time t + r using the forward rate f t ,r .
One period forward rates are of particular interest. The one-period forward rate at time t (agreed at time 0) is denoted ft = ft ,1 . We define f0 = y 1 . Comparing an amount of £1 invested for t years at the spot rate y t , and the same investment invested 1 year at a time with proceeds reinvested at the appropriate one-year forward rate, we have

(1 + y t )t = (1 + f0 )(1 + f1)(1 + f2 )

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The one-year forward rate, f t , is therefore the rate of interest from time t to time t + 1 .
It can be expressed in terms of spot rates:

(1 + f t ) =

(1 + yt +1 ) t +1
(1 + yt ) t

Question 14.4
If the n year spot rates can be approximated by the function 0.09 − 0.03e −0.1n , calculate the one-year forward rate at time 10.

Question 14.5
The 3, 5 and 7-year spot rates are 6%, 5.7% and 5% pa respectively. The 3-year forward rate from time 4 is 5.2% pa. Calculate:
(i)

f3

(ii)

f5,2

(iii)

y4

(iv)

f3,4

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2

Continuous time rates

2.1

Continuous time spot rates
The continuous time spot rate is the force of interest that is equivalent to an effective rate of interest equal to the spot rate.
Let Pt be the price of a unit zero coupon bond of term t. Then the t-year spot force of interest is Yt where:
Pt = e -Yt t

1 fi Yt = - log Pt t This is also called the continuously compounded spot rate of interest or the continuous-time spot rate. Yt and its corresponding discrete annual rate y t are connected in the same way as δ and i; an investment of £1 for t years at a discrete spot rate y t accumulates to (1 + y t )t ; at the continuous time rate it accumulates to eYt t ; these must be equal, so y t = eYt - 1 .

Question 14.6
Calculate the present value of a payment of £800 made at time 12 years given the following information:



2.2

the five-year spot rate is equal to 6% pa the seven-year spot force of interest is equal to 7% pa

Continuous time forward rates
The continuous time forward rate Ft ,r is the force of interest equivalent to the annual forward rate of interest ft ,r .
A £1 investment of duration r years, starting at time t, agreed at time 0 £ t accumulates using the annual forward rate of interest to (1 + ft ,r )r at time t + r .
Using the equivalent forward force of interest the same investment accumulates to e

Ft ,r r

.

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Hence the annual rate and continuous-time rate are related as:

ft ,r = e

Ft ,r

-1

The relationship between the continuous time spot and forward rates may be derived by considering the accumulation of £1 at a continuous time spot rate of
Yt for t years, followed by the continuous time forward rate of Ft ,r for r years.
Compare this with an investment of £1 at a continuous time spot rate of Yt + r for

t + r years. The two investments are equivalent, so the accumulated values must be the same. Hence:

e tYt e fi rFt ,r

= e (t + r )Yt + r

tYt + rFt ,r = (t + r )Yt + r fi Ft ,r =

(t + r )Yt + r - tYt r 1
Also, using Yt = - log Pt , we have: t Ft ,r =

Ê P ˆ
1
log Á t ˜ r Ë Pt + r ¯

Once again we can represent the connection between the continuous time spot and forward rates on a time line. e tYt

e

rFt ,r

t +r

t e ( t +r )Yt + r

Question 14.7
The prices for zero coupon bonds of various terms are as follows:
1 year = £94%

5 years = £70%

10 years = £47%

15 years = £30%

Calculate Y10 and F5,10 .

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2.3

CT1-14: Term structure of interest rates

Instantaneous forward rates
The instantaneous forward rate Ft is defined as:

Ft = lim Ft ,r r Æ0

The instantaneous forward rate may broadly be thought of as the forward force of interest applying in the instant of time t Æ t + Dt .

Ê P ˆ
1
log Á t ˜ r Æ0 r
Ë Pt + r ¯

Ft = lim

(1)

log Pt - log Pt + r r Æ0 r = lim

log Pt + r - log Pt r r Æ0

(2)

=-

d log Pt dt (3)

=-

1 d
Pt
Pt dt

(4)

= - lim

We also find, by integrating both sides of (3) and using the fact that P0 = 1 (as the price of a unit zero coupon bond of term zero years must be 1), that: t Pt = e

- Ú Fsds
0

This formula should look very familiar to you. Earlier in the course we defined v (t ) to be the discounted present value of 1 due at time t and we expressed v (t ) in terms of δ (t ) , the force of interest per unit time at time t:

z t v (t ) = e

− δ ( s ) ds
0

As you can see, Pt is equivalent to v ( t ) and Fs is equivalent to δ ( s) .

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Question 14.8
The price at time 0, Pt , of a zero-coupon bond for 2 < t < 4 is given by the equation:
Pt = (100 − 2t 2 )%
Calculate the instantaneous forward rate F3 .

Note
We have described in this unit the initial term structure, where everything is fixed at time 0. In practice the term structure varies rapidly over time, and the 5-year spot rate tomorrow may be quite different from the 5-year spot rate today. In more sophisticated treatments we model the change in term structure over time.
In this case all the variables we have used, ie:

Pt

yt

ft ,r

Yt

Ft ,r

need another argument, v, say, to give the “starting point”. For example, y v ,t would be the t-year discrete spot rate of interest applying at time v; Fv ,t ,r would be the force of interest agreed at time v, applying to an amount invested at time v + t for the r-year period to time v + t + r .

Question 14.9
Using the above notation, if y0,5 = 6% , y5,5 = 7½% , F0,5,5 = 7% and F5,5,5 = 8¼% , calculate the value of a ten-year zero-coupon bond issued at time 0.

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3

Theories of the term structure of interest rates

3.1

Why interest rates vary over time
The prevailing interest rates in investment markets usually vary depending on the time span of the investments to which they relate. This variation determines the term structure of the interest rates which we have already mentioned and which we will define more precisely later.
The variation arises because the interest rates that lenders expect to receive and borrowers are prepared to pay are influenced by the following factors which are not normally constant over time:

Supply and demand
Interest rates are determined by market forces ie the interaction between borrowers and lenders. If cheap finance is easy to obtain or if there is little demand for finance, this will push interest rates down.

Base rates
In many countries there is a central bank that sets a base rate of interest which provides a reference point for other interest rates. For example, an interest rate in the UK may be expressed as the Bank of England’s base rate plus 4%. Investors will have a view on how this rate is likely to move in the future.

Interest rates in other countries
The interest rates in a particular country will also be influenced by the cost of borrowing in other countries because major investment institutions have the alternative of borrowing from abroad.

Expected future inflation
Lenders will expect the interest rates they obtain to outstrip inflation. So periods of high inflation tend to be associated with high interest rates.

Tax rates
If tax rates are high, interest rates may also be high, because investors will require a certain level of return after tax.

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Risk associated with changes in interest rates
In general, rates of interest tend to increase as the term increases because the risk of loss due to a change in interest rates is greater for longer-term investments.
The following Core Reading illustrates how these factors can lead to three possible patterns of term structure.

3.2

The theories
Some examples of typical (spot rate) yield curves are given below.

Figure 1: Decreasing yield curve
In Figure 1 the long-term bond yields are lower than the short-term bonds. Since price is a decreasing function of yield, an interpretation is that long-term bonds are more expensive than short-term bonds.
There are several possible explanations – for example it is possible that investors believe that they will get a higher overall return from long-term bonds, despite the lower current yields, and the higher demand for long-term bonds has pushed up the price, which is equivalent to pushing down the yield, compared with short-term bonds. Other explanations for different yield curve shapes are given below.

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Figure 2: Increasing yield curve
In Figure 2 the long-term bonds are higher yielding (or cheaper) than the shortterm bonds.

Figure 3: Humped yield curve
In Figure 3 the short-term bonds are generally cheaper than the long bonds, but the very short rates (with terms less than 1 year) are lower than the 1-year rates.
The three most popular explanations for the fact that interest rates vary according to the term of the investment are:
1.

Expectations Theory

2.

Liquidity Preference

3.

Market Segmentation

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Expectations Theory
The relative attraction of short and longer-term investments will vary according to expectations of future movements in interest rates. An expectation of a fall in interest rates will make short-term investments less attractive and longer-term investments more attractive. In these circumstances yields on short-term investments will rise and yields on long-term investments will fall.
An
expectation of a rise in interest rates will have the converse effect.
In Figure 1 it appears that the demand for long-term bonds may be greater than for short, implying an expectation that interest rates will fall. By buying longterm bonds investors can continue getting higher rates after a future fall in interest rates, for the duration of the long bond.
In Figure 2 the demand is higher for short-term bonds – perhaps indicating an expectation of a rise in interest rates.

Liquidity Preference
Consider a ten-year and a twenty-year zero-coupon bond. If the spot rate for all terms is

.
.
.
5% then the prices of the bonds are 105−10 = 6139% and 105−20 = 37.69% respectively. If interest rates rise to 6% then the price of both bonds will fall:
The ten-year bond price falls to 55.84%, a 9% drop.
The twenty-year bond price falls to 31.18%, a 17% drop.
Longer dated bonds are more sensitive to interest rate movements than shortdated bonds. It is assumed that risk averse investors will require compensation
(in the form of higher yields) for the greater risk of loss on longer bonds. This might explain some of the excess return offered on long-term bonds over shortterm bonds in Figure 2.

Later in this chapter we will look at measures that allow us to quantify the effects of changes in interest rates.

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CT1-14: Term structure of interest rates

Market Segmentation
Bonds of different terms are attractive to different investors, who will choose assets that are similar in term to their liabilities. The liabilities of banks, for example, are very short term (investors may withdraw a large proportion of the funds at very short notice); hence banks invest in very short-term bonds. Many pension funds have liabilities that are very long term, so pension funds are more interested in the longest dated bonds. The demand for bonds will therefore differ for different terms.
The supply of bonds will also vary by term, as governments, and companies’ strategies may not correspond to the investors’ requirements.

Remember that governments and companies issue bonds because they need to borrow money, not because they are kind hearted and want to give investors something to invest in. More bonds will be supplied if more money needs to be borrowed. This will put downward pressure on prices.
The market segmentation hypothesis argues that the term structure emerges from these different forces of supply and demand.
These theories are covered in more detail in Subject CA1, Core Applications
Concepts.

Question 14.10
What happens to yields of fixed interest securities if:
(i)

bond prices fall

(ii)

demand for fixed-interest securities falls

(iii)

the government issues many more stocks

(iv)

institutional investors suddenly decide to invest less in equities and more in fixed-interest securities

(v)

bond prices rise

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3.3

Page 17

Yields to maturity
The yield to maturity for a coupon paying bond (also called the redemption yield) has been defined as the effective rate of interest at which the discounted value of the proceeds of a bond equal the price. It is widely used, but has the disadvantage that it depends on the coupon rate of the bond, and therefore does not give a simple model of the relationship between term and yield.
In the UK, yield curves plotting the average (smoothed) yield to maturity of coupon paying bonds are produced separately for “low coupon”, “medium coupon” and “high coupon” bonds.

Question 14.11
The current annual term structure of interest rates is:
(6%, 6%, 6%, 6%, 7%)
Calculate the gross redemption yield of a five-year fixed-interest security redeemable at par if the annual coupon is (i) 2% and (ii) 4%.

3.4

Par yields
You are already familiar with the yield to maturity or the redemption yield for a fixed interest investment. This is just the constant interest rate that satisfies the equation of value. For a zero coupon bond, this is the same as the spot rate.
The n-year par yield represents the coupon per £1 nominal that would be payable on a bond with term n years, which would give the bond a current price under the current term structure of £1 per £1 nominal, assuming the bond is redeemed at par.
That is, if ycn is the n-year par yield:
2
3
1 = ycn (v y 1 + v y + v y +
2
3

n n + v y ) + 1v y n n

The par yields give an alternative measure of the relationship between the yield and term of investments. The difference between the par yield rate and the spot rate is called the “coupon bias”.

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Question 14.12
Calculate the 5 year par yield if the annual term structure of interest rates is:
(6%,6¼%,6½%,6¾%,7%,… )

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Duration, convexity and immunisation
One of the key factors a manager responsible for the investment of a fixed interest portfolio will be concerned about is how the portfolio would be affected if there was a change in interest rates and, in particular, whether such a movement might compromise the ability of the fund to meet its liabilities.
In this section we consider simple measures of vulnerability to interest rate movements. We will also look at the technique of immunisation, which is a method of minimising the risks relating to interest rates.
For simplicity we assume a flat yield curve, and that when interest rates change, all change by the same amount, so that the curve stays flat. A flat yield curve implies that y t = ft ,r = i for all t, r and Yt = Ft ,r = Ft = d for all t, r.

4.1

Interest rate risk
Suppose an institution holds assets of value VA , to meet liabilities of value VL .
Since both VA and VL represent the discounted value of future cashflows, both are sensitive to the rate of interest. We assume that the institution is healthy at time 0 so that currently VA ≥ VL .

If V A > VL , then we say that there is a surplus in the fund which is equal to V A − V L . If
V A < VL then the fund is in deficit.
If rates of interest fall, both VA and VL will increase. If rates of interest rise then both will decrease. We are concerned with the risk that following a downward movement in interest rates the value of assets increases by less than the value of liabilities, or that, following an upward movement in interest rates the value of assets decreases by more than the value of the liabilities.

In other words for a fund currently in surplus we are concerned that after a movement in interest rates the fund moves into deficit.
In order to examine the impact of interest rate movements on different cashflow sequences we will use changes in the yield to maturity to represent changes in the underlying term structure. This is approximately (but not exactly) the same as assuming a constant movement of similar magnitude in the one-period forward rates.

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CT1-14: Term structure of interest rates

Before we can look at a technique used to minimise this risk we must be familiar with the measures: effective duration, duration and convexity.

4.2

Effective duration
One measure of the sensitivity of a series of cashflows, to movements in the interest rates, is the effective duration (or volatility). Consider a series of cashflows {Ct k } for k = 1,2,… , n . Let A be the present value of the payments at rate (yield to maturity) i, so that:

A=

n

 Ct

k =1

t

k

vi k

Then the effective duration is defined to be:

ν (i ) = −

1 d
A′
A= −
A di
A

Ê
Á
1
Á
=Á n t Á
Ct k v i k
Á
Ë k =1

Â

ˆ
˜
˜
˜
˜
˜
¯

(4.1)

Ê n
ˆ
t +1
Ctk t k v i k ˜
Á
Ë k =1
¯

Â

The last relationship may not be that obvious to you. It is important to realise that you need to differentiate with respect to i and not v. So if: n n

k =1

k =1

A = ∑ Ctk vtk = ∑ Ctk (1 + i ) −tk then: n

n

k =1

k =1

A′ = ∑ Ctk ( − t k )(1 + i ) − t k −1 = ( −1) ∑ Ct k t k v tk +1
This is a measure of the rate of change of value of A with i, which is independent of the size of the present value. Equation (4.1) assumes that the cashflows do not depend on the rate of interest.

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For a small movement ε in interest rates, from i to i + ε , the relative change in value of the present value is approximately − εν (i ) so the new present value is approximately A(1 − ε ν (i )) .

4.3

Duration
Another measure of interest rate sensitivity is the duration, also called Macauley
Duration or discounted mean term (DMT). This is the mean term of the cashflows {Ct k } , weighted by present value. That is, at rate i, the duration of the cashflow sequence {Ct k } is: n τ=

t

Σ t k Ct k v i k k =1 n t

Σ Ct k v i k k =1

The discounted mean term for a continuously payable payment stream (or a mixture of discrete and continuous payments) is calculated similarly, but with the summations replaced by integrals for the continuous payments.
Comparing this expression with the equation for the effective duration it is clear that: τ = (1 + i )ν (i )

Question 14.13
Does this relationship also hold for cashflows involving continuous streams of payment? Justify your answer.
Note the following points:
1.

The discounted mean term is dependent on the interest rate used to calculate the present value, as well as the amounts and timings of the cashflows.

2.

Because it is calculated as an average, the DMT of a combined series of cashflows must take a value that is intermediate between the DMTs of the separate cashflows.

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CT1-14: Term structure of interest rates

Example
Evaluate the discounted mean term of a bond redeemable at par in 10 year’s time with annual coupons of 8% at interest rates of 5%, 10% and 15%. Hence sketch a graph of the discounted mean term as a function of the interest rate over the range 5% to 15%.

Solution
Here, the discounted mean term, calculated at interest rate i , is:

DMT (i ) =

F t × 8v
GH ∑
10

t

+ 10 × 100v 10

t =1

I F 8v
JK GH ∑
10

t

+ 100v 10

t =1

I
JK

In terms of compound interest functions, this is:

DMT (i ) =

8( Ia ) 10| + 1000v 10
8a10| + 100v 10

Evaluating this expression for the three interest rates given:

DMT(5%) =

928.90
= 7.54
12317
.

DMT(10%) =

617.83
= 7.04
87.71

DMT(15%) =

42317
.
= 6.52
64.87

The graph of the discounted mean term over this range of interest rates is approximately linear, as shown in the graph below.
7.6
7.4

DMT

7.2
7
6.8
6.6
6.4
4

6

8

10

12

14

16

i

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Question 14.14
Derive an expression for the DMT of a level 10-year annuity of 1 pa payable annually in arrears and sketch a graph of the DMT in this case as a function of the interest rate.
Another way of deriving the Macauley duration is in terms of the force of interest, d: t =-

1 d di A= n (i )
A dd dd i = ed - 1 fi

di
= ed dd fi t = ed n (i ) = (1 + i )n (i )
The equation for τ in terms of the cashflows Ct k may be found by differentiating
A with respect to δ , recalling that v i k = e -d t k . t The duration of an n year coupon paying bond, with coupons of D payable annually, redeemed at R, is:

τ=

D(Ia)n + Rnv n
Dan + Rv n

We deduced this result in the last example.
The duration of an n year zero coupon bond of nominal amount 100, say, is:

τ=

100nv n
100v n

=n

This last result should be intuitively obvious. The average term of a series of cashflows that has only one payment must be the time of that cashflow.
Both the volatility and the discounted mean term provide a measure of the average
“life” of an investment. This is important when considering the effect of changes in interest rates on investment portfolios since an investment with a longer term will in general be affected more drastically by a change in interest rates than an investment with a shorter term.

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CT1-14: Term structure of interest rates

Example
How will the price of a conventional gilt that is redeemable at par with an annual coupon of 3% be affected if future rates of interest over all terms increase from 7% to
8%, if the term of the gilt is (a) 5 years and (b) 25 years? Comment on your results.

Solution
Using the formula P = 3a n| + 100v n with n = 5 and n = 25 , we find that:
The price of the 5 year stock would fall from £83.60 to £80.04 ie a fall of 4.3%
The price of the 25 year stock would fall from £53.39 to £46.63 ie a fall of 12.7%.
The change in interest rate has a greater effect on the longer 25 year stock, which has a
DMT of 14.9 years (based on 7% interest), than it has on the shorter 5 year stock, which has a DMT of 4.7 years.
Roughly speaking, a change in interest rates has the same effect on the present value of a cashflow series as it would have on a zero coupon bond with the same discounted mean term or volatility.

4.4

Convexity
To determine more precisely the effect of a change in the interest rate (which we will need to do to carry out immunisation calculations) we need another quantity called the convexity. The convexity of the cashflow series {Ct k } is defined as:

c (i ) =

1 d2
A ¢¢
A=
A di 2
A

Ê
ˆ
Á
˜Ê n
1
t +2 ˆ
=Á n
˜ Á S Ct k t k (t k + 1) v i k ˜
¯
Á S C v tk ˜ Ë k =1
Á
Ë k =1 tk i ˜
¯

The convexity always has a positive value. For cashflow series with the same term, a cashflow series consisting of payments paid close together will have a low convexity, whereas a series that is more spread out over time will have a higher convexity.

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Example
Find an expression for the convexity of the cashflows arising from a lump sum deposited in a bank account that pays simple interest at a constant rate at the end of each year. Assume that the lump sum is invested for an indefinite term.

Solution
If the account pays interest at 100 j% per annum the PV of the cashflows will be:

P(i ) = ja ∞| =

j i So the convexity will be:
P ′′(i ) 2 j
= 3
P (i ) i j 2
=
i i2

Question 14.15
Calculate, using 10% pa interest, the convexity of the following assets, each of which has a discounted mean term of 11 years. Comment on your answers.


Asset A is an 11-year zero coupon bond.



Asset B will provide a lump sum payment of £9,663 in 5 years’ time and a lump sum payment of £26,910 in 20 years’ time.



Asset C is a level annuity of 1 pa payable annually in arrears for 50 years.

Combining convexity and duration gives a more accurate approximation to the change in A following a small change in interest rates. For small ε :

A(i + e ) - A(i ) ∂A 1
∂2 A 1
=
¥ ¥ e + ½ ¥ 2 ¥ ¥ e2 +
A
∂i A
A
∂i
ª -en (i ) + e 2 ¥ ½ ¥ c (i )

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CT1-14: Term structure of interest rates

This last result comes from directly applying Taylor’s formula. In case you’re a bit rusty, this can be given in the form:

h2 h3 f ( x + h) = f ( x ) + h f ′ ( x ) + f ′′( x ) + f ′′′( x ) +
2!
3!
See FAC for a more detailed discussion of Taylor’s formula.
Convexity gives a measure of the change in duration of a bond when the interest rate changes. Positive convexity implies that τ (i ) is a decreasing function of i.
This means, for example, that A increases more when there is a decrease in interest rates than it falls when there is an increase of the same magnitude in interest rates.

Why is it called “convexity”?
“Convexity” refers to the shape of the graph of the present value as a function of the interest rate. The following graph shows the present value of the three assets in
Question 13.14 (all scaled to have a present value of 1 unit at 5% interest). You can see that Asset C (which has the highest convexity) has the most “curved” graph, while
Asset A (which has the lowest convexity) has the “flattest” graph.
4

A
B
C

PV

3

2

1

0
0

10

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CT1-14: Term structure of interest rates

4.5

Page 27

Immunisation
Suppose an organisation has liabilities that will require a known series of cashflows
(which we will assume are all negative) and holds assets that will generate a known series of cashflows (which we will assume are all positive) to meet these liabilities.
If it were possible to select a portfolio of assets that generated cashflows that exactly matched the liabilities of the fund (in terms of timing and amount), then the fund would be completely protected against any changes in interest rates. However, this is an idealised scenario and, apart from in very simple cases, perfect matching of this kind cannot be achieved.
It may, however, be possible to choose an asset portfolio that offers the fund a milder form of protection. Suppose the present value of the fund’s liabilities and assets, calculated at a valuation rate of interest i which reflects the interest rate in the market, are V L (i ) and V A (i ) . Then the fund would consider that it has a surplus of
S (i ) = V A (i ) − V L (i ) . We can then consider how this surplus would be affected by changes in the interest rate i . In particular, we will be concerned about the downside risk if a change in market interest rates causes the surplus to become negative ie a deficit. In simple cases it is possible to select an asset portfolio that will protect this surplus against small changes in the interest rate. This is known as immunisation. In the 1950s the actuary Frank Redington derived three conditions that are required to achieve immunisation. Consider a fund with asset cashflows { At k } and liability cashflows {Lt k } . Let

VA (i ) be the present value of the assets at effective rate of interest i and let VL (i ) be the present value of the liabilities at rate i; let ν A (i ) and ν L (i ) be the volatility of the asset and liability cashflows respectively, and let c A (i ) and cL ( i ) be the convexity of the asset and liability cashflows respectively.
At rate of interest i 0 the fund is immunised against small movements in the rate of interest of ε if and only if VA (i 0 ) = VL (i 0 ) and VA (i 0 + ε ) ≥ VL (i 0 + ε ) .

In words, a fund is said to be immunised against small changes in the interest rate if


the surplus in the fund at the current interest rate is zero and



any small change in the interest rate (in either direction) would lead to a positive surplus. The Actuarial Education Company

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CT1-14: Term structure of interest rates

Redington’s conditions
Then consider the surplus S(i ) = VA (i ) − VL (i ) .
From Taylor’s theorem:

S (i0 + ε ) = S (i0 ) + ε S ′(i0 ) +

ε2
2

S ′′(i0 ) +

Consider the terms on the right hand side. We know that S (i0 ) = 0 .

This is because the definition of immunisation requires that the surplus is zero at the current interest rate ie S (i0 ) = V A (i0 ) − V L (i0 ) = 0 . In other words, V A (i0 ) = V L (i0 ) , ie the present value of the assets at the original interest rate will exactly match the present value of the liabilities.
The second term, εS ′ (i 0 ) , will be equal to zero for any values of ε (positive or negative) if and only if S ′ (i 0 ) = 0 , that is if VA ( i 0 ) = VL (i 0 ) .



Since we have already assumed that V A (i0 ) = V L (i0 ) , this is equivalent to:



V A (i0 )
V ′ (i )

=− L 0
V A (i0 )
V L (i0 )

In other words, the assets and the liabilities must have the same volatility.
This is equivalent to requiring that ν A (i ) = ν L (i ) or (equivalently) that the durations of the two cashflow series are the same.

ε2

is always positive, regardless of the sign of e . Thus, if we
2
ensure that S ′′(i0 ) > 0 , then the third term will also always be positive.

In the third term,

This is equivalent to requiring that VA (i 0 ) > VL (i 0 ) , which is equivalent to
′′
′′ requiring that c A (i ) > cL (i ) .
For small ε the fourth and subsequent terms in the Taylor expansion will be very small. Hence, given the three conditions above, the fund is protected against small movements in interest rates. This result is known as Redington’s immunisation after the British actuary who developed the theory.

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The conditions for Redington’s immunisation may be summarised as follows:
1.

VA (i0 ) = VL (i0 ) – that is, the value of the assets at the starting rate of interest is equal to the value of the liabilities.

2.

The volatilities of the asset and liability cashflow series are equal, that is, ν A (i 0 ) = ν L (i 0 ) .

3.

The convexity of the asset cashflow series is greater than the convexity of the liability cashflow series – that is, c A (i0 ) > cL (i0 ) .

As mentioned in the above reasoning, the second condition could be replaced by one of the following equivalent conditions:


The discounted mean terms of the asset and liability cashflow series are equal



V A (i0 ) = VL (i0 )



The third condition could be replaced by the condition:


V A (i0 ) > V L (i0 )
′′
′′

Example
A fund must make payments of £50,000 at the end of the sixth and eighth years. Show that, if interest rates are currently 7% pa at all durations, immunisation to small changes in interest rates can be achieved by holding an appropriately chosen combination of a 5year zero-coupon bond and a 10-year zero-coupon bond.

Solution
Let £ P and £ Q denote the maturity values of the 5 year and 10 year zero coupon bonds.
Then the present value of the assets is:
VA (0.07) = Pv5 + Qv10 @ 7%
By Redington’s first condition, this must equal the PV of the liabilities, which is:
V L (0.07) = 50,000(v 6 + v 8 ) @7% = £62,418

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This gives us our first equation for finding P and Q :
Pv5 + Qv10 = 62, 418

(1)

The negative of the derivative of the PV of the assets is given by:

-VA (0.07) = P ¥ 5v 6 + Q ¥ 10v11 @ 7%
¢
So the volatility of the assets is:

VA (0.07) 5Pv 6 + 10Qv11 5 Pv 6 + 10Qv11
¢
=
=
VA (0.07)
VL (0.07)
62, 418
By Redington’s second condition, this must equal the volatility of the liabilities, which is:

V L (0.07) 50,000(6v 7 + 8v 9 ) 404,398


=
=
V L (0.07)
62,418
62,418
This gives us our second equation for finding P and Q :
5 Pv 6 + 10Qv11 = 404,398

(2)

Solving equations (1) and (2), we find that P = £53,710 and Q = £47,454 .
Note that we have left the v terms in the equations to make it easier to solve the simultaneous equations.

This determines the portfolio of assets we require. We now need to check Redington’s third condition. With these values of P and Q , the convexity of the assets is:

,
V A (0.07) P × 30v 7 + Q × 110v12 3,321152
′′
=
=
= 53.21
V A (0.07)
62,418
62,418
The convexity of the liabilities is:

V L (0.07) 50,000(42v 8 + 72v 10 ) 50,000 × 61045
′′
.
=
=
= 48.90
62,418
62,418
V L (0.07)

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Since the convexity of the assets exceeds the convexity of the liabilities, all three of
Redington’s conditions are now satisfied and the fund is immunised against small changes in the interest rate around 7% pa.

Question 14.16
Verify numerically that this fund is in fact immunised by calculating the surplus for interest rates of 6½% pa and 7½% pa.
In practice there are difficulties with implementing an immunisation strategy based on these principles. For example the method requires continuous rebalancing of portfolios to keep the asset and liability volatilities equal.

The asset portfolio required to provide Redington immunisation normally depends on the initial interest rate. Once the interest rates have moved away from the initial rate, it may be necessary to “rebalance” the portfolio so that it is once again immunised at the new rate. This throws a spanner into the practical application of the technique except in very simple situations.
Other limitations of immunisation include:


There may be options or other uncertainties in the assets or in the liabilities, making the assessment of the cashflows approximate rather than known.



Assets may not exist to provide the necessary overall asset volatility to match the liability volatility.



The theory relies upon small changes in interest rates. The fund may not be protected against large changes.



The theory assumes a flat yield curve and requires the same change in interest rates at all terms. In practice this is rarely the case.



Immunisation removes the likelihood of making large profits.

Despite these problems, immunisation consideration in the selection of assets.

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theory

remains

an

important

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CT1-14: Term structure of interest rates

In practice, actuaries making investment decisions are aware of Redington’s theory in a general sense. For example, they are aware of the consequences of investing “long” (ie holding assets with a higher DMT than the liabilities), but they would not normally apply the theory directly. A more open-ended technique called asset-liability modelling is often used instead. You will meet this in later subjects.

Question 14.17
Would it be possible to immunise the cashflows in this example using some combination of these two zero coupon bonds if current interest rates were 15%?

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Page 33

Exam-style questions
It’s quite common to get a long (and tricky) question in the exam on duration, volatility, convexity and immunisation. The other topics covered in this chapter are also examined frequently.
Have a go at the following two (short) questions yourself before looking at the solution.

Question 1
At time t = 0 , the 2-year spot rate is 4% per annum effective, the 3-year spot rate is 5% per annum effective and the 4-year spot rate is 6% per annum effective. Calculate the
2-year continuous time forward rate from time t = 2 .
[2]

Question 2
At 1 July 2004, an investor has a liability of £20,000 to be paid on 1 January 2008 and a liability of £18,000 to be paid on 1 July 2010 The investor currently holds assets with a present value equal to the present value of the liabilities.
The investor wishes to immunise its position by investing in two zero coupon bonds with outstanding terms of four years and seven years. Determine whether or not this is possible assuming an effective interest rate of 10% per annum.
[6]

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CT1-14: Term structure of interest rates

Solution 1
Let yt equal the t-year spot rate, f t , r equal the annual forward rate of interest from time t to time t + r and Ft , r equal the continuous time forward rate from time t to time t +r.
(1 + f 2 ,2 ) 2 =

(1 + y4 ) 4
(1 + y2 ) 2

=

1064
.
104 2
.

= 116723
.

Therefore: f 2 ,2 = 8.0385% .
The continuous time forward rate is the force of interest equivalent to the annual forward rate of interest:
F2 ,2 = log(1 + f 2 ,2 ) = log 1080385 = 7.73%
.

Solution 2
Let X be the present value of the investment in the zero-coupon bond with a term of 4 years. Let Y be the present value of the investment in the zero-coupon bond with a term of 7 years. The present values of the assets and liabilities are equal. Therefore:

X + Y = 20,000v 3.5 + 18,000v 6 = £24,488
For immunisation the discounted mean terms must also be equal or since the present values are equal this is equivalent to:

4 X + 7Y = 35 × 20,000v 3.5 + 6 × 18,000v 6 = £111,108
.
Solving the above two equations simultaneously gives:
X = £20,103 and Y = £4,385

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We must now check whether the convexity of the assets is greater than the convexity of the liabilities.
Since the convexity of a series of cashflows of Ctk at time tk is: n  Ct tk (tk + 1)v

k =1

k

n

 Ct v

k =1

 tk (tk + 1) (Ct vt n tk + 2

=

k =1

tk

k

k

k

)v

2

n

 Ct vt k =1

k

k

and the denominator is just the present value calculated above, we have:
Conv A =

4 × 5 × Xv 2 + 7 × 8 × Yv 2
= 2186
.
24,488

Alternatively, first write X = Av 4 and Y = Bv 7 where A and B are the nominal amounts of the stocks, differentiate twice and substitute back to get X and Y.
Conv L =

.
35 × 4.5 × 20,000v 5.5 + 6 × 7 × 18,000v 8
= 22.02
24,488

Since the convexity of the liabilities is greater than the convexity of the assets, immunisation using the stocks given is not possible.

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CT1-14: Term structure of interest rates

This page has been left blank so that you can keep the chapter summaries together for revision purposes.

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Chapter 14 Summary
The yield on a unit zero coupon bond with term n years is called the n-year spot rate of interest. The variation by term of interest rates is often referred to as the term structure of interest rates. The curve of spot rates is an example of a yield curve.
The discrete time forward rate, ft,r , is the annual interest rate agreed at time 0 for an investment made at time t > 0 for a period of r years.
There is a direct relationship between forward rates of interest and spot rates:
(1 + f t ,r ) r =

(1 + yt +r ) t + r
(1 + yt ) t

=

Pt
Pt +r

Usually the term structure of interest rates in fixed interest investments is not constant.
The three most popular explanations for the fact that interest rates vary according to the term of the investment are:
1.
2.
3.

Expectations Theory
Liquidity Preference
Market Segmentation

The performance of a fixed interest investment can be assessed by its yield to maturity or its par yield.
The n-year par yield represents the coupon per £1 nominal that would be payable on a bond with term n years, which would give the bond a current price under the current term structure of £1 per £1 nominal, assuming the bond is redeemed at par.
The effects of changes in interest rates on the cashflows generated by an asset or required by a liability can be quantified by calculating the discounted mean term
(duration), the volatility and the convexity. The discounted mean term and the volatility are related.
Volatility = −

n
A′
= ∑ Ck t k v tk +1
A k =1

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n

∑ Ck v t

k

k =1

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CT1-14: Term structure of interest rates

DMT =

n

∑ t k Ck v tk

k =1

n

∑ Ck v t

k

= (1 + i ) × Volatility .

k =1

n
A′′
Convexity =
= ∑ Ck t k ( t k + 1) v t k +2
A k =1

n

∑ Ck v t

k

.

k =1

The surplus in a fund can be immunised to local changes in interest rates if Redington’s conditions can be met. These require the assets and liabilities to have the same present value and discounted mean term (or volatility) and for the convexity of the assets to exceed that of the liabilities. The conditions are:
1.

V A (i0 ) = V L (i0 )

ie

PV ( Assets) = PV ( Liabilities)

2.

V A (i0 ) = V L (i0 )



ie

Volatility ( Assets) = Volatility ( Liabilities) or DMT ( Assets) = DMT ( Liabilities)

3.

V A (i0 ) > V L (i0 )
′′
′′

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Convexity ( Assets) > Convexity ( Liabilities)

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Chapter 14 Solutions
Solution 14.1
The yield is the value of i that solves the equation:
0.54 = v15

fi (1 + i ) = 0.54 -1/15 fi i = 4.19%

Solution 14.2
Arbitrageurs would spot the anomaly and seek to make risk-free profits by selling holdings of the fixed-interest security (at an inflated price) and buying an appropriate combination of zero-coupon bonds. The future cashflows of the investors would not change but they would make an immediate risk-free profit.
The increased demand for the zero-coupon bonds relative to the fixed-interest stock would result in the price anomaly being removed.

Solution 14.3
The gross redemption yield is the value of i that solves the equation:
92.25 = 6a5| + 100v 5
@7.5%

RHS=93.93

@8%

RHS=92.01

Therefore, by interpolation, the gross redemption yield is 7.9%.

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CT1-14: Term structure of interest rates

Solution 14.4
Using the formula given, the 10-year and 11-year spot rates are: y10 = 0.09 − 0.03e −0.1×10 = 0.07896 and: y11 = 0.09 − 0.03e −0.1×11 = 0.08001
Therefore:
(1 + f10 ) =

(1 + y11 )11

(1 + y10 )10

= 10906
.

ie the one-year forward rate at time 10 is 9.06%.

Solution 14.5
We know that:

y3 = 6%,

y5 = 5.7%,

y7 = 5%,

f 4,3 = 5.2%

On a time line this looks as follows:
5% pa

5.7% pa

1

2
6% pa

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3

4

5

6

7

Time

5.2% pa

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(i)

Page 41

We have:

(1 + y3 )3 (1 + f3 )(1 + f 4,3 )3 = (1 + y7 )7 fi 1 + f3 = fi (ii)

1.057
1.063 ¥ 1.0523

= 1.0148

f3 = 1.48%

We have:

(1 + y5 )5 (1 + f5,2 ) 2 = (1 + y7 )7
1

fi 1 + f5,2 fi (iii)

Ê 1.057 ˆ 2

˜ = 1.0665
Ë 1.0575 ¯

f5,2 = 3.27%

We have:
(1 + y4 ) 4 (1 + f 4,3 )3 = (1 + y7 )7
1

Ê 1.057 ˆ 4 fi 1 + y4 = Á
˜ = 1.0485
Ë 1.0523 ¯ fi (iv)

y4 = 4.85%

We have:

(1 + y3 )3 (1 + f3,4 ) 4 = (1 + y7 )7
1

fi 1 + f3,4 fi Ê 1.057 ˆ 4

˜ = 1.0426
Ë 1.063 ¯

f3,4 = 4.26%

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Solution 14.6
There is not enough information given in the question to find the present value for a payment at time 12. The five-year spot rate can only be used for the time period from 0 to 5, and the seven-year spot force of interest can only be used from 0 to 7. We do not know the interest rate from time 7 to time 12.

Solution 14.7
Y10 = =-

1 log P
10
10
1
log 0.47
10

= 7.55%
F5,10 =

ÊP ˆ
1
log Á 5 ˜
10
ËP ¯
15

=

1
Ê 0.7 ˆ log Á
Ë 0.3 ˜
¯
10

= 8.47%

Solution 14.8
Ft = =-

1 d
Pt
Pt dt
1
(100 - 2t 2 )

( -4t )

and so:

F3 = -

1
(100 - 2 ¥ 32 )

( -4 ¥ 3)

= 14.6%

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Solution 14.9
P = (1 + y0,5 ) -5 e
10

-5 F0,5,5

= 1.06 -5 e -5¥0.07
= 52.66%

Solution 14.10
(i)

prices fall



yields rise

(ii)

demand falls ⇒

prices fall



yields rise

(iii)

more issued



prices fall



yields rise

(iv)

more demand ⇒

prices rise



yields fall

(v)

prices rise



yields fall

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CT1-14: Term structure of interest rates

Solution 14.11
(i)

2% coupon rate
5
P = 2a4|6% + 102v7%

= 2 ¥ 3.4651 + 102 ¥ 0.71299 = 79.66
The GRY is found from the equation:
P = 2a5| + 100v5 which gives GRY = 6.96% .
(ii)

4% coupon rate
5
P = 4a4|6% + 104v7%

= 4 ¥ 3.4651 + 104 ¥ 0.71299 = 88.01

The GRY is found from the equation:
P = 4a5| + 100v5 which gives GRY = 6.92% .

Solution 14.12
The 5-year par yield yc5 is found from the equation:

yc5 (v y1 + v 22 + … + v55 ) + v55 = 1 y y y ie:

yc5 (1.06 -1 + 1.0625-2 + 1.065-3 + 1.0675-4 + 1.07 -5 ) + 1.07 -5 = 1

ie:

yc5 ¥ 4.1401 + 0.71299 = 1

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yc5 = 6.93%

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CT1-14: Term structure of interest rates

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Solution 14.13
If we incorporate a continuous stream of payments payable at rate r (t ) at time t , in addition to the discrete cashflows, the PV becomes: n •

n

A = Â Ctk v + Ú r (t ) v dt = Â Ctk (1 + i ) tk k =1

t

- tk

k =1

0



+ Ú r (t ) (1 + i) - t dt
0

The derivative then becomes: n A¢ = Â Ctk ( -tk ) (1 + i )

- tk -1

k =1



+ Ú r (t ) ( -t ) (1 + i ) -t -1 dt
0


È n
˘
tk
= - (1 + i ) Í Â tk Ctk v + Ú t r (t ) vt dt ˙
Í k =1
˙
0
Î
˚
-1

and again we find that the volatility satisfies the relationship:

n (i ) = -


= (1 + i ) -1 ¥ t (i )
A

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CT1-14: Term structure of interest rates

Solution 14.14
The present value is: v + v 2 + v3 +

+ v10 = a10

The DMT is:

 tcvt
DMT (i ) =
 cvt
=
=

v + 2v 2 + 3v3 + v + v 2 + v3 +

+ 10v10
+ v10

( Ia )10 a10 Calculating this as various rates of interest we get:

DMT (5%) = 5.10

DMT (7%) = 4.95

DMT (10%) = 4.73

DMT (12%) = 4.58

DMT (15%) = 4.38
So the graph of the DMT as a function of the interest rate looks like this:

5.2
5.1
5

DMT

4.9
4.8
4.7
4.6
4.5
4.4
4.3
0

2

4

6

8

10

12

14

16

interest rate

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Solution 14.15
The present value of £100 nominal of Asset A at interest rate i is:
PA (i ) = 100(1 + i ) -11
So its convexity is:
PA (i ) 100( -11)( -12)(1 + i ) -13 11 ¥ 12
¢¢
=
=
PA (i )
100(1 + i ) -11
(1 + i ) 2
When i = 10% , this gives:
Convexity A =

11 ¥ 12
1.102

= 109.1

The present value of Asset B at interest rate i is:
PB (i ) = 9, 663(1 + i ) -5 + 26,910(1 + i ) -20
So its convexity is:
PB (i ) 9, 663( -5)( -6)(1 + i ) -7 + 26,910( -20)( -21)(1 + i ) -22
¢¢
=
PB (i )
9, 663(1 + i ) -5 + 26,910(1 + i ) -20
When i = 10% , this gives:
ConvexityB =

148, 759 + 1,388, 430
= 153.7
6, 000 + 4, 000

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CT1-14: Term structure of interest rates

The present value of Asset C at interest rate i is:

PC (i ) =

1 - (1 + i ) -50 i Differentiating this:

PC (i ) =
¢
=

(

i ¥ 50(1 + i ) -51 - 1 - (1 + i ) -50 i2 )

50i (1 + i ) -51 - 1 + (1 + i ) -50 i2 Differentiating again:

(

i 2 50i ¥ -51(1 + i ) -52 + (1 + i ) -51 ¥ 50 - 50(1 + i ) -51
PC (i ) =
¢¢

(

)

)

- 50i (1 + i ) -51 - 1 + (1 + i ) -50 ¥ 2i i4 So its convexity is:

(

3
-52
- 2i 50i (1 + i ) -51 - 1 + (1 + i ) -50
PC (i ) -2,550i (1 + i )
¢¢
=
PC (i ) i 3 1 - (1 + i ) -50

(

)

)

When i = 10% , this gives:
ConvexityC = 174.1
As expected, the convexity is lowest for Asset A because it consists of a single payment and highest for Asset C because the payments are spread out over a long time period.

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Solution 14.16
The surplus, calculated at interest rate i , is:
VA (i ) - VL (i ) = (53, 710v5 + 47, 454v10 ) - 50, 000(v 6 + v8 )
So the surplus, calculated using the interest rates given, is:
At 6½%:

VA (0.065) - VL (0.065) = 64, 482 - 64, 478 = 4 > 0

At 7½%:

VA (0.075) - VL (0.075) = 60, 437 - 60, 433 = 4 > 0

So a ½% movement in interest rates in either direction will result in a positive surplus.
(Note that the change in the surplus is tiny in relation to the value of the assets. Here, a
½% change in interest rates leads to a surplus of just 0.006% . This is typical for an immunised portfolio.)

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CT1-14: Term structure of interest rates

Solution 14.17
Yes. If we repeat the calculations using 15% interest, we get the two equations:
Pv5 + Qv10 = 37,961
5 Pv 6 + 10Qv11 = 226, 486 which lead to:

P = £47,930 and Q = £57,170 .
With these values of P and Q , the convexities of the assets and liabilities are:

VA (0.15)
¢¢
= 45.20 and
VA (0.15)

VL (0.15)
¢¢
= 41.53
VL (0.15)

Again, the convexity of the assets exceeds the convexity of the liabilities. So the fund can also be immunised against small changes in the interest rate around 15% (but with different sized holdings of the two bonds).
(In fact, the liabilities in this example can be immunised with these two zero coupon bonds at any interest rate.)

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CT1-15: Stochastic interest rate models

Page 1

Chapter 15
Stochastic interest rate models
Syllabus objective
(xiv)

Show an understanding of simple stochastic models for investment returns.
1.

Describe the concept of a stochastic interest rate model and the fundamental distinction between this and a deterministic model.

2.

Derive algebraically, for the model in which the annual rates of return are independently and identically distributed and for other simple models, expressions for the mean value and the variance of the accumulated amount of a single premium.

3.

Derive algebraically, for the model in which the annual rates of return are independently and identically distributed, recursive relationships which permit the evaluation of the mean value and the variance of the accumulated amount of an annual premium.

4.

Derive analytically, for the model in which each year the random variable (1+ i ) has an independent log-normal distribution, the distribution functions for the accumulated amount of a single premium and for the present value of a sum due at a given specified future time.

5.

Apply the above results to the calculation of the probability that a simple sequence of payments will accumulate to a given amount at a specific future time.

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CT1-15: Stochastic interest rate models

Introduction
The calculations we have done so far have been based on the assumption that future interest rates will take definite values that are known in advance. This is the deterministic approach.
In this chapter we will study stochastic interest rate models where future interest rates are assumed to be random. We cannot specify in advance precisely what interest rates will apply. Instead, we can make an assumption about the statistical distribution of future interest rates considered as a random variable.
The deterministic approach can provide only a single fixed answer to a problem. This answer will be correct only if the assumptions made about future interest rates turn out to be correct. In practice, an interest rate that errs on the cautious side may be chosen to allow for uncertainty.
The stochastic approach is a more general method that allows us to determine both the expected value, as a best estimate of the quantity of interest, and the variance, which gives an indication of the likely spread of values. The stochastic approach can give unreliable results if the statistical distribution used is not appropriate.
This chapter requires you to know many statistical techniques including means, variances, probabilities and the normal distribution. If you have not studied statistics before or are not simultaneously studying Subject CT3 then we strongly recommend that you spend some time looking at statistical techniques before you study this chapter.
If you have no material to study from, you can either purchase StatsPack from ActEd or any A-Level (or Higher Level) statistics text book.

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1

Simple models

1.1

Page 3

Preliminary remarks
Financial contracts are often of a long-term nature. Accordingly, at the outset of many contracts there may be considerable uncertainty about the economic and investment conditions which will prevail over the duration of the contract. Thus, for example, if it is desired to determine premium rates on the basis of one fixed rate of interest, it is nearly always necessary to adopt a conservative basis for the rate to be used in any calculations.
An alternative approach to recognising the uncertainty that in reality exists is provided by the use of stochastic interest rate models. In such models no single interest rate is used. Variations in the rate of interest are allowed for by the application of probability theory. Possibly one of the simplest models is that in which each year the rate of interest obtained is independent of the rates of interest in all previous years and takes one of a finite set of values, each value having a constant probability of being the actual rate for the year.

For example, the effective annual rates of return that will apply during each of the next n years might be i1 , i2 ,..., in , where ik , k = 1,2,..., n are random variables with the following discrete distribution:

Ï0.06
Ô
i k = Ì 0.08
Ô0.10
Ó

with probability 0.2 with probability 0.7 with probability 0.1

Question 15.1
Calculate the mean, j, and the standard deviation, s, of ik .
Alternatively, the rate of interest may take any value within a specified range, the actual value for the year being determined by some given probability density function. For example we might assume that the annual rates of return are uniformly distributed between 5% and 10%.

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1.2

CT1-15: Stochastic interest rate models

Fixed interest rate model
At this stage we consider briefly an elementary example, which – although necessarily artificial – provides a simple introduction to the probabilistic ideas implicit in the use of stochastic interest rate models.
Suppose that an investor wishes to invest a lump sum of P into a fund which grows under the action of compound interest at a constant rate for n years. This constant rate of interest is not known now, but will be determined immediately after the investment has been made.
The accumulated value of the sum will, of course, be dependent on the rate of interest. In assessing this value before the interest rate is known, it could be assumed that the mean interest rate will apply. However, the accumulated value using the mean rate of interest will not equal the mean accumulated value. In algebraic terms: n k


⎛ k

P ⎜1+
(i j p j ) ⎟ ≠ P ⎜ p j (1 + i j )n ⎟



⎟ j =1


⎝ j =1






where:

i j is the jth of k possible rates of interest p j is the probability of the rate of interest i j

The above result is easily demonstrated with a simple numerical example:
Example
Calculate the expected accumulated value at the end of 5 years of an initial investment of £5,000 if the returns from the investment are assumed to conform to the fixed interest rate model with the distribution of interest rates specified in Question 15.1.
Also calculate the accumulated value at the mean rate of interest.

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Solution
The expected accumulated value must be calculated directly:
5, 000 E ( S5 ) = 5, 000 E[(1 + i )5 ]
= 5, 000(0.2 ¥ 1.065 + 0.7 ¥ 1.085 + 0.1 ¥ 1.105 )
= 5, 000 ¥ 1.4572
= £7, 286
The mean rate of interest was found in Question 15.1 to be 7.8%.
Therefore the accumulated value at the mean rate of interest is:

5,000 × 10785 = £7,279
.

Question 15.2
What is the variance of the accumulated value of this investment?
This model, where the effective annual interest rate of return is a single unknown rate i and will apply throughout the next n years, is often known as the fixed interest rate model. For the fixed interest rate model, the mean and variance of the accumulated value of an investment must be calculated from first principles.

1.3

Varying interest rate model
In our previous example the effective annual rate of interest was fixed throughout the duration of the investment. A more flexible model is provided by assuming that over each single year the annual yield on invested funds will be one of a specified set of values or lie within some specified range of values, the yield in any particular year being independent of the yields in all previous years and being determined by a given probability distribution.

This model is often called the varying interest rate model. The main difference between this and the fixed interest rate model is that, in the varying rate model, the interest rates can be different in each future year, whereas, in the fixed rate model, the same
(unknown) interest rate will apply in each future year.

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CT1-15: Stochastic interest rate models

Measure time in years.

Consider the time interval [0, n ] subdivided into

successive periods [0,1],[1,2],… ,[ n − 1, n ] .

For t = 1,2,… , n let it be the yield

obtainable over the tth year, ie the period [t − 1, t ] .

Assume that money is

invested only at the beginning of each year. Let Ft denote the accumulated amount at time t of all money invested before time t and let Pt be the amount of money invested at time t. Then, for t = 1,2,3,… :
Ft = (1 + it )(Ft − 1 + Pt − 1 )

(1.1)

It follows from this equation that a single investment of 1 at time 0 will accumulate at time n to:
Sn = (1 + i1 )(1 + i2 )… (1 + in )

(1.2)

Similarly a series of annual investments, each of amount 1, at times
0,1,2,… , n − 1 will accumulate at time n to:
An = (1 + i1 )(1 + i2 )(1 + i3 )… (1 + in )

+ (1 + i2 )(1 + i3 )… (1 + in )
+

(1.3)
+ (1 + in − 1 )(1 + in )
+ (1 + in )

Note that An and Sn are random variables, each with its own probability distribution function.
For example, if the yield each year is 0.02, 0.04, or 0.06 and each value is equally likely, the value of Sn will be between 1.02n and 1.06n . Each of these extreme values will occur with probability (1 3)n .

Question 15.3
What is the probability that Sn will take the value 102 × 104 n −1 ?
.
.

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In general, a theoretical analysis of the distribution functions for An and Sn is somewhat difficult. It is often more useful to use simulation techniques in the study of practical problems. However, it is perhaps worth noting that the moments of the random variables An and Sn can be found relatively simply in terms of the moments of the distribution for the yield each year. This may be seen as follows.

Moments of Sn
Let’s consider the kth moment of Sn .
From Equation (1.2) we obtain:
(Sn )k =

n

∏ (1 + it )k t =1

and hence: k E [Sn ] = E

n

∏ (1 + it )k t =1

=

n

∏ E [(1 + it )k ]

(1.4)

t =1

since (by hypothesis) i1 , i2 ,… , in are independent. Using this last expression and given the moments of the annual yield distribution, we may easily find the moments of Sn .
For example, suppose that the yield each year has mean j and variance s 2 .
Then, letting k = 1 in Equation (1.4), we have:
E [Sn ] =

n

∏ E [(1 + it )]

t =1

=

n

∏ (1 + E [it ]) t =1

= (1 + j )n

(1.5)

since, for each value of t, E [ it ] = j .

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With k = 2 in Equation (1.4) we obtain:
2
E [Sn ] =

=

n

’ E [(1 + 2it + it2 )] t =1 n ’ (1 + 2E [it ] + E [it2 ]) t =1

= (1 + 2 j + j 2 + s 2 )n

(1.6)

since, for each value of t:

E [ it2 ] = (E [ it ])2 + var[ it ] = j 2 + s 2
The variance of Sn is:
2
var[Sn ] = E [Sn ] - (E [Sn ])2

= (1 + 2 j + j 2 + s 2 )n - (1 + j )2n

(1.7)

from Equations (1.5) and (1.6).

or equivalently: n 2
E[ S n ] = ’ E[(1 + it ) 2 ] t =1 n (

)

= ’ var[(1 + it )] + E 2 [(1 + it )] t =1 n (

= ’ s 2 + (1 + j ) 2 t =1

(

= s 2 + (1 + j ) 2

)

)

n

Question 15.4
Explain the steps in the proof above.

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This means that: var[ Sn ] = [(1 + j ) 2 + s2 ]n − (1 + j ) 2 n
These arguments are readily extended to the derivation of the higher moments of
Sn in terms of the higher moments of the distribution of the annual rate of interest. Example
Calculate the variance of the accumulated value of the investment in the example on page 4, assuming the returns conform to the varying interest rate model with the specified distribution.

Solution
Using the formula gives:

var(5, 000S5 ) = 5, 0002 var( S5 )
= 5, 0002 [((1 + j ) 2 + s 2 )5 - (1 + j )10 ]
= 5, 0002 [(1.0782 + 0.000116)5 - (1.078)10 ]
= (£163) 2

Question 15.5
Calculate the mean and variance of the accumulated value of an initial investment of
£40,000 at the end of 25 years if the annual rates of return are assumed to conform to the varying interest rate model and follow a Gamma (16,200) distribution. (You can find formulae for the mean and variance of a Gamma ( α , λ ) distribution on page 12 of the Tables.)

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CT1-15: Stochastic interest rate models

Moments of An
Remember that An is a random variable that represents the accumulated value at time n of a series of annual investments, each of amount 1, at times 0,1, 2,… , n - 1 .

i1 , i2 , … , in are independent random variables, each with a mean j and a variance s2 .
From Equation (1.3):
An -1 = (1 + i1 )(1 + i2 ) … (1 + in -1 )
+ (1 + i2 ) … (1 + in -1 )

+ (1 + in - 2 )(1 + in -1 )
+ (1 + in -1 ) and: An = (1 + i1 )(1 + i2 ) … (1 + in -1 )(1 + in )
+ (1 + i2 ) … (1 + in )

+ (1 + in -1 )(1 + in )
+ (1 + in )
It follows from Equation (1.3) (or from Equation (1.1)) that, for n ≥ 2 :

An = (1 + i n )(1 + An − 1)

(1.8)

Equation (1.8) can also be deduced easily by general reasoning.
An−1 is the accumulated value at time n − 1 of a series of annual payments, each of amount 1, at times 0, 1, 2, … , n − 2 . The value, at time n − 1 , of the same series of payments together with an extra payment at time n − 1 is 1 + An−1 . Accumulating this value forward to time n gives (1 + in )(1 + An −1 ) and this is equivalent to An .

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The usefulness of Equation (1.8) lies in the fact that, since An - 1 depends only on the values i1, i2 ,… , in - 1 , the random variables in and An - 1 are independent. (By assumption the yields each year are independent of one another.) Accordingly,
Equation (1.7) permits the development of a recurrence relation from which may be found the moments of An . We illustrate this approach by obtaining the mean and variance of An .
Let:

mn = E [ An ] and let:
2
mn = E [ An ]

Since:

A1 = 1 + i1 it follows that:

E[ A1 ] = E[1 + i1 ] = 1 + E[i1 ] = 1 + j
⇒ µ1 = 1 + j and: 2
2
m1 = E[ A1 ] = E[(1 + i1 ) 2 ] = 1 + 2 E[i1 ] + E[i1 ]

m1 = 1 + 2 j + j 2 + s 2 where, as before, j and s 2 are the mean and variance of the yield each year.
Taking expectations of Equation (1.8), we obtain (since in and An - 1 are independent): µ n = (1 + j )(1 + µ n − 1)

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CT1-15: Stochastic interest rate models

Applying the recursive formula repeatedly for each year, gives:

mn = (1 + j )[1 + mn -1 )] = (1 + j ) + (1 + j ) mn -1
= (1 + j ) + (1 + j ) 2 [1 + mn - 2 ] = (1 + j ) + (1 + j ) 2 + (1 + j ) 2 mn - 2
=
= (1 + j ) + (1 + j ) 2 + (1 + j )3 +
+ (1 + j ) n which is the formula for sn| , calculated at the expected interest rate j .
This equation, combined with initial value µ 1 , implies that, for all values of n:

µ n = sn|

at rate j

(1.9)

Thus the expected value of An is simply s n| , calculated at the mean rate of interest. Since:
2
2
2
An = (1 + 2i n + i n )(1 + 2An − 1 + An − 1)

by taking expectations we obtain, for n ≥ 2 :

mn = (1 + 2 j + j 2 + s 2 )(1 + 2µ n − 1 + mn − 1)

(1.10)

As the value of µ n − 1 is known (by Equation (1.9)), Equation (1.10) provides a recurrence relation for the calculation successively of m2 , m3 , m4 ,… . The variance of An may be obtained as:
2
2 var[ An ] = E [ An ] - (E [ An ])2 = mn - mn

(1.11)

In principle the above arguments are fairly readily extended to provide recurrence relations for the higher moments of An .

The arguments are also easily extended to provide recurrence relations for other series of investments. Letting Ft again represent the accumulated amount at time t of all money invested before time t and let Pt be the amount of money invested at time t. We stated in Equation (1.1) that:

Ft = (1 + it )( Ft −1 + Pt −1 )

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Therefore the mean of Ft can be found from the recursive relationship:
E ( F0 ) = 0
E ( Ft ) = (1 + j )( E ( Ft -1 ) + Pt -1 )

Question 15.6
Explain what is wrong with the following derivation a student has used.
We know that: Ak = (1 + ik )(1 + Ak −1 )

(1)

So, finding the variance of both sides gives:

var( Ak ) = var[(1 + ik )(1 + Ak −1 )]

(2)

By independence, this is:

var( Ak ) = var[(1 + ik )] var[(1 + Ak −1 )]

(3)

But we know that var[(1 + ik )] = var(ik ) = s2 and var[(1 + Ak −1 )] = var( Ak −1 )

(4)

So this gives: var( Ak ) = s2 var( Ak −1 )

(5)

Applying this equation recursively gives: var( An ) = s 2 var( An -1 ) = s 4 var( An - 2 ) =

= s 2 n var( A0 )

(6)

Since there is definitely no money in the fund at the outset, we know that:

var( A0 ) = 0

(7)

var( An ) = s2 n × 0 = 0

(8)

So:

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Example
A company considers that on average it will earn interest on its funds at the rate of 4% pa. However, the investment policy is such that in any one year the yield on the company’s funds is equally likely to take any value between 2% and 6%.
For both single and annual premium accumulations with terms of 5, 10, 15, 20, and 25 years and single (or annual) investment of £1, find the mean accumulation and the standard deviation of the accumulation at the maturity date. (Ignore expenses.)

Solution
The annual rate of interest is uniformly distributed on the interval [0.02,0.06] .
The corresponding probability density function is constant and equal to 25 (ie
1 (0.06 - 0.02) ). The mean annual rate of interest is clearly:

j = 0.04 and the variance of the annual rate of interest is:

s2 =

1
4
(0.06 − 0.02)2 = 3 × 10−4
12

We are required to find

E [ An ] ,

1

(var[ An ]) 2 ,

E [Sn ] , and

1

(var[Sn ]) 2

for

n = 5,10,15,20 and 25 .
Substituting the above values of j and s 2 in Equations (1.5) and (1.7), we immediately obtain the results for the single premiums.

For example:
E[ S5 ] = 1.045 = 1.21665

var[ S5 ] = (1 + 0.08 + 0.042 + 4 10 -4 )5 - 1.0410 = 0.000913
3
fi standard deviation [ S5 ] = 0.03021
For the annual premiums we must use the recurrence relation (1.10) (with µ n − 1 = sn − 1| at 4%) together with Equation (1.11).

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Equation (1.9) is used to calculate E[ An ] .
The results are summarised in Table 1. It should be noted that, for both annual and single premiums, the standard deviation of the accumulation increases rapidly with the term.

Table 1
Term
(years)

Single premium £1

Annual premium £1

5

Mean accumulation (£)
1.21665

Standard deviation (£)
0.03021

Mean accumulation (£)
5.63298

Standard deviation (£)
0.09443

10

1.48024

0.05198

12.48635

0.28353

15

1.80094

0.07748

20.82453

0.57899

20

2.19112

0.10886

30.96920

1.00476

25

2.66584

0.14810

43.31174

1.59392

Question 15.7
Check the values given in the table for E[ A5 ] and standard deviation [ A5 ] .

Question 15.8
An investor invests 1 unit at time t = 0 and a further 2 units at time t = 2 . Use recursive formulae to calculate the accumulated value of the fund at time t = 5 , assuming the varying interest rate model applies and that the expected interest rate for each year is 10%.
Calculate the accumulated value based on the corresponding deterministic model and comment on your answer.

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CT1-15: Stochastic interest rate models

The log-normal distribution
In general a theoretical analysis of the distribution functions for An and Sn is somewhat difficult, even in the relatively simple situation when the yields each year are independent and identically distributed. There is, however, one special case for which an exact analysis of the distribution function for Sn is particularly simple. Because of the compounding effect of interest, the accumulated value of an investment bond grows multiplicatively. This makes the log-normal distribution a natural choice for modelling the annual growth factors 1+ i , since a log-normal random variable can take any positive value and has the following multiplicative property:
2
If X 1 ~ log N ( µ 1, σ 12 ) and X 2 ~ log N ( µ 2, σ 2 ) are independent random variables, then:

2
X 1 X 2 ~ log N ( µ 1 + µ 2, σ 12 + σ 2 )

The graph below illustrates the shape of the PDF of a typical log-normal distribution used to model annual growth rates. log N (0.075, 0.12 )

Question 15.9
What are the mean and standard deviation of the log-normal distribution shown in the graph? Suppose that the random variable log(1 + it ) is normally distributed with mean µ and variance σ 2 . In this case, the variable (1 + it ) is said to have a log-normal distribution with parameters µ and σ 2 .

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Page 17

Equation (1.2) is equivalent to:

log Sn =

n

∑ log(1 + it )

t =1

The sum of a set of independent normal random variables is itself a normal random variable.
Hence, when the random variables (1 + it ) (t ≥ 1) are independent and each has a log-normal distribution with parameters and µ and

σ 2 , the random variable Sn has a log-normal distribution with parameters nµ and nσ 2 .

ie

log Sn ~ N (nµ , nσ 2 )

or

log Sn − nµ
~ N (0,1) σ n

Since the distribution function of a log-normal variable is readily written down in terms of its two parameters, in the particular case when the distribution function for the yield each year is log-normal we have a simple expression for the distribution function of Sn .
Similarly for the present value of a sum of 1 due at the end of n years:

Vn = (1 + i1 )-1 … (1 + in )-1 fi logVn = - log(1 + i1 ) -

- log(1 + in )

Since, for each value of t, log(1 + it ) is normally distributed with mean m and variance σ 2 , each term on the right hand side of the above equation is normally distributed with mean - m and variance s 2 . Also the terms are independently distributed. So, logVn is normally distributed with mean - n m and variance nσ 2 .
That is, Vn has log-normal distribution with parameters −nµ and nσ 2 .
By statistically modelling Vn , it is possible to answer questions such as:


to a given point in time, for a specified confidence interval, what is the range of values for an accumulated investment



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CT1-15: Stochastic interest rate models

Although outside the scope of this Subject, it is interesting to note that such techniques may be extended readily to model and predict the behaviour of portfolios of investments. These techniques are referred to as “Value at Risk” or
“VaR” methods and are covered in more detail in subjects CT8, Financial
Economics, and ST6, Finance and Investment Specialist Technical B. One possible definition of Value at Risk is a portfolio’s maximum loss from an adverse market movement, within a specified confidence interval and over a defined period of time.
As with all statistical modelling techniques, the results of VaR can only be as good as the statistical model of the performance of the underlying investments.
In all investment markets, even seemingly efficient ones, it continues to prove very difficult to choose a reliable statistical model which is robust over even short periods of time.

Example
If the annual growth factors 1 + ik for individual years have a log N ( µ , σ 2 ) distribution, then the distribution functions for the accumulated value Sn and the discounted value
Vn can be expressed in terms of Φ( x ) , the distribution function of the standard normal distribution: Ê log s - n m ˆ
P(Sn £ s) = F Á and Ë s n ˜
¯

Ê - log s - n m ˆ
P (Vn £ s ) = 1 - F Á
˜
Ë s n
¯

Proof
The formula for the distribution function of Sn follows immediately from the result that

Sn has a log-normal distribution with parameters nµ and nσ 2 .
Vn is log-normally distributed with parameters −nµ and nσ 2 and so:
Ê log s - ( - n m ) ˆ
P (Vn < s ) = F Á
˜
Ë
¯
s n
Ê - log s - n m ˆ
= 1- FÁ
˜
Ë s n
¯
These results enable us to calculate probabilities for the range of the accumulated amount of a single payment.

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Example
Find the upper and lower quartiles for the accumulated value at the end of 5 years of an initial investment of £1,000, using the varying interest rate model and assuming that the annual growth rate has a log-normal distribution with parameters µ = 0.075 and

σ 2 = 0.0252 .
Solution
By definition, the accumulated amount X = 1,000S5 will exceed the upper quartile u with probability 25%, ie:

0.75 = P( X ≤ u) = P(1,000S5 ≤ u) = P( S5 ≤ u 1,000)
So, from the formula for the distribution function:
Ê log(u 1, 000) - n m ˆ
0.75 = P( S5 £ u 1, 000) = F Á
˜
Ë
¯
s n
From the tables of the distribution function of the normal distribution on page 162 of the Tables, we find that F(0.6745) = 0.75 . So, we must have: log(u 1, 000) - n m
= 0.6745 s n

ie u = 1, 000e5 m +0.6745s

5

= £1,511

Similarly, the lower quartile is:

l = 1, 000e5 m -0.6745s

5

= £1, 401

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CT1-15: Stochastic interest rate models

Question 15.10
A man now aged exactly 50 has built up a savings fund of £400,000. In order to retire at age 60, he will require a fund of at least £600,000 at that time. Calculate the probability that, if he makes no further contributions to the fund, he will be able to retire at age 60. Assume that annual growth rates vary independently from year to year and have the log-normal distribution shown in the graph on page 16.

Question 15.11
Derive expressions for the mean and variance of the accumulated value of 1 unit after n years for the fixed interest rate model, assuming that the annual growth rate has a lognormal distribution with parameters µ and σ 2 .

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3

Page 21

Exam-style question
Before you finish, try the following two exam-style questions on stochastic interest rates. The solutions are over the page.

Question 1
Interest rates are fixed at either 4% pa with probability 0.2 or 5% pa with probability
0.8. What is the standard deviation of the present value of a payment of £25,000 in 5 years’ time?
[3]

Question 2
A lump sum of $14,000 will be invested at time 0 for 4 years at a constant annual rate of interest i. (1 + i ) has a log-normal distribution with mean 1.05 and variance 0.007.
What is the probability that the investment will accumulate to more than $20,000 in 4 years’ time?
[5]

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CT1-15: Stochastic interest rate models

Solution 1
The present value will either be
25, 000
1.055

25, 000
1.045

= 20548.18 with probability 0.2, or

= 19588.15 , with probability 0.8.

The mean of these values is 20,548.18 ¥ 0.2 + 19,588.15 ¥ 0.8 = 19, 780.16 .
The standard deviation of these values is:
20,548.182 ¥ 0.2 + 19,588.152 ¥ 0.8 - 19780.162 = £384

Solution 2
We first need to find the values of the parameters for the lognormal distribution. By using the formulae for the mean and variance from the Tables:
1
exp µ + 2 σ 2 = 105
.

and

exp 2 µ + σ 2 (exp σ 2 − 1) = 0.007

Ê 0.007 ˆ fi 1.052 (exp s 2 - 1) = 0.007 fi s 2 = ln Á
+ 1 = 0.006329
Ë 1.052 ˜
¯
which gives us that µ = 0.04563 .
We know that (1 + i ) ~ logN ( µ , σ 2 ) . So, if the accumulated value of an investment of
1 after n years is S n , then we have that S n ~ logN (nµ , n 2σ 2 ) .
P(14,000S 4 > 20,000) , where:

Here we need

S 4 ~ logN (4 µ ,16σ 2 ) = logN (01825,01013)
.
.
We require the probability:

P(14,000S 4 > 20,000) = P( S 4 > 1429) = P(ln S 4 > 0.3567)
.

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If Z ~ N (0,1) , then:
0.3567 - 0.1825 ˆ
Ê
P (ln S 4 > 0.3567) = P Á Z >
˜ = P( Z > 0.547) = 0.292
Ë
¯
0.1013

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4

CT1-15: Stochastic interest rate models

End of Part 4
Congratulations, you have now completed Part 4 of the Subject CT1 Notes.

Review
Before looking at the Question and Answer Bank we recommend that you review the key topics from Chapters 14 and 15, or maybe re-read the chapter summaries.

Question and Answer Bank
You should now be able to answer the questions in Part 4 of the Question and Answer
Bank. We recommend that you work through several of these questions before attempting the assignment.

Assignments
On completing this part, you should be able to attempt the questions in the X4
Assignment.

End of the course
This is also the end of the course. You should therefore now start your revision. This should include all of the following:


re-reading all Chapter Summaries



re-reading all the Core Reading and if necessary the whole course



looking at each syllabus item



doing all the questions in the Q&A Bank



doing the Mock Exam and having it marked



looking at the specimen paper



reviewing old assignments



looking at old 102 and Subject A1 exam questions

Good luck!

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Page 25

Chapter 15 Summary
A stochastic interest rate model provides information about the distribution of financial outcomes. This distribution can be used to find best estimates and probabilities.
The varying interest rate model and the fixed interest rate model provide formulae for the mean and variance of the accumulated amount of a fund or the present value of a future payment.
Varying interest rate model (single premium):

E ( Sn ) = (1 + j )n

var( S n ) = [(1 + j )2 + s 2 ]n - (1 + j ) 2 n

Fixed interest rate model (single premium):

E ( S n ) = E[(1 + i ) n ]

var( Sn ) = E[(1 + i ) 2 n ] - ( E[(1 + i ) n ]) 2

For the varying interest rate model, the variance and higher moments of the accumulated amount of a series of payments can be calculated using recursive formulae.
Varying interest rate model (annual premium)

E ( An ) = sn|

at rate j

Recursive formulae for E ( An ) :

E ( A0 ) = 0 and: E ( Ak ) = (1 + j )[1 + E ( Ak -1 )] ( k = 1, 2,..., n )
Recursive formulae for var( An ) :
2
E ( A0 ) = 0

and:
2
2
E ( Ak ) = [(1 + j ) 2 + s 2 ][1 + 2 E ( Ak -1 ) + E ( Ak -1 )]

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( k = 1, 2,..., n )

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CT1-15: Stochastic interest rate models

Then:
2
var( An ) = E ( An ) - [ E ( An )]2

For the fixed interest rate model with annual premiums, calculate the mean and variance directly from the definitions.
The log-normal distribution can be used to model the annual growth rate. This allows probabilities to be determined in terms of the distribution function of the normal distribution. The lognormal model formulae for the varying interest rate model are:

log Sn - n m

s n

~ N (0,1)

Ê log s - n m ˆ
P ( Sn £ s) = F Á
Ë s n ˜
¯

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Ê - log s - n m ˆ
P ( Sn 1 £ s) = 1 - F Á
˜
Ë s n
¯

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CT1-15: Stochastic interest rate models

Page 27

Chapter 15 Solutions
Solution 15.1
The mean is:

j = E (ik ) = 0.2 ¥ 0.06 + 0.7 ¥ 0.08 + 0.1 ¥ 0.10 = 0.078
2
We can calculate the variance using the formula var(ik ) = E (ik ) - [ E (ik )]2 :
2
E (ik ) = 0.2 ¥ 0.062 + 0.7 ¥ 0.082 + 0.1 ¥ 0.102 = 0.0062
2
fi s 2 = var(ik ) = E (ik ) - [ E (ik )]2 = 0.0062 - 0.0782 = 0.000116 = 0.01082

So, the mean is 7.8% and the standard deviation is 1.08%.

Solution 15.2
The variance of the accumulated value must be calculated directly:

var = 5, 0002 ( E[(1 + i )10 ] - ( E[(1 + i )5 ]) 2 )
= 5, 0002 [(0.2 ¥ 1.0610 + 0.7 ¥ 1.0810 + 0.1 ¥ 1.1010 )
-(0.2 ¥ 1.065 + 0.7 ¥ 1.085 + 0.1 ¥ 1.105 ) 2 ]
= 5, 0002 ¥ (2.128791 - 1.4572262 ) = (£363)2
(You need to keep a few extra decimals in this last calculation to avoid losing accuracy, since the calculation involves subtracting two numbers of similar magnitude.)

Solution 15.3
There are n different years in which the 2% could fall and so:
1 Ê 1ˆ
Probability = n ¥ ¥ Á ˜
3 Ë 3¯

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n -1

=

n
3n

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CT1-15: Stochastic interest rate models

Solution 15.4
The first step, namely: n 2
E[ S n ] = ’ E[(1 + it ) 2 ] t =1 n (

)

= ’ var[(1 + it )] + E 2 [(1 + it )] t =1

comes straight from the definition of the variance: var[ g (i )] = E[ g 2 (i )] - E 2 [ g (i )] fi E[ g 2 (i )] = var[ g (i )] + E 2 [ g (i )]
The second step, namely:

’ (var[(1 + it )] + E 2[(1 + it )]) n t =1 n (

= ’ s 2 + (1 + j ) 2 t =1

)

comes from the facts that: var[(1 + it )] = var[it ] = s 2

and

E 2 [(1 + it )] = (1 + E[it ]) = (1 + j ) 2
2

The last step follows since the factor s 2 + (1 + j ) 2 is independent of n.

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Solution 15.5
Let ik be the return in year k. Therefore: ik ∼ Gamma (16, 200)
Using the formulae for the mean and variance of the gamma distribution from page 12 of the Tables:

j = E (ik ) =

α 16
=
= 0.08 λ 200

s 2 = var(ik ) =

a
16
=
= (0.02) 2
2
2 l 200

So, the mean of the accumulated amount is:
40, 000 E ( S25 ) = 40, 000(1 + j ) 25 = 40, 000 ¥ 1.0825 = £273,900 and the variance is:

var(40, 000 S25 ) = 40, 0002 var( S25 )

(

= 40, 0002 [(1 + j ) 2 + s 2 ]25 - (1 + j )50

)

= 40, 0002 [(1.082 + 0.022 ) 25 - (1.08)50 ]
= (£25, 400) 2
(If you keep exact values during the calculation, you should get £273,939 and
(£25, 417) 2 .)

Solution 15.6
Step (3) is invalid. Independence doesn’t allow you to “factorise” variances (which is
2
why we had to work out E ( Ak ) instead).

In fact, this step would only be valid if the terms (1 + ik ) and (1 + Ak -1 ) are constant. If this was the case, we would be dealing with a deterministic model, and the calculation then gives the correct answer of zero, since there is no uncertainty in the final amount.

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CT1-15: Stochastic interest rate models

Solution 15.7
E[ A5 ] = s5|4%

1.045 - 1
=
= 5.63298
0.04 1.04

In order to calculate the standard deviation of A5 , we first need to calculate m5 from the recursive formula: mn = (1 + 2 j + j 2 + s 2 )(1 + 2 mn -1 + mn -1 )
The values required are tabulated below: n mn

1
2
3
4
5

1.08173
4.50189
10.54158
19.50853
31.73933018

mn
1.04
2.1216
3.24646
4.41632
5.6329755

var[ A5 ] = m5 - ( E[ A5 ])5 = 31.73933018 - 5.63297552 = 0.0089172
The standard deviation is:
0.0089172½ = 0.09443
Warning: this answer is very sensitive to rounding. Try to carry forward as many decimal places as possible.

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Page 31

Solution 15.8
Let An be the accumulated amount at the end of n years of the series of cashflow given.
We can use a recursive approach similar to before to obtain:
E ( A0 ) = 0
E ( A1 ) = 1.1 ¥ [1 + E ( A0 )] = 1.1 ¥ (1 + 0) = 1.1
E ( A2 ) = 1.1 ¥ E ( A1 ) = 1.1 ¥ 1.1 = 1.21
E ( A3 ) = 1.1 ¥ [2 + E ( A2 )] = 1.1 ¥ (2 + 1.21) = 3.531
E ( A4 ) = 1.1 ¥ E ( A3 ) = 1.1 ¥ 3.531 = 3.8841
E ( A5 ) = 1.1 ¥ E ( A4 ) = 1.1 ¥ 3.8841 = 4.27251
If we had used the corresponding deterministic model (ie an “ordinary” actuarial calculation) based on the expected interest rate of 10%, we would have got:
1.15 + 2 ¥ 1.13 = 1.61051 + 2 ¥ 1.331 = 4.27251
So the corresponding deterministic model gives the same answer.

Solution 15.9
The mean and variance of the
2

log N ( m , s 2 )

distribution are

e m +½s

2

and

2

e 2 m +s (es - 1) . These formulae are given in the Tables on page 14.
So, with m = 0.075 and s 2 = 0.12 , the mean is 1.0833 and the variance is 0.10862 ie the annual rate of growth will have a mean value of 8.33% and a standard deviation of
10.86%.

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CT1-15: Stochastic interest rate models

Solution 15.10
If he makes no further contributions, the accumulated fund at age 60 will be
400,000 S10 .
So, the probability that the fund will be sufficient for him to retire is:

600, 000 ˆ
Ê
P (400, 000 S10 ≥ 600, 000) = 1 - P Á S10 £
Ë
¯
400, 000 ˜
Ê log1.5 - 10 m ˆ
= 1- FÁ
˜
Ë s 10
¯
= 1 - F( -1.0895) = F(1.0895) = 0.862

Solution 15.11
For the fixed interest rate model:

S n = (1 + i ) n
If 1 + i ~ log N ( m , s 2 ) , then: log(1 + i ) ~ N ( m , s 2 )
So:
log(1 + i ) n = n log(1 + i ) ~ N (n m , n 2s 2 )
So:

S n = (1 + i )n ~ log N (n m , n 2s 2 )
Using the formulae for the mean and variance of the log-normal distribution:

E ( Sn ) = e

n m + 1 n 2s 2
2

2 2

2 2

var( S n ) = e2 n m + n s (e n s - 1)

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CT1: Q&A Bank Part 1 – Questions

Page 1

Part 1 – Questions
Introduction
The Question and Answer Bank is divided into five parts. The first four parts of the
Question and Answer Bank include a range of short and long questions to test your understanding of the corresponding part of the Course Notes, whilst the last part contains a set of exam-style questions covering the whole course. For each part the questions may require knowledge from earlier parts of the course.
The Question and Answer Bank contains a mixture of shorter questions and longer exam-style questions. Some of the shorter questions are not exam standard, but they are a good test of your understanding and will aid your progress towards the correct level.
We strongly recommend that you use these questions to practise the techniques necessary to pass the exam. Do not use them as a set of material to learn but attempt the questions for yourself under strict exam style conditions, before looking at the solutions provided.
This distinction represents the difference between active studying and passive studying.
Given that the examiners will be aiming to set questions to make you think (and in doing so they will be devising questions you have not seen before) it is much better if you practise the skills that they will be testing.
It may also be useful to you if you group a number of the questions together to attempt under exam time conditions. Ideally three hours would be set aside, but anything from one hour (ie 35 marks) upwards will help your time management.

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CT1: Q&A Bank Part 1 – Questions

Question 1.1
A rate of interest of 4% pa convertible monthly is equivalent to what annual effective rate of discount?
[2]

Question 1.2
Calculate the total present value as at 1 June 2008 of payments of £100 on 1 January
2009 and £200 on 1 May 2009, assuming a rate of interest of 12% pa convertible quarterly. [2]

Question 1.3
(i)

Define a real rate of interest.

[1]

(ii)

Define a money rate of interest.

[1]

(iii)

Under what circumstances would you expect the real rate of interest to be lower than the money rate of interest?
[1]
[Total 3]

Question 1.4
Would you use a real rate of interest or a money rate of interest to calculate the present value of repayments on a student loan? The repayment on the loan is £100 per month for the next five years.
[1]

Question 1.5
I pay £100 into an account today. The account pays simple interest at a rate of 4% pa.
How much will I have in my account in five year’s time?
[1]

Question 1.6
A man invests £ X on the birth date of his daughter to provide a payment of £20 on her birthday forever. If the annual effective rate of interest is 5%, calculate X .
[2]

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Page 3

Question 1.7
State which of the following equations are correct:
(i)

d ( p ) = p[1 - (1 - d )1/ p ]

(ii)

i( p) =

(iii)

È d ( p) ˘ v = Í1 ˙ p ˙
Í
Î
˚

1
[(1 + i ) p - 1] p [½]
[½]

p

an| - nv n

[½]

(iv)

( Ia ) n| =

(v)

sn +1| = 1 + sn|

[½]

(vi)

( Ia ) n| = an| + v( Ia ) n -1|

[½]

(vii)

an = 1 + an

[½]

i

[½]

[Total 3½]

Question 1.8
An annuity provides payments of $40 at the end of each month forever. If the interest rate is 10% pa convertible quarterly, calculate the present value of the annuity.
[2]

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Question 1.9
An annuity certain is payable monthly in advance for 40 years. The annuity is to be paid at the rate of £100 pa for the first 20 years, £120 pa for the next 5 years and £200 pa for the last 15 years. The following expressions (calculated at the effective annual interest rate) have been suggested for finding the present value of the payments as at the commencement date of the annuity. State which of them are correct.

i

(i)

(100a20| + 120v 20 a5| + 200v 25 a15| )

(ii)

(12)
(12)
(12)
200a40| - 80a25| - 20a20|

[1]

(iii)

(12)
(12)
(12)
100(1 + a39| ) + 20v 20 (1 + a19| ) + 80v 25 (1 + a14| )

[1]

i

(12)

[1]

[Total 3]

Question 1.10
A continuous payment stream is such that the level rate of payment in year t is
100 ¥ 1.05t -1 , t = 1, 2,… ,10 . Calculate the present value of the payment stream as at its commencement date, assuming a rate of interest of 10% pa.
[3]

Question 1.11
An inflation index Q(t ) is such that Q(1.1.03) = 724 and Q(1.1.08) = 913 . Calculate the average annual rate of inflation over the period from 1 January 2003 to 1 January
2008.
[2]

Question 1.12
Calculate the accumulated value of £6.34, assuming a force of interest of 9% pa, after:
(i)

3 months

[1]

(ii)

3 years

[1]

(iii)

7 years and 5 days

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[1]
[Total 3]

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Question 1.13
When i = 0.07 Calculate:

(i)

i (6)

[1]

(ii)

d (6)

[1]

(iii)

i (4)

[1]

(iv)

d (2)

[1]
[Total 4]

Question 1.14
A loan of £3,000 is repayable in 91 days at a simple discount rate of 4% per annum.
Calculate the amount repayable in 91 days’ time.
[2]

Question 1.15
An investor is considering two investments. One is a 3-month deposit account which pays a rate of return of 4% pa convertible half-yearly. The second is a 3-month government bill. Calculate the annual simple rate of discount available from the government bill if both investments provide the same effective rate of return.
[3]

Question 1.16
A series of payments is to be received annually in advance. The first payment will be
£10. Thereafter, payments increase by £2 per annum. The last payment will be made at the beginning of the tenth year. Which of the following are correct expressions for the present value of the annuity?
9

9

t =0

(i)

t =0

∑ 8vt + 2∑ tvt

(ii)

10 a10 + 2( Ia )9

(iii)

8 a10 + 2( Ia )10

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[3]

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CT1: Q&A Bank Part 1 – Questions

Question 1.17
Write down simple (but exact) formulae expressing:
(i)

v in terms of d

[½]

(ii)

d in terms of d

[½]

(iii)

i in terms of d

[½]

(iv)

v in terms of i

[½]

(v)

d in terms of i

[½]

(vi)

i in terms of v

[½]

(vii)

d in terms of i

[½]

(viii)

d in terms of d

[½]
[Total 4]

Question 1.18
(i)

Calculate the effective annual rate of interest corresponding to:
(a)
(b)

(ii)

a nominal rate of 11% pa convertible half yearly a nominal rate of interest of 12% pa convertible monthly

[2]

Calculate the rate of interest convertible monthly corresponding to:
(a)

an effective rate of 14.2% pa

(b)

a nominal rate of 11% pa convertible three times a year

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[2]
[Total 4]

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Question 1.19
A woman who has won a prize is offered a lump sum of £100,000 to invest now or
£55,000 to invest at the end of this year and another £55,000 to invest at the end of the following year. If all investments are assumed to earn 7% pa, which should she choose if she intends to withdraw the money after:
(i)

4 years

(ii)

2 years

[3]

Question 1.20
Let X denote the present value of an annuity consisting of payments of £2,000 payable at the end of each of the next 8 years, valued using an interest rate of 8% pa convertible quarterly and let Y denote the present value of an annuity consisting of payments of
£4,000 payable at the end of every fourth year for the next 16 years, valued using an interest rate of 8% pa convertible half yearly. Calculate the ratio X / Y .
[6]

Question 1.21
Calculate the effective annual rate of interest for:
(i)

a transaction in which £200 is invested for 18 months to give £350.

(ii)

a transaction in which £100 is invested for 24 months and another £100 for 12 months (starting 12 months after the first investment) to give a total of £350.
[2]
[Total 3]

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[1]

© IFE: 2009 Examinations

Page 8

CT1: Q&A Bank Part 1 – Questions

Question 1.22
(i)

Evaluate the following at i = 9% :
(a)
(b)

( Ia )60|

(c)

( Ia )100|

(d)
(ii)

( Ia )60|

( Ia )100|

[4]

Evaluate the following at i = 7% :
(a)

( Is )1|

(b)

( Is )10|

[2]
[Total 6]

Question 1.23
(i)

Evaluate the following functions at i = 9% :
(a)
(b)

(4) a4| (c)
(ii)

(4) a3| s10|

[3]

Evaluate the following functions at i = 25% :
(a)

(12) a10| (b)

a (12)
1 |
6

[2]

2

[Total 5]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 1 – Questions

Page 9

Question 1.24
Calculate the accumulated value as at 1 January 2008 of a series of payments of £100 paid every two years from 1 January 1968 to 1 January 2006 inclusive, which accumulate at 12% pa.
[2]

Question 1.25
Calculate the present value as at 1 June 2008 of 41 monthly payments each of £100 commencing on 1 January 2009, assuming a rate of interest of 10% pa convertible half yearly. [3]

Question 1.26
I invest £450.00 for 13 months at i = 0.09 , then switch to an investment that pays interest at a force of d = 0.086178 for 11 months and then d = 0.113329 for two years.
How much has my investment accumulated to?
[3]

Question 1.27
An account pays interest at an effective annual rate of interest of:
15% on Mondays, Tuesdays and Fridays
12% on Wednesdays and Thursdays
10% at weekends
Calculate the equivalent level effective annual rate of interest (for transactions that last for a whole number of weeks).
[2]

Question 1.28
An investor deposits £2,000, then withdraws level annual payments starting one year after the deposit was made. Immediately after the 11th annual drawing, the investor has
£400 left in the account. Calculate the amount of each withdrawal, given that the annual rate of interest is 8%.
[4]

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© IFE: 2009 Examinations

Page 10

CT1: Q&A Bank Part 1 – Questions

Question 1.29
If d = 0.01t for 0 < t < 10 , calculate the value of ( Ia )10| .

[4]

Question 1.30
Calculate the present value of an annuity payable annually in advance for a term of 20 years such that the payment is £500 in year 1, £550 in year 2, £600 in year 3 etc.
Assume a rate of interest of 5% pa for the first twelve years and 7% pa thereafter.
[5]

Question 1.31
A 90-day government bill was bought by an investor for a price of £91 per £100 nominal. After 30 days the investor sold the bill to a second investor for a price of
£93.90 per £100 nominal. The second investor held the bill to maturity when it was redeemed at par.
Determine which investor obtained the higher annual effective rate of return.

[3]

Question 1.32
The force of interest δ (t ) is:

δ (t ) = 0.005t + 0.0001t 2 for all t
(i)

At t = 8 , calculate the accumulated value of an investment of £100 made at time t = 0.
[3]

(ii)

Calculate the constant annual effective rate of interest over the eight year period.
[2]
[Total 5]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 1 – Questions

Page 11

Question 1.33
From first principles, derive expressions, which may involve an| , i and i ( p ) (but no other compound interest functions), for each of the following functions:
(i)

( an|p ) (assuming n is divisible by p )

an| − nv n

[4]

(ii)

( Ia ) n| =

(iii)

the present value of an annuity certain payable annually in arrears for n years,

δ

where the payment at the end of the year t is t 2 .

[4]

[7]
[Total 15]

Question 1.34
(i)

Calculate the combined present value of an immediate annuity payable monthly in arrears such that payments are £1,000 pa for the first 6 years and £400 pa for the next 4 years, together with a lump sum of £2,000 at the end of the 10 years.
[3]

(ii)

Calculate the amount of the level annuity payable continuously for 10 years having the same present value as the payments in (i).
[3]

(iii)

Calculate the accumulated values of the first 7 years’ payments at the end of the
7th year for the payments in (i) and (ii).
[3]

Basis: Assume an interest rate of 12% pa convertible monthly.

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[Total 9]

© IFE: 2009 Examinations

Page 12

CT1: Q&A Bank Part 1 – Questions

Question 1.35
(i)

Define the accumulation factor A(t , t + h) and give a formula for the force of interest d (t ) per unit time in terms of the accumulation factor.
[2]

(ii)

The force of interest d (t ) at time t (measured in years) is given by d (t ) = 0.01t + 0.04 .
(a)

Calculate the corresponding nominal rate of interest for the period t = 1 to t = 2 .

(b)

If an investment of 1 is made at time t =

1
2

, calculate the value to which

it will have accumulated by time t = 6 .
(iii)

[6]

Calculate the accumulated value after 6 months of an investment of £100 at time 0 at the following rates of interest:
(a)

a force of interest of 0.05 pa

(b)

a rate of interest of 5% pa convertible monthly

(c)

an effective rate of interest of 5% pa.

[3]
[Total 11]

Question 1.36
(i)

Calculate the present value of a continuously payable annuity that is initially paid at a rate of £200 pa but decreases linearly to £100 pa after 10 years, assuming a force of interest of d (t ) = 0.2 - 0.01t , where t is the time from commencement (measured in years).
[5]

(ii)

Calculate the accumulated value of the annuity after 10 years at the same force of interest.
[2]
[Total 7]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 1 – Questions

Page 13

Question 1.37
(i)

Assuming a rate of interest of 6% pa, calculate the present value as at 1 January
2008 of the following annuities, each with a term of 25 years:
(a)

(b)

(ii)

an annuity payable annually in advance from 1 January 2009, initially of
£3,000 pa, and increasing by £500 pa on each subsequent 1 January an annuity as in (i), but only 10 increases are to be made, the annuity then remaining level for the remainder of the term
[6]

An investor is to receive a special annual annuity for a term of 10 years in which payments are increased by 5% compound each year to allow for inflation. The first payment is to be £1,000 on 1 November 2009. Calculate the accumulated value of the annuity payments as at 31 October 2026 if the investor achieves an effective rate of return of 4% per half year.
[4]
[Total 10]

Question 1.38
A payment stream, where the payment at time t is 1 + 0.3t , is received continuously between times 7 and 9.
Calculate the present value of this payment stream at time 5 if the force of interest at time t is given by:

⎧0.002 + 0.01t + 0.0004t 2 0 ≤ t < 6

δ (t ) = ⎨
0.01 + 0.003t
6 ≤ t < 10

0.04 t ≥ 10

[8]

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CT1: Q&A Bank Part 1 – Questions

Question 1.39
The force of interest at time t is given by:

0≤t 6 , the present value is:

100e

-0.268

È t
˘
exp Í - Ú 0.06 ds ˙
Í 6
˙
Î
˚

{

= 100e -0.268 exp - [0.06 s ] t6

}

= 100e -0.268 exp [ -0.06t + 0.36]
= 100 exp [0.092 - 0.06t ]
(ii)

[2]

The accumulated value is:

100 A(3,8) + 250 A(5,8)
È4
˘
È6
˘
È8
˘
2
= 100 exp Í Ú (0.01 + 0.01s ) ds ˙ exp Í Ú (0.15 - 0.003s ) ds ˙ exp Í Ú 0.06 ds ˙
Í3
˙
Í4
˙
Í6
˙
Î
˚
Î
˚
Î
˚
È6
˘
È8
˘
+ 250 exp Í Ú (0.15 - 0.003s 2 ) ds ˙ exp Í Ú 0.06 ds ˙
Í5
˙
Í6
˙
Î
˚
Î
˚

{

4

6

= 100 exp È 0.01s + 0.005s 2 ˘ + È 0.15s - 0.001s 3 ˘ + [0.06s ] 8
6
Î
˚3 Î
˚4

{

6

+ 250 exp È 0.15s - 0.001s 3 ˘ + [0.06s ] 8
6
Î
˚5

}

[1]

}
[1]

= 100 exp [0.04 + 0.08 - (0.03 + 0.045) + (0.9 - 0.216) - (0.6 - 0.064) + 0.12]
+ 250 exp [0.9 - 0.216 - (0.75 - 0.125) + 0.12]

[1]

= 100 exp(0.313) + 250 exp(0.179)
= 136.752 + 299.005 = 435.76

© IFE: 2009 Examinations

[1]

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CT1: Q&A Bank Part 1 – Solutions

(iii)

Page 25

The accumulated value needs to be worked out separately for the two payment streams. We get:
4

6

3

5

2
Ú (10 + 10t ) A(t ,10) dt + Ú (50 - t ) A(t ,10) dt
10
È4
˘
È6
˘
2
= Ú (10 + 10t ) exp Í Ú (0.01 + 0.01s ) ds ˙ exp Í Ú (0.15 - 0.003s ) ds + Ú 0.06 ds ˙ dt
Ít
˙
Í4
˙
3
6
Î
˚
Î
˚
4

6
È6
˘
È10
˘
+ Ú (50 - t 2 ) exp Í Ú (0.15 - 0.003s 2 ) ds ˙ exp Í Ú 0.06 ds ˙ dt
Ít
˙
Í6
˙
5
Î
˚
Î
˚

[2]

{
{

4

4

}
}

= Ú (10 + 10t ) exp È 0.01s + 0.005s 2 ˘ e0.9-0.216-0.6+0.064+ 0.6-0.36 dt
Î
˚t
3

6

6

+ Ú (50 - t 2 ) exp È 0.15s - 0.001s 3 ˘ exp [0.6 - 0.36] dt
Î
˚t
5

[1]

4

= -1, 000 Ú ( -0.01 - 0.01t ) exp È 0.04 + 0.08 - 0.01t - 0.005t 2 ˘ exp [0.388] dt
Î
˚
3

6

-

1, 000
2
3
Ú (-0.15 + 0.003t ) exp È0.9 - 0.216 - 0.15t + 0.001t ˘ exp [0.24] dt
Î
˚
3 5
[1]
4

= -1, 000 È exp(0.12 - 0.01t - 0.005t 2 ) ˘ exp [0.388]
Î
˚
3

-

6
1, 000 È exp(0.684 - 0.15t + 0.001t 3 ) ˘ exp [0.24]
˚5
3 Î

(

)

= -1, 000 e0 - e0.045 e0.388 = 67.846 + 25.753 = 93.60

(

[1]

)

1, 000 0 0.059 0.24 e -e e 3
[1]

Alternatively, we could accumulate the payment streams to times 4 and 6 respectively and then accumulate to time 10 separately.

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You must take care of your study material to ensure that it is not used or copied by anybody else.

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CT1: Q&A Bank Part 2 – Questions

Page 1

Part 2 – Questions
Question 2.1
The following data relate to the assets of a small trust fund:
Date
1 January 2007
1 April 2007
1 July 2007
1 October 2007
1 January 2008

Market value
£25,000
£29,000
£30,000
£32,000
£31,500

The only cashflow during 2007 that was not generated from the assets of the fund was an injection of £5,000 on 31 March. Calculate:
(i)

the money-weighted rate of return for the fund for 2007

[4]

(ii)

the time-weighted rate of return for the fund for 2007

[1]

(iii)

the linked internal rate of return for the fund for 2007, using quarterly subintervals. [1]
[Total 6]

Question 2.2
A loan of £1,000 is to be repaid by level monthly instalments over 10 years using an interest rate of 10% pa. What is the capital repaid in the sixth year?
[4]

Question 2.3
A woman repays a loan of £3,000 by monthly payments of £100 for 3 years. Calculate the APR for this arrangement.
[5]

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CT1: Q&A Bank Part 2 – Questions

Question 2.4
A man borrows £7,500 to buy a car. He repays the loan by 24 monthly instalments in arrears. The flat rate of interest is 9% pa.
(i)

What is his monthly payment?

(ii)

[1]

What is the APR on this transaction?

[4]
[Total 5]

Question 2.5
An investor borrows money at an effective rate of interest of 10% pa to invest in a 6year project with an internal rate of return of 18.7% pa. The cashflows for the project are: ●

an initial outlay of £25,000



regular income of £10,000 pa during the first 5 years (assumed to be payable continuously) ●

regular expenditure of £2,000 pa during the first 5 years (assumed to be payable continuously) ●

a decommissioning expense of £5,000 at the end of the 6th year.

Calculate the discounted payback period for this project.

[2]

Question 2.6
A borrower repays a loan effected on 1 January 2008 by making six equal monthly payments starting on 31 July 2008. If interest on the loan is calculated using a flat rate of 7½% pa, what is the APR for this arrangement?
[5]

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CT1: Q&A Bank Part 2 – Questions

Page 3

Question 2.7
A businessman is considering an investment which requires an initial outlay of £60,000 and a further outlay of £25,000 in eight months time.
Starting two years after the initial outlay, it is estimated that income will be received continuously for four years at a rate of £5,000 per annum, increasing to £9,000 per annum for the next four years, then increasing to £13,000 per annum for the following four years and so on, increasing by £4,000 per annum every four years until the payment stream stops after income has been received for 20 years (ie 22 years after the initial outlay). At the point when the income ceases, the investment can be sold for
£50,000.
Calculate the net present value of the project at a rate of interest of 9% per annum effective. [7]

Question 2.8
An investor borrows £1,000 in order to invest in a project. She takes out an interest only loan at an effective rate of interest of 6% pa. The loan is to be redeemed after 2 years
(with no early repayment option) and interest on the money borrowed is paid at the end of each month. The project will provide income of £50 per month for 24 months and she can invest spare funds at 5% pa. Calculate the borrower’s accumulated profit at the end of 2 years.
[4]

Question 2.9
An investment house issues a 10-year loan for £100,000 to a businessman. The loan is to be repaid by annual repayments (payable in arrears) calculated using 8% pa interest.
The repayment schedule has been designed so that half the capital will have been repaid by the end of the term. The remaining £50,000 will be repaid with funds from other sources. Calculate the annual repayment.
[3]

Question 2.10
A customer borrows £4,000 under a consumer credit loan. Repayments are calculated to give an APR of 15.4%. Instalments are paid monthly in arrears for 5 years.
Calculate the flat rate of interest.
[4]

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CT1: Q&A Bank Part 2 – Questions

Question 2.11
A loan of £30,000 is repaid by a level annuity certain payable annually in arrears for
20 years. Determine, on the basis of an interest rate of 15% pa:
(i)

the annual repayment

[1]

(ii)

the first repayment for which the capital content exceeds the interest content. [3]
[Total 4]

Question 2.12
(i)

In respect of an investment project, define:
(a)
(b)

(ii)

the discounted payback period the payback period

[3]

Discuss why both the discounted payback period and the payback period are inferior measures compared with the net present value for determining whether to proceed with an investment project.
[3]
[Total 6]

Question 2.13
The following table gives information concerning an investment fund:
Calendar year
Value of fund on 1 January before cash flow
Net cash flow received on 1 January
Value of fund on 31 December

2006
£ million
100
20
80

2007
£ million
80
30
200

2008
£ million
200
10
200

Calculate the effective time weighted rate of return per annum over the three year period. [3]

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CT1: Q&A Bank Part 2 – Questions

Page 5

Question 2.14
A loan of £50,000 is being repaid over a period 10 years by a series of level monthly instalments. Interest is charged on the loan at the rate of 8% pa effective.
(i)

Calculate the monthly repayment.

[2]

(ii)

Calculate the amount of interest paid in the first year.

[3]

(iii)

After the payment at the end of 7 years, the borrower takes a 2-month payment break, ie he does not pay the next 2 monthly instalments. Calculate the extra amount he has to pay each month in order to clear his debt by the end of the 10th year. [4]
[Total 9]

Question 2.15
On 1 September 2005, a company placed part of its assets with two fund managers.
Manager P was given £80,000 and Manager Q was given £140,000. Both managers received a net cashflow of £15,000 on 1 September 2006, bringing their total fund values to £103,000 and £183,000, respectively.
A further net cashflow of £20,000 was received by each manager on 1 September 2007.
This brought their total fund values to £143,600 and £239,600, respectively.
On 31 August 2008, the value of Manager P’s fund was £172,320 and the value of
Manager Q’s fund was £263,560.
(i)

For the period from 1 September 2005 to 31 August 2008, calculate for each fund manager:
(a)
(b)

the time weighted rate of return the money weighted rate of return.

[8]

(ii)

By examining the growth factors between cashflows, describe the performance of each manager over the three-year period. Hence, explain why the moneyweighted rate of return for Manager P is higher than that of Manager Q.
[3]

(iii)

Comment briefly on the relative performance of the two fund managers.
[2]
[Total 13]

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Page 6

CT1: Q&A Bank Part 2 – Questions

Question 2.16
On 1 January 2008 an investor has a choice of two projects A and B.
Project A involves an initial cost of £100,000 and provides income annually in arrear of
£5,000 at the end of the first year, inflating at 7% pa, the last payment being on
31 December 2015. The investor will be able to sell the ongoing rights to the project at
31 December 2015 after the payment then due for £130,662.
Project B also involves an initial cost of £100,000 and provides no income, but the investor will be able to sell the rights to the project at 31 December 2015 for £197,750.
(i)

Calculate the internal rates of return for each project as at 1 January 2008, correct to the nearest 0.1%.
[6]

(ii)

The investor has no capital for investing in either project, but can borrow
£100,000 from a bank at 7% pa interest payable annually in arrear. The loan would be repayable on 31 December 2015 at par with no early repayment option.
If further loans are required they will also be granted at 7% pa repayable at par on
31 December 2015 with no early repayment option. However, interest on further loans is rolled up to 31 December 2015. If the investor has any surplus proceeds after paying interest on the original loan as it becomes due, these can be invested at an interest rate of 4% pa effective up to 31 December 2015.
Calculate the accumulated profit on each project at 31 December 2015.

[14]
[Total 20]

Question 2.17
A loan is to be repaid by an immediate annuity. The annuity starts at a rate of £100 pa and increases by £10 per annum. The annuity is paid for 20 years. Repayments are calculated using a rate of interest of 8% pa effective.
(i)

Calculate the amount of the loan.

(ii)

Construct a loan schedule showing the capital and interest elements in and the amount of loan outstanding after the 6th and 7th payments.
[5]

(iii)

Find the capital and interest element of the last instalment.

© IFE: 2009 Examinations

[3]

[2]
[Total 10]

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CT1: Q&A Bank Part 2 – Questions

Page 7

Question 2.18
A football club is considering buying a player on 1 January for £1,000,000. The player’s wages will be £2,000 per month higher than those of the man he will replace.
The manager expects the purchase to generate a level increase in attendances, which will yield an extra income in the first year of £10,000 from each home match. The manager also expects the new player to increase the club’s chance of reaching the Cup
Final in any one year from 10% to 40%. The extra amount generated for club funds by an appearance in a Cup Final on 30 April is £200,000.
The club plays a home match on the second of each month throughout the year, but all
Cup matches are played away from home. Wages are paid at the end of each month.
Wages, ticket prices and the reward for reaching a Cup Final rise at 5% pa, the increases taking place on 1 January.
If the player is purchased, the cost will be borrowed from a bank, which will charge interest at 1% per month and will accept repayment at any time. The owner of the club insists that any purchase should show a profit if the manager’s expectation are borne out in practice.
(i)

If the manager expects that he will keep the player for 9½ years until he retires, calculate the net present value of the cashflow, in order to assess whether or not the purchase should go ahead.
[12]

(ii)

The purchase goes ahead. Attendances rise as expected, but the club does not reach the Cup Final and twelve months after being bought the player is sold again.
The owner of the club calculates that he has made a profit at that time of £75,898.
Calculate the sale price.
[5]
[Total 17]

Question 2.19
An investor borrows £120,000 at an effective interest rate of 7% per annum. The investor uses the money to purchase an annuity of £14,000 per annum payable halfyearly in arrears for 25 years. Once the loan is paid off, the investor can earn interest at an effective rate of 5% per annum on money invested from the annuity payments.
(i)

Determine the discounted payback period for this investment.

(ii)

Determine the profit the investor will have made at the end of the term of the annuity. [7]
[Total 12]

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[5]

© IFE: 2009 Examinations

Page 8

CT1: Q&A Bank Part 2 – Questions

Question 2.20
A man takes out a loan over 25 years. He makes monthly repayments in arrears. The first payment is £50 and the remaining payments increase by £5 each time until the payments reach £770. The remaining payments are then all £770. The annual effective rate of interest is 7.5%.
(i)

Show that the amount of money that the man borrowed is £66,767.

[6]

(ii)

Calculate the amount of capital outstanding after 6 years.

[4]

(iii)

Calculate the amount of capital repaid in the 6th year.

[3]

(iv)

Calculate the interest paid in the 73rd payment. Comment on your answers to parts (i), (ii), (iii) and (iv).
[3]

(v)

At the end of the sixth year (after the payment then due) the man decides to reschedule the loan over a term of ten years with level monthly payments. He negotiates a deal with the loan company to reduce the interest he pays by
0.1% pa. Calculate how much he saves (compared to his original loan) under this arrangement.
[7]
[Total 23]

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CT1: Q&A Bank Part 2 – Questions

Page 9

Question 2.21
A company is considering investing in the following project. The company has to make an initial investment of three payments, each of £105,000. The first is due at the start of the project, the second six months later, and the third payment is due one year after the start of the project.
After 15 years it is assumed that a major refurbishment of the infrastructure will be required, costing £200,000.
The project is expected to provide no income in the first year, an income received continuously of £20,000 in the second year, £23,000 in the third year, £26,000 in the fourth year and £29,000 in the fifth year. Thereafter the income is expected to increase by 3% per annum (compound) at the start of each year.
The income is expected to cease at the end of the 30th year from the start of the project.
The cash flow within each year is assumed to be received at a constant rate.
(i)

Calculate the net present value of the project at a rate of interest of 8% pa effective. [8]

(ii)

Show that the discounted payback period does not fall within the first 15 years, assuming an effective rate of interest of 8% pa.
[5]

(iii)

Calculate the discounted payback period for the project, assuming an effective rate of interest of 8% pa.
[5]
[Total 18]

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You may not hire out, lend, give out, sell, store or transmit electronically or photocopy any part of the study material.

You must take care of your study material to ensure that it is not used or copied by anybody else.

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These conditions remain in force after you have finished using the course.

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CT1: Q&A Bank Part 2 – Solutions

Page 1

Part 2 – Solutions
Solution 2.1
(i)

The money-weighted rate of return is found from the equation:
25(1 + i ) + 5(1 + i ) ¾ = 31.5

(working in £000s)

[1]

Using an approximation:

25(1 + i ) + 5(1 + ¾i ) = 31.5
⇒ 28.75i = 1.5
⇒ i = 5%

[1]

Using trial and improvement:
At 5%, the left-hand side is 25(1 + i ) + 5(1 + i ) ¾ = 31.436

At 5.5%, the left-hand side is 25(1 + i ) + 5(1 + i ) ¾ = 31.580

[1]

Interpolating between these values we get:

i - 5%
31.5 - 31.436
=
fi i = 5.2%
5.5% - 5% 31.580 - 31.436
(ii)

[1]

The time-weighted rate of return is found from the equation:

1+ i =

29 - 5 31.5
¥
= 1.0428 fi i = 0.0428
25
29

[1]

So the TWRR is 4.28%.
(iii)

The linked quarterly rate of return can be found by combining the “growth factors” for each quarter:

1+ i =

29 - 5 30 32 31.5
¥
¥ ¥
= 1.0428
25
29 30 32

[1]

So the linked quarterly rate of return is the same as the time-weighted rate of return ie 4.28%, as the cashflows coincide with the linking points.

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Page 2

CT1: Q&A Bank Part 2 – Solutions

Solution 2.2
Let X be the monthly payment, then:
12 Xa (12) = 1, 000
10



X = 12.98

[1]

The capital outstanding at the start of the sixth year (ie with 5 years still to run) is:
12 ¥ 12.98a (12) = 617.05

[1]

5

Similarly, the capital outstanding at the end of the sixth year (ie with 4 years still to run) is: 12 ¥ 12.98a (12) = 515.98

[1]

4

So the capital repaid during the sixth year is 617.05 - 515.98 = £101.07 .

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[1]

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CT1: Q&A Bank Part 2 – Solutions

Page 3

Solution 2.3
The equation of value for the loan is:
3, 000 = 1, 200a (12)
3

fi 2.5 = a (12)

[1]

3

To find a first guess for the APR we can calculate the flat rate of interest using:

Total interest paid = flat rate ¥ term ¥ amount of the loan
Letting the flat rate equal f , this gives:
3 ¥ 12 ¥ 100 - 3000 = f ¥ 3 ¥ 3000 fi

f =62%
3

[1]

The APR is approximately twice the flat rate of interest. Try a first guess of 13.0%:

a (12) = 2.4987

[1]

3

This is slightly too low so we need to decrease the interest rate. Try a second guess of
12.9%.

a (12) = 2.5019

[1]

3

Since the 13.0% value is closer to 2.5 than the 12.9% value, the APR is 13.0%.

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Page 4

CT1: Q&A Bank Part 2 – Solutions

Solution 2.4
(i)

If the flat rate is 9%, then the total interest payable is 0.09 × 7,500 × 2 = 1,350 .
The monthly instalment is therefore

(ii)

7,500 + 1,350
= £368.75 .
24

[1]

The APR is the rate of interest which solves the equation of value:
(12)
12 × 368.75a2| = 7,500

ie:

a (12 ) = 16949
.
|

[1]

2

The APR is roughly twice the flat rate, so as a first guess we can try 17%:
(12)
a2| @17% = 1.7052

Then 18%:
(12)
a2| @18% = 1.6909

[1]

Interpolating gives 17.7%. Try 17.7% and 17.8%:
(12)
a2| @17.7% = 1.6952
(12)
a2| @17.8% = 1.6938

[1]

Since the 17.7% value is closer to 1.6949 than the 17.8% value, the APR is
17.7%.
[1]

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CT1: Q&A Bank Part 2 – Solutions

Page 5

Solution 2.5
We need to find the point at which the accumulated profit (calculated at the borrowing rate of 10%) equals zero, ie we need to find the value of t for which:
-25, 000(1 + i )t + (10, 000 - 2, 000) st| = 0

[1]

which can be simplified to: at@10% =
|

25, 000
= 3.125
8, 000

So the DPP is:
1 - 1.1- t
= 3.125 log1.1 fi

t = 3.71

ie 3 years 8½ months

[1]

The internal rate of return is not needed.

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Page 6

CT1: Q&A Bank Part 2 – Solutions

Solution 2.6
We have:

Total interest paid = flat rate ¥ term ¥ amount of the loan
So, in this case, the amount of interest payable, assuming that the loan is for an amount
1, will be 0.075.
The monthly repayment is the total amount paid out divided by the number of payments: 1.075
6

[1]

since the loan is for one year.
Working in years, the equation of value is:

12 ¥

1.075 0.5 (12)
¥ v a0.5| = 1
6

[1]

We need to solve this using trial and error.
Try 9.5%:

12 ¥

1.075 0.5 (12)
¥ v a0.5| = 1.0006
6

[1]

1.075 0.5 (12)
¥ v a0.5| = 0.9998
6

[1]

Try 9.6%:

12 ¥

Since the 9.6% value is closer to 1 than the 9.5% value, the APR is 9.6%.

[1]

Note that here, the flat rate is 7.5% and the APR turns out to be 9.6%. So, the APR is therefore quite a lot less than 15% (twice the flat rate), which is our usual first guess.
The APR ≈ 2 × Flat Rate approximation relies on an assumption that loan repayments start to be made straight away. So here, where the repayments on the loan are deferred for 6 months (the first repayment is in July 2008), this approximation is not very good.

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CT1: Q&A Bank Part 2 – Solutions

Page 7

Solution 2.7
This question is taken from Subject 102, April 2003, Question 3.
The following timeline illustrates the cashflows involved in the project:
5,000
-60,000 -25,000

0

8/12

13,000

9,000

2

6

10

21,000 50,000

17,000 pa pa

pa

pa

14

pa

18

Cashflows
22 Time (years)

The value at time 0 of the investment required is:
8

12
60, 000 + 25, 000v9% = 60, 000 + 25, 000 × 0.94417 = 83, 604.20

[2]

Working in a time period of four years, the income can be divided into two parts:


a level payment of 4,000 per period for 5 periods,



a payment of 16,000 in the first period, 32,000 in the second period, increasing by 16,000 in each period with a final payment of 80,000 in the 5th period.

The first part can be valued using a level annuity, and the second part using an increasing annuity. Each annuity will have a term of 5.
Using our time period equal to 4 years, the value at time 2 of the income from the project is:
5
4, 000a5 k + 16, 000 ( I a )5 k + 50, 000vk where k = (1.09 ) − 1 = 0.41158

4

So, the value of the income at time 2 is:

4, 000 × 2.3834 + 16, 000 × 5.5860 + 50, 000 × 0.17843 = 107,830

[3]

The value of the income at time 0 is:
2
107,830v9% = 107,830 × 0.84168 = 90, 759

[1]

So the net present value of the project is:

−83, 604 + 90, 759 = £7,155

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Page 8

CT1: Q&A Bank Part 2 – Solutions

Solution 2.8
The amount of each monthly payment required under the loan is:
1, 000(1.061/12 - 1) = £4.87

[1]

So the investor’s net monthly income is:

50 - 4.87 = £45.13 pm

[1]

The accumulated profit at the end of 2 years (when the £1,000 borrowed must be repaid) will be:
AV = 12 ¥ 45.13s

(12)@5%
2|

- 1, 000

= 12 ¥ 45.13 ¥ 2.05 ¥ 1.022715 - 1, 000 = £135

[2]

Solution 2.9
The annual repayment R can be found from the equation of value for the investment house’s cashflows:
-100, 000 + Ra10| + 50, 000v10 = 0 @8%

[1]

-100, 000 + 6.7101R + 50, 000 ¥ 0.46319 = 0

[1]

ie:

So:

R=

76,840
= £11, 451
6.7101

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CT1: Q&A Bank Part 2 – Solutions

Page 9

Solution 2.10
This question is taken from Subject A1, September 1998, Question 9.
First calculate the amount of each monthly repayment, M, from an appropriate equation of value:
(12)
4, 000 = 12 M a5| @15.4%

= 12 M

1 − v5

i (12)
= 42.5877 M

= 12 M

[1]

1 − 1.154−5
12(1.1541/12 − 1)

⇒ M = £93.92

Flat rate =

5 ×12 × 93.92 − 4000
= 8.2% pa
5 × 4000

[1]
[2]

Solution 2.11
(i)

The annual repayment X satisfies:
Xa20| = 30, 000@15%

So:

X=
(ii)

30, 000
= £4, 793
6.2593

[1]

When the capital content exceeds the interest content then the capital component must be more than half of the payment (ie £2,396). The easiest way to obtain the capital content is probably to calculate the difference between the capital outstanding after each payment.
[1]
Since the capital outstanding under a loan decreases very slowly at first, but very rapidly in the last few years we should try at least 10 years.

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CT1: Q&A Bank Part 2 – Solutions

Let L be the amount of the loan outstanding, then:
13 yrs L = 4, 793a7 = 19,941
14 yrs L = 4, 793a6 = 18,139 fi capital repaid in 14th payment = 1,802
15 yrs L = 4, 793a5 = 16, 067 fi capital repaid in 15th payment = 2, 072
16 yrs L = 4, 793a4 = 13, 684 fi capital repaid in 16th payment = 2,383
17 yrs L = 4, 793a3 = 10,943 fi capital repaid in 17th payment = 2, 741 [1]
So the first repayment where the capital content exceeds the interest content is the 17th.
[1]

Solution 2.12
This question is taken from Subject 102, September 2003, Question 13 (i) and (ii).
(i)(a) The discounted payback period is the smallest time t for which the accumulated value of the returns up to time t exceeds the accumulated value of the costs up to time t .
[2]
(i)(b) The payback period is the same as the discounted payback period, except that the accumulation is carried out using an interest rate of 0. In other words, it is the earliest time for which the monetary value of the returns exceeds the monetary value of the costs.
[1]
(ii)

Unlike the NPV, neither the DPP nor the PP give any indication of how profitable a project is, as they ignore cashflows after the accumulated value of zero is reached.
[1]
There may not be one unique time when the balance in the investor’s account changes from negative to positive. However, the NPV can always be calculated.
[1]
The PP can give misleading results, as it does not take into account the time value of money.
[1]

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CT1: Q&A Bank Part 2 – Solutions

Page 11

Solution 2.13
This question is taken from Subject 102, September 2003, Question 2.
The time-weighted rate of return is the solution, i , of the equation:

80 200 200
3
¥
¥
= (1 + i )
120 110 210

[2]

Solving this gives: i = 1.15441/ 3 - 1 = 4.90%

[1]

Solution 2.14
(i)

Let P denote the monthly repayment. Then, working in months:
Pa120 = 50, 000

[1]

where the annuity is calculated using the effective monthly interest rate: i = 1.081/12 - 1 = 0.00643403
Solving for P :

P=

50, 000
= £599.29
83.43239

[1]

Alternatively, you could work in years and use the equation of value:
12 Pa (12) @8% = 50, 000
10

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Page 12

(ii)

CT1: Q&A Bank Part 2 – Solutions

The amount outstanding at the end of the first year is given by:
599.29a108 = 599.29 ¥ 77.67310 = £46,548.71

[1]

The capital repaid in the first year is then:

50, 000 - 46,548.71 = £3, 451.29

[1]

and the interest paid in the first year is:

(12 ¥ 599.29) - 3, 451.29 = £3, 740.19

[1]

Alternatively, working in years, the amount outstanding is given by:
(12 ¥ 599.29)a (12) @8%
9

(iii)

The amount outstanding after 7 years is:
599.29a36 = 599.29 ¥ 32.043333 = £19, 203.25

[1]

As no payments are made for the next 2 months, the capital outstanding after 7 years and 2 months is:
19, 203.25 ¥ 1.082 /12 = 19, 451.15

[1]

This is to be repaid in equal monthly instalments of Q over the next 34 months.
So:

Qa34 = 19, 451.15 fiQ= 19, 451.15
= £638.78
30.45056

[1]

The extra monthly payment is therefore 638.78 - 599.29 = £39.49

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CT1: Q&A Bank Part 2 – Solutions

Page 13

Solution 2.15
(i)(a) For Manager P:

88 123.6 172.32
×
×
= 1.584
80 103
143.6

(1 + i )3 =

⇒ i = 16.6%

[1]

For Manager Q:

(1 + i )3 =

168 219.6 263.56
×
×
= 1.584
140 183
239.6

⇒ i = 16.6%

[1]

(i)(b) For Manager P the equation of value is:
80(1 + i )3 + 15(1 + i ) 2 + 20(1 + i ) = 172.32

[1]

Using a binomial approximation to obtain a first guess:

80(1 + 3i ) + 15(1 + 2i ) + 20(1 + i ) = 172.32
⇒ i = 0.1977

[½]

This first guess will be too high since the terms in i 2 and above have been omitted from the LHS in the binomial approximation. By trial and error, we obtain: i = 0.19 ⇒ LHS = 179.85 i = 0.18 ⇒ LHS = 175.93 i = 0.17 ⇒ LHS = 172.06

[½]

Using linear interpolation:

i = 0.17 +

172.32 − 172.06
(0.18 − 0.17) = 17.1%
175.93 − 172.06

[1]

Similarly, for Manager Q the equation of value is:
140(1 + i )3 + 15(1 + i ) 2 + 20(1 + i ) = 263.56

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CT1: Q&A Bank Part 2 – Solutions

Using a binomial approximation to obtain a first guess:

140(1 + 3i ) + 15(1 + 2i ) + 20(1 + i ) = 263.56
⇒ i = 0.1884

[½]

Again, the value of i will be less than 18.84%. By trial and error, we find:

i = 0.18 ⇒ LHS = 274.51 i = 0.17 ⇒ LHS = 268.16 i = 0.16 ⇒ LHS = 261.91

[½]

Then, by linear interpolation:

i = 0.16 +
(ii)

263.56 − 261.91
(0.17 − 0.16) = 16.3%
268.16 − 261.91

[1]

For Manager P the growth in each year is 10%, 20% and 20%.

[½]

For Manager Q the growth in each year is 20%, 20% and 10%.

[½]

Because they each have the same overall growth rate the TWRR is the same, however Manager P performed worse during the first year, whereas Manager Q performed worse during the third year.
[1]
The MWRR puts more emphasis on the money (hence the name money weighted). Since for both managers the greatest money in the fund occurs during the third year, the MWRR will give this final year more weight than the other years. Manager Q’s poorer performance in this year will be more heavily penalised. Consequently, manager Q has a lower MWRR.
[1]
(iii)

The MWRR is higher for Manager P than for Manager Q, whereas the TWRR is the same for both managers.
The TWRR is considered the better measure of performance as it ignores the effects of the cashflows, which are beyond the managers’ control.
[1]
Therefore, on this basis, both fund managers should be considered as having performed equally.
[1]

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CT1: Q&A Bank Part 2 – Solutions

Page 15

Solution 2.16
(i)

The IRR for Project B is the interest rates that satisfies the equation of value:
-100, 000 + 197, 750v8 = 0

[1]

Rearranging gives: i = 1.977501/ 8 - 1 = 0.08897
So the IRR for Project B (correct to the nearest 0.1%) is 8.9%.

[1]

The IRR for Project A is the interest rates that satisfies the equation of value:

-100, 000 +

5, 000 * a | + 130, 662v8 = 0
1.07 8

[1]

1+ i
.
1.07
Trying 9.05% (assuming that the IRRs for the two projects are similar) gives: where the annuity is calculated at a rate of interest of 1 + i* =

LHS = -100, 000 +
= -100, 000 +

5, 000 @1.91589% a| + 130, 662v8@9.05%
8
1.07
5, 000
¥ 7.3521 + 130, 662 ¥ 0.50009 = -309.64
1.07

[1]

Trying 8.95% (assuming that the IRRs for the two projects are similar) gives:

LHS = -100, 000 +
= -100, 000 +

5, 000 @1.8224% a + 130, 662v8@8.95%
1.07 8|
5, 000
¥ 7.3819 + 130, 662 ¥ 0.50371 = 310.70
1.07

So the IRR for Project A (correct to the nearest 0.1%) is 9%.
(ii)

[1]
[1]

For Project B, the investor will need to take out the loan for £100,000, which requires interest payments of £7,000 each year and repayment of the £100,000 on 31 December 2015. In order to pay the interest payments, the investor will have to take out 8 extra loans for £7,000 each. (Note that for the extra loans, all the interest is paid at the end of the term, rather than in the form of annual payments.) The Actuarial Education Company

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CT1: Q&A Bank Part 2 – Solutions

On 31 December 2015 the investor will need to repay the £100,000 from the original loan and the accumulated value of £7,000 from each of the 8 extra loans required, making a total of:
100, 000 + 7, 000 s8| @ 7% = £171,819

[2]

There will be no surplus funds to invest. The project will provide a payment of
£197,750 on 31 December 2015. So the accumulated profit will be:

197, 750 - 171,819 = £25,931

[1]

For Project A, the investor will again need to take out the loan for £100,000, which will require interest payments of £7,000 each year and repayment of the
£100,000 on 31 December 2015. The income from the project increases by 7%
(compound) each year.
So the cashflows in each year will be as follows:
31 December
2008
2009
2010
2011
2012
2013
2014
2015

Interest due on original loan
7,000
7,000
7,000
7,000
7,000
7,000
7,000
7,000

Income from project
5,000.00
5,350.00
5,724.50
6,125.50
6,553.98
7,012.75
7,503.64
8,028.90
[5]

During the first 5 years, the income will not be sufficient to pay the interest on the original loan. So extra loans will be required, which will need to be repaid on 31 December 2015. The accumulated amount of the debt can be calculated as: (7, 000a @ 7% |
5

5, 000 * a ) ¥ 1.078
1.07 5|

= (7, 000 ¥ 4.1002 -

5, 000
¥ 5) ¥ 1.078 = £9,170
1.07

[3]

(Since the interest rate equals the accumulation rate here, the starred annuity factor is calculated at 0%.)

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CT1: Q&A Bank Part 2 – Solutions

Page 17

In the last 3 years there will be surplus funds to invest, which will accumulate (at
4%) to:
12.75 ¥ 1.042 + 503.64 ¥ 1.04 + 1, 028.90 = £1,566

[2]

So the overall accumulated profit will be:

130, 662 - 100, 000 - 9,170 + 1,566 = £23, 058

[1]

Solution 2.17
This question is taken from Subject A1, September 1998, Question 12.
(i)

Assuming that the loan repayments are made at the end of each year, the amount of the loan, L, is found from the equation of value:
L = 90a20| + 10( Ia ) 20|

= 90 × 9.8181 + 10 ×

[1]

10.6036 − 20 × 0.21455
0.08

[1]

= £1, 672.71
(ii)

[1]

First calculate the capital outstanding immediately after the 5th payment:

140a15| + 10( Ia)15| = 140 × 8.5595 + 10 ×

9.2442 − 15 × 0.31524
0.08

= £1, 762.78

[1]

This is higher than the original loan since the interest charges in the early years are greater than the repayments.
The 6th payment is £150 and the 7th payment £160.
The interest element of the 6th payment is:

0.08 × 1, 762.78 = £141.02

[1]

Hence the capital repayment is: 150 − 141.02 = £8.98 and the capital outstanding after the 6th payment is: 1, 762.78 − 8.98 = £1, 753.80 .
[1]
Similarly, the interest element of the 7th payment is £140.30 and the capital element is £19.70. The capital outstanding is therefore £1,734.10.
[2]

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Page 18

CT1: Q&A Bank Part 2 – Solutions

Expressing these amounts in a schedule gives:

Payment

Interest element
(£)

Capital element
(£)

Capital outstanding after payment (£)

5

1,762.78

6

8.98

1,753.80

7

(iii)

141.02
140.30

19.70

1,734.10

As there are 19 increases after the first repayment, the last payment is £290. The capital outstanding immediately after the penultimate payment is therefore:
290v = £268.52

[1]

This must also be the capital element of the last payment if that payment is to pay off the loan. Thus the interest element is:
290 − 268.52 = £21.48

[1]

Solution 2.18
(i)

The expected changes in the cashflows in the first year if the player is bought are: •



extra wages of £2,000 per month, paid at the end of each month extra ticket sales of £10,000 per month, paid at the start of each month extra winnings of (40% - 10%) ¥ 200, 000 = £60, 000 in the Cup Final on
30 April.
[2]

So the increase in the net present value of the payments in year 1 (working in months) is:
-2, 000a12| + 10, 000a12| + 60, 000v 4 @1% = £148,825

[2]

The increase in the net present values of the payments in years 2 to 9 (as at the beginning of each year) will be the same, but increased by 5% (compound) each year. © IFE: 2009 Examinations

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Page 19

In year 10, the cashflows for wages and ticket sales will only apply for 6 months.
So the increase in the net present values of the payments in year 10 (as at the beginning of year 10) will be:
-2, 000a6| + 10, 000a6| + 60, 000v 4 @1% = £104, 601

[2]

So the overall increase in the net present value will be:
DNPV = 148,825(1 + 1.05v12 + 1.052 v 24 + … + 1.058 v96 ) + 104, 601 ¥ 1.059 v108
*
= 148,825a9| + 104, 601 ¥ 1.073167 -9

= 148,825 ¥ 6.8987 + 104, 601 ¥ 0.52966 = £1, 082,103

[4] where the starred annuity factor is calculated at a rate of interest of
1.0112 /1.05 - 1 = 7.3167% .

[1]

Since this exceeds the purchase price of £1,000,000, the purchase should go ahead. [1]
(ii)

If the player is sold for an amount P , the equation of value (ignoring any change in the team’s chances of winning in the Cup Final, which do not affect the actual profit) becomes:
-1, 000, 000 - 2, 000a12| + 10, 000a12| + Pv12 = 75,898v12

[2]

Rearranging to find P gives:
P = 75,898 + 1.0112 ¥ (1, 000, 000 + 2, 000 ¥ 11.2551 - 10, 000 ¥ 11.3676)
= £1, 099,995
[2]
So the sale price was £1.1m.

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CT1: Q&A Bank Part 2 – Solutions

Solution 2.19
This question is taken from Subject 102, April 2004, Question 9.
(i)

Let us assume that we need n years of the annuity to pay off the loan. Then the equation of value is:
120, 000 = 14, 000a (2)

[1]

n

We must evaluate this using 7% interest since we have not yet paid off the loan.
Rearranging this:

120
1 − 1.07 − n
=
14 2 1.070.5 − 1

(

)

⇒ 1.07 − n = 0.410148

Taking logs of both sides, we get:

− n ln1.07 = ln 0.410148 ⇒ n =

ln 0.410148
= 13.17 ln1.07 [3]

Since we need just over 13 years worth of payments, the discounted payback period is 13.5 years (since the discounted payback period must be a multiple of 6 months). [1]
(ii)

We need to find out how much money is left over at time 13.5, since we will not have a balance of zero at that time. During the period from time 0 to time 13.5 we need to use an interest rate of 7%. At time 13.5, we have:
−120, 000(1.07)13.5 + 14, 000s (2)

13.5 @ 7%

[2]

= −299,131.57 + 303, 688.92
= 4,557.35

[1]

The accumulated value of this at the end of the term is (using 5% interest as we have now paid off the loan and are investing the money):
4,557.35(1.05)11.5 = 7,987.10

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Page 21

The remaining annuity payments after time 13.5 will accumulate to:
14, 000 s (2)

= 213,323.26

11.5 @5%

[2]

So the total profit is: 7,987.10 + 213,323.26 = 221,310

[1]

Solution 2.20
(i)

We will work in months using a monthly effective rate of interest of:
1

1.07512 - 1 = 0.0060449

[½ ]

The timeline showing the payments is:
50

0

55

60

65

Payment

1

2

3

4

300 Time

The payment reaches £770 at time t where:

50 + 5(t - 1) = 770 fi t = 145

[½ ]

ie at time 145 months.

The present value at time 0 of the payments up to and including time 145 is:
45a145 + 5( Ia )145

[½ ]

The present value at time 145 of the remaining payments is:
770a155

[½ ]

So the total present value at time 0 is:
45a145 + 5( Ia )145 + 770v145 a155

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Page 22

CT1: Q&A Bank Part 2 – Solutions

Calculating the values:

a145 =

1 - (1 + i ) -145
= 96.38981 i a155 =

1 - (1 + i ) -155
= 100.42767 i ( Ia )145

1 - (1 + i ) -145
- 145(1 + i ) -145 i 1+ i
=
= 6, 031.41708 i [2]

So the present value (ie the amount of money borrowed) is:

45 ¥ 96.38981 + 5 ¥ 6, 031.41708 + 770 ¥ 0.417331 ¥ 100.42767 = £66, 766.57 [1]
Alternatively, this can be evaluated as:
45a144 + 5( Ia )144 + 770v144 a156
(ii)

The 73rd repayment (ie the next payment after 6 years) is:
50 + 5 ¥ 72 = £410

[1]

Therefore the capital outstanding after 6 years is:
405a73 + 5( Ia )73 + 770a155 v 73

[1]

Calculating the values: a73 =

( Ia )73

1 - (1 + i ) -73
= 58.881156 i 1 - (1 + i ) -73
- 73(1 + i ) -73 i 1+ i
=
= 2, 021.55082 i [1]

So the capital outstanding is:

405 ¥ 58.881156 + 5 ¥ 2, 021.55082 + 770 ¥ 100.42767 ¥ 0.644068

= £83, 759.97

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[1]

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CT1: Q&A Bank Part 2 – Solutions

Page 23

Alternatively, this can be calculated as:
405a72 + 5( Ia )72 + 770a156 v 72
(iii)

The sixth year is between time 5 years and time 6 years. The 61st repayment is:
50 + 5 ¥ 60 = £350

Therefore the capital outstanding after 5 years is:
345a85 + 5( Ia )85 + 770a155 v85

[½ ]

Calculating the values: a85 =

( Ia )85

1 - (1 + i ) -85
= 66.314670 i 1 - (1 + i ) -85
- 85(1 + i ) -85 i 1+ i
=
= 2, 611.981357 i [½ ]

So the capital outstanding is:

345 ¥ 66.314670 + 5 ¥ 2, 611.981357 + 770 ¥ 100.42767 ¥ 0.599133
= £82, 269.02

[1]

Hence the capital repaid is:

82, 269.02 - 83, 759.97 = -£1, 490.95
(iv)

[1]

The interest due in the 73rd repayment is:

0.0060449 ¥ 83, 759.97 = £506.32

[1]

The 73rd repayment is £410. Therefore the payment is not sufficient to meet the interest and so the capital outstanding will increase.
[1]
This is why the capital outstanding in part (ii), ie £83,759.97 is higher than the original amount borrowed, ie £66,766.57 and why the capital repaid in part (iii) is negative.
[1]

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Page 24

(v)

CT1: Q&A Bank Part 2 – Solutions

Let the new monthly payment be Y . We have:
Ya120 = 83, 759.97

[1]
1

This is evaluated at an interest rate of 1.07412 − 1 = 0.0059669 , to give:
Y = 979.46

[1]

The total repayment under the new loan is:

979.46 × 120 = 117,534.88

[1]

The total repayment for the period outstanding under the original loan is:
410 + 415 +

+ 770 + (155 × 770)

[1]

The first few terms form an arithmetic progression. We will sum it using the formula: n
(2a + (n − 1)d )
2

There are 73 terms in the progression. Therefore the total repayment is:
73
(410 × 2 + 72 × 5) + 155 × 770 = 43, 070 + 119,350 = 162, 420
2

[2]

Alternatively, you could have evaluated this as:

410 + 415 +

+ 765 + (156 × 770) =

72
(410 × 2 + 71 × 5) + 156 × 770 = 162, 420
2

The overall saving is therefore:
162, 420 − 117,534.88 = £44,885.12

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[1]

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CT1: Q&A Bank Part 2 – Solutions

Page 25

Solution 2.21
This question is taken from Subject A1, September 1997, Question 13.
(i)

The PV of the outgo (in £000s) is:
PV outgo = 105(1 + v½ + v) + 200v15
= 105 × 2.888176 + 200 × 0.315242
= 303.259 + 63.048
= 366.307

[2]

The PV of the income is:
PV income = 20 v a1| + 23 v 2 a1| + 26 v3 a1| + 29 v 4 a1|

+ 29 ×1.03 v5 a1| + 29 × 1.032 v 6 a1| +

+ 29 ×1.0325 v 29 a1|
[1]

This can be simplified to:
PV income = (20v + 23v 2 + 26v3 ) a1|

+ 29v 4 (1 + 1.03v + 1.032 v 2 +

+ 1.0325 v 25 ) a1|

[1]

The first part can be calculated directly and the second part can be summed as a geometric series, to give:

PV income = 58.877 × 0.962488
1 − 1.0326 v 26
+ 29 × 0.735030 ×
× 0.962488
1 − 1.03v
= 370.608
So the NPV is: 370,608 − 366,307 = +£4,301

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[3]
[1]

© IFE: 2009 Examinations

Page 26

(ii)

CT1: Q&A Bank Part 2 – Solutions

The PV of the income for the first 15 years can be calculated similarly as:

PV income(15 yrs ) = 58.877 × 0.962488
+ 29 × 0.735030 ×
= 56.668 + 20.516 ×
= 236.732

1 − 1.0311 v11
× 0.962488
1 − 1.03v

1 − 1.0311 v11
1 − 1.03v

[3]

As this is less than the PV of the outgo up to the end of the 15th year (£303,259 before the refurbishment payment) and the initial outlay (the 3 payments of
£105,000) precedes the start of the income, the accumulated net value is negative throughout the first 15 years.
[2]
(iii)

Equating the PV of the income for the first n years ( n > 15 ) to the PV of the outgo gives the following equation (which will be exact only if n is a whole number): PV income(n yrs) = 56.668 + 20.516 ×

1 − 103n−4 v n−4
.
= 366.307
1 − 103v
.

[1]

Solving this, we find that:
(103v ) n−4 = 0.30127 ⇒ n = 29.31 (approx)
.

[1]

To find the exact value we need to value the payments in the final part year (of length t , say) exactly, which gives the equation:

56.668 + 20.516 ×
⇒ at| = 0.303

1 − 1.0325 v 25
+ 29 × 1.0325 v 29 at| = 366.307
1 − 1.03v
⇒ t = 0.307

So the discounted payback period is 29.31 years.

© IFE: 2009 Examinations

[1]
[1]
[1]

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CT1: Q&A Bank Part 3 – Questions

Page 1

Part 3 – Questions
Question 3.1
Dividends payable on a certain share are assumed to increase at a compound rate of 3% per half-year. A dividend of £5 per share has just been paid. Dividends are paid halfyearly.
Calculate the value of the share to the nearest £1, assuming an effective rate of interest of 8% pa.
[3]

Question 3.2
Describe the main features of Eurobonds.

[3]

Question 3.3
Describe the investment characteristics of debentures.

[2]

Question 3.4
An asset has a current price of £1.20. Given a risk-free rate of interest of 5% pa effective and assuming no arbitrage, calculate the forward price to be paid in 91 days.[2]

Question 3.5
A fixed-interest security with a 6% annual coupon payable half-yearly in arrears is purchased at a price that gives a gross effective yield of 10% pa by an investor who is subject to capital gains tax at 30%. It is redeemable at par after 15 years. Calculate the amount of the capital gains tax payable per £100 nominal.
[3]

Question 3.6
State the main differences between a preference share and an ordinary share.

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[3]

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CT1: Q&A Bank Part 3 – Questions

Question 3.7
An ordinary share pays annual dividends. The next dividend is expected to be 10p per share and is due in exactly 9 months’ time. It is expected that subsequent dividends will grow at a rate of 5% pa compound and that inflation will be 3% pa. The price of the share is 250p and dividends are expected to continue forever.
Calculate the expected effective real rate of return pa for an investor who purchases the share. [5]

Question 3.8
An investor is considering the purchase of 100 ordinary shares in a company.
Dividends from the share will be paid annually. The next dividend is due in one year and is expected to be 8p per share. The second dividend is expected to be 8% greater than the first dividend and the third dividend is expected to be 7% greater than the second dividend. Thereafter, dividends are expected to grow at 5% pa compound.
Calculate the present value of this dividend stream at a rate of interest of 7% pa effective. [5]

Question 3.9
In a South American country, which uses dollars as its national currency, price inflation has been running at 20% pa for the last 10 years. Calculate the average annual real rate of return for each of the following investments:
(i)

A set of gold coins purchased for $14,000 on 1 January 2001 and sold for
$20,000 on 31 December 2003.
[1]

(ii)

A painting purchased for $3,000 on 1 March 2004 and sold for $3,200 on
1 September 2004.
[1]

(iii)

A diamond purchased for $13,000 on 1 July 2005 and sold for $10,000 on
1 July 2007.
[1]

(iv)

A statuette purchased for $7,500 on 1 November 2000 and sold for $19,000 on
31 December 2007.
[1]
[Total 4]

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CT1: Q&A Bank Part 3 – Questions

Page 3

Question 3.10
A stock with a term of 9½ years has a coupon of 5% pa payable half-yearly in arrears and is redeemable at 105%. An investor who is not subject to tax purchases the stock at
85 per 100 nominal immediately after the coupon payment. Calculate the yield obtained by the investor.
[5]

Question 3.11
(i)

Describe the risk characteristics of a government-issued, conventional, fixed interest bond.
[2]

(ii)

A particular government bond is structured as follows:
Annual coupons are paid in arrears of 8% of the nominal value of the bond.
After five years, a capital payment is made, equal to half of the nominal value of the bond. The capital is repaid at par. The repayment takes place immediately after the payment of the coupon due at the end of the fifth year. After the end of the fifth year, coupons are only paid on that part of the capital that has not been repaid. At the end of the tenth year, all the remaining capital is repaid.
Calculate the purchase price of the bond per £100 nominal, at issue, to provide a purchaser with an effective net rate of return of 6% per annum. The purchaser pays tax at a rate of 30% on coupon payments only.
[5]
[Total 7]

Question 3.12
On 1 January 2000 an investor purchased £10,000 nominal of a stock that pays coupons on 30 June and 31 December at the rate of 6% pa and is redeemable at par on
31 December 2012. The investor, who has no unused tax allowances, is liable for income tax payable at the rate of 40% on each 1 August in respect of coupons received during the previous calendar year. If the investor’s net redemption yield on this investment is 5% pa effective, calculate the price paid for the holding.
[5]

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CT1: Q&A Bank Part 3 – Questions

Question 3.13
An index-linked zero-coupon bond was issued on 1 January 2003 for redemption at par on 31 December 2007. The redemption payment was linked to a price inflation index with a 6-month time lag. Calculate the average money and real rates of return obtained by an investor who purchased £10,000 nominal of the stock on 1 January 2005 for
£10,250 and held it until redemption. You are given the following values for the price index: Date
01.01.02
01.07.02
01.01.03
01.07.03
01.01.04
01.07.04
01.01.05
01.07.05

Index
144
148
155
160
162
168
175
177

Date
01.01.06
01.07.06
01.01.07
01.07.07
01.01.08

Index
181
182
188
193
201

[4]

Question 3.14
Define the following terms:
(i)

arbitrage

[3]

(ii)

hedging.

[1]
[Total 4]

Question 3.15
A ten-year zero-coupon bond is issued on 1 February 2001 at a price of £79%. On
1 February 2003 an investor entered into forward contract to buy £1,000 nominal of the bond in 5 years’ time. The price of the bond was £83% on 1 February 2003 and £92% on 1 February 2008.
Calculate the profit or loss made by the investor on 1 February 2008 if the risk-free force of interest was 3% pa.
[2]

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CT1: Q&A Bank Part 3 – Questions

Page 5

Question 3.16
Company shares with a current dividend yield of 3% have a share price of £7.50 per share. Calculate the forward price of a five-year forward contract on the shares if the risk-free effective rate of interest is 6% pa. State any assumptions that you make.
[2]

Question 3.17
Outline the differences between government bonds, unsecured loan stock and
Eurobonds.
[6]

Question 3.18
(i)

State what is meant by a “forward contract”. Your answer should include reference to the terms “short forward position” and “long forward position”. [3]

(ii)

A 3-month forward contract is issued on 1 February 2008 on a stock with a price of £150 per share. Dividends are received continuously and are constantly reinvested such that the dividend yield is 3% pa. In addition, it is anticipated that a special dividend of £30 per share will be paid on 1 April 2008. Assuming a risk-free force of interest of 5% pa and no arbitrage, calculate the forward price per share of the contract.
[3]
[Total 6]

Question 3.19
An equity pays half-yearly dividends. A dividend of d per share is due in exactly 3 months’ time. Subsequent dividends are expected to grow at a compound rate of g per half year forever.
(i)

If i denotes the annual effective rate of return on the equity, show that P , the price per share, is given by:
P=

(ii)

d (1 + i )¼
(1 + i )½ - (1 + g )

[3]

The current price of a share is £3.60, dividend growth is expected to be 2% per half year and the next dividend payment in 3 months is expected to be 12p.
Calculate the expected annual effective rate of return for an investor who purchases the share.
[3]
[Total 6]

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CT1: Q&A Bank Part 3 – Questions

Question 3.20
An investor purchases £100 nominal of a fixed interest stock, which pays coupons of
7% pa half-yearly in arrears. The stock is redeemable at par and can be redeemed at the option of the investor at any time between 5 and 10 years from the date of issue. The investor is subject to tax at the rate of 40% on income and 25% on capital gains.
(i)

Calculate the maximum price that the investor should pay in order to obtain a net yield of 6% pa.
[6]

(ii)

Given that this was the price paid by the investor, calculate his net annual running yield.
[1]
[Total 7]

Question 3.21
A woman purchased a government bond on 1 January 2000. The bond pays coupons of
6% pa six monthly in arrears on 30 June and 31 December. The bond is due to be redeemed at 105% at the end of the year 2010. The woman expects to achieve a net yield of 5% pa effective interest on her investment. She pays income tax at the rate of
23% on 1 April for any coupon payments received in the previous year (1 April to
31 March). She also pays capital gains tax on that date at the rate of 40% on any capital gains she realised in the previous year. Calculate the price she paid for the bond.
[7]

Question 3.22
A man bought a 5-year forward contract on 1 May 2006 to buy £400 nominal of a stock that pays coupons of 4% pa payable quarterly on 31 March, 30 June, 30 September and
31 December. The stock is also expected to pay out a lump sum of £50% on 1 August
2010. The stock is expected to yield 4.5% pa effective if purchased on 1 May 2006 and held forever.
(i)

Calculate the forward price for the contract, given that the risk free rate of interest is 5%.
[5]

(ii)

What is the value of the forward contract on 1 September 2008 when the stock price is £140%?
[4]
[Total 9]

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CT1: Q&A Bank Part 3 – Questions

Page 7

Question 3.23
(i)

A fixed interest stock is redeemable at 106% in 15 years’ time and pays interest at 9% pa payable half-yearly in arrears. What price should an investor pay to obtain a gross redemption yield of 9% pa?
[2]

(ii)

Instead of purchasing the stock, the investor decides to agree a forward contract to buy the security in six years’ time, immediately after the coupon payment then due. Calculate the forward price based on a risk-free rate of return of 6% pa effective and no arbitrage. The current price of the stock is that calculated in part (i).
[3]

(iii)

Three years later, the price of the security is such that the gross redemption yield is still 9%. Calculate the value of the forward contract if the risk-free yield has not changed.
[6]

(iv)

Calculate the yield obtained if the investor sold the forward contract after three years. [2]
[Total 14]

Question 3.24
In a particular country, income tax and capital gains tax are both collected on 1 April each year in relation to gross payments made during the previous 12 months.
A fixed interest bond is issued on 1 January 2003 with term of 25 years and is redeemable at 110%. The security pays a coupon of 8% per annum, payable half-yearly in arrears.
An investor, who is liable to tax on income at a rate of 25% and on capital gains at a rate of 30%, bought £10,000 nominal of the stock at issue for £9,900.
(i)

Assuming an inflation rate of 3% per annum over the term of the bond, calculate the net real yield obtained by the investor if he holds the stock to redemption. [9]

(ii)

Without doing any further calculations, explain how and why your answer to (i) would alter if tax were collected on 1 June instead of 1 April each year.
[2]
[Total 11]

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CT1: Q&A Bank Part 3 – Questions

Question 3.25
(i)

Explain what is meant by the ‘no arbitrage’ assumption in financial mathematics. [2]

(ii)

A fixed interest security will be redeemed at £110% nominal on 1 July 2012. It has a coupon rate of 4% pa payable half yearly and interest payments are made on 1 January and 1 July each year. It is priced in the market to give a gross redemption yield 5% pa effective. Calculate the market price of this security on
1 April 2007.
[3]

(iii)

On 1 April 2007 a two-year forward contract is issued to buy the security in (ii).
If the risk-free rate of return in the market is 3.5% pa effective, calculate the forward price of this contract.
[4]

(iv)

On 1 July 2008, immediately after the payment of the coupon due on this date, the market price of the security is £105% nominal and the risk-free rate of return is 4% pa effective. Calculate the market price of the forward contract in (iii). [3]
[Total 12]

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CT1: Q&A Bank Part 3 – Questions

Page 9

Question 3.26
On 15 May 2007 the government of a country issued an index-linked bond of term
15 years. Coupons are payable half-yearly in arrears, and the annual nominal coupon rate is 4%.
Interest and capital repayments are indexed by reference to the value of a retail price index with a time lag of 8 months. The retail price index value in September 2006 was
200 and in March 2007 was 206.
The issue price of the bond was such that, if the retail price index were to increase continuously at a rate of 7% pa from March 2007, a tax-exempt purchaser of the bond at the issue date would obtain a real yield of 3% pa convertible half-yearly.
(i)

Derive the formula for the price of the bond at issue to a tax-exempt investor. (b)
(ii)

(a)

Show that the issue price of the bond is £111.53%.

[12]

An investor purchases a bond at the price calculated in (i) and holds it to redemption. The retail price index increases continuously at 5% pa from March
2007. A new tax is introduced such that the investor pays tax at 40% on any real capital gain, where the real capital gain is the difference between the redemption money and the purchase price revalued according to the retail price index to the redemption date. Tax is only due if the real capital gain is positive.
Calculate the real annual yield convertible half-yearly actually obtained by the investor. [7]
[Total 19]

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CT1: Q&A Bank Part 3 – Questions

Question 3.27
A loan of nominal amount of £100,000 is to be issued bearing interest payable quarterly in arrear at a rate of 8% pa. Capital is to be redeemed at £105% on a coupon date between 15 and 20 years after the date of issue, inclusive, the date of redemption being at the option of the borrower.
(i)

An investor who is liable to income tax at 40% and tax on capital gains at 30% wishes to purchase the entire loan at the date of issue. What price should she pay to ensure a net effective yield of at least 6% pa?
[9]

(ii)

Exactly 10 months after issue the loan is sold to an investor who pays income tax at 20% and capital gains tax at 30%. Calculate the price this investor should pay to achieve a net yield of 6% pa on the loan:
(a)
(b)

(iii)

assuming redemption at the earliest possible date assuming redemption at the latest possible date

[6]

Explain which price the investor should pay to achieve a yield of at least 6% pa.
[2]
[Total 17]

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CT1: Q&A Bank Part 3 – Solutions

Page 1

Part 3 – Solutions
Solution 3.1
This question is taken from Subject A1, September 1997, Question 4.
The value of the share is:
PV = 5 ×

1.03
1.08½

+ 5×

1.032
1.033
+ 5×
+
1.08
1.081½

Summing this as an infinite geometric series, using S∞ =

PV = 5 ×

1.03
1.08½

[1]

a
, gives:
1− r

1.03 ⎞

⎜1 −
⎟ = £557.93 = £558 (to the nearest £)
⎝ 1.08½ ⎠

[2]

An alternative approach is to write the present value as an annuity valued at an interest rate j, ie:

(

PV = 5 v + v 2 + v3 + …

)

= 5a∞| @ j %

PV =

where v =

1
1.03
=
⇒ j = 0.8962%
1 + j 1.08½

5
= £558
0.008962

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Page 2

CT1: Q&A Bank Part 3 – Solutions

Solution 3.2
Eurobonds are a form of corporate or government borrowing.

[½]

They are usually unsecured.

[½]

They pay regular interest payments and are redeemable at par.

[½]

They are issued in many different currencies, eg Yen, DM, Sterling.

[½]

They are traded internationally through banks, and not in the traditional bond market.
[½]
Yields depend on risk and marketability, but will typically be slightly lower than for conventional unsecured loan stocks of the same issuer.
[½]

Solution 3.3
Debentures are part of the loan capital of companies. They are usually long-term investments. [½]
The issuing company provides some form of security to holders of debentures.

[½]

They are more risky than government bonds and are usually less marketable. The yield required by investors in debentures will be higher than the yield required on government bonds. [1]

Solution 3.4
This question is taken from Subject 102, September 2000, Question 3.
Assuming that the asset generates no income then the forward price, K, is calculated from the current price, S0 , using the formula:
K = S0ed t = S0 (1 + i )t
= 1.20 ¥ 1.0591/ 365
= £1.21

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[1]

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CT1: Q&A Bank Part 3 – Solutions

Page 3

Solution 3.5
The gross redemption yield solves the equation of value with no tax. So the price per
£100 nominal is:
(2)
P = 6a15| + 100v15 @10%

= 6 ¥ 1.024404 ¥ 7.6061 + 100 ¥ 0.23939 = 70.69

[2]

Since the stock is redeemed at par, the capital gains tax payable is:

0.30(100 - 70.69) = £8.79

[1]

Solution 3.6
This question is taken from Subject 102, September 2000, Question 4.
Preference shares are less common than ordinary shares.

[½]

Preference share dividends are limited to a set amount whereas ordinary share dividends are not fixed but are set by the directors of the company.
[½]
Preference shareholders rank above ordinary shareholders both for dividends and on winding up.
[½]
Therefore, for any given dividend distribution, no ordinary dividend can be paid if there are any outstanding preference dividends.
[½]
The expected return on preference shares is likely to be lower than on ordinary shares because of the risk of holding preference shares is lower.
[½]
Marketability of preference shares is likely to be worse than for ordinary shares.

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[½]

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CT1: Q&A Bank Part 3 – Solutions

Solution 3.7
This question is taken from Subject 102, September 2000, Question 7.
Since inflation is constant, first calculate the effective nominal rate of return. We need to find the value of i that solves the equation of value:
3

(

)

250 = 10v 4 1 + 1.05v + 1.052 v 2 + 1.053 v3 +

[1]

Summing this as a geometric progression:
1

1

1
10(1 + i ) 4 10(1 + i) 4
250 = 10v
=
=
1 - 1.05v 1 + i - 1.05 i - 0.05
3
4

[1]

Alternatively, this series could be summed using a perpetuity.
Solving this using trial and error:
@10%
@9%
@9.1%

RHS = 20.482
RHS = 25.544
RHS = 24.927

Therefore using interpolation i = 9.09% .

[1]
[1]

The real rate of return i¢ is such that:

1 + i¢ =

1.0909 fi i ¢ = 5.91%
1.03

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CT1: Q&A Bank Part 3 – Solutions

Page 5

Solution 3.8
This question is taken from Subject 102, April 2000, Question 7.
The dividends (in pence) per share are:
8
8 ¥ 1.08
8 ¥ 1.08 ¥ 1.07
8 ¥ 1.08 ¥ 1.07 ¥ 1.05

after 1 year after 2 years after 3 years after 4 years

8 ¥ 1.08 ¥ 1.07 ¥ 1.052

after 5 years

etc
The present value is:
8 È v + 1.08v 2 + 1.08 ¥ 1.07v3 + 1.08 ¥ 1.07 ¥ 1.05v 4 + 1.08 ¥ 1.07 ¥ 1.052 v5 +
Î
= 8 È v + 1.08v 2 + 1.08 ¥ 1.07v3 + 1.08 ¥ 1.07v3 (1.05v + 1.052 v 2 +
Î
Since v =


˚

˘
˚
[2]

1
, the present value is:
1.07

1.05v ˘
È
8 Í v + 2 ¥ 1.08v 2 + 1.08v 2
1 - 1.05v ˙
Î
˚
= 7.4766 + 15.0930 + 396.1918
= 418.76

[2]

This is the present value in pence for 1 share, so the total present value for 100 shares is
£418.76.
[1]

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Page 6

CT1: Q&A Bank Part 3 – Solutions

Solution 3.9
First, calculate the average nominal rates of return, which are:

(i)

13

for the set of gold coins in A:

Ê 20, 000 ˆ
Á 14, 000 ˜
Ë
¯

for the painting in B:

Ê 3, 200 ˆ
Á 3, 000 ˜ - 1 = 13.78%
Ë
¯

for the diamond in C:

Ê 10, 000 ˆ
Á 13, 000 ˜
Ë
¯

for the statuette in D:

Ê 19, 000 ˆ
Á 7,500 ˜
Ë
¯

- 1 = 12.62%

2

(ii)
(iii)
(iv)

12

- 1 = -12.29%
2
1 7 12

- 1 = 13.85%

The real rates of return, j , can then be calculated using an equation of the form:

1+ j =

1+ i i − 0.2
⇒ j=
1.2
1.2

where i is the nominal rate of return.
So the answers are:
(i)

–6.15%

[1]

(ii)

–5.19%

[1]

(iii)

–26.91%

[1]

(iv)

–5.13%

[1]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 3 – Solutions

Page 7

Solution 3.10
We will work in years, assuming a holding of £100 nominal.
The yield is the interest rate satisfying:
85 = 5a (2) + 105v9.5

[1]

9.5

For a rough guess we will use the total payment at a ‘typical’ time:
85 = (9.5 ¥ 5 + 105)v 7

fi i

8.7%

Using trial and error:
At 8%:

5a (2) + 105v9.5 = 33.0505 + 50.5433 = 83.5938
9.5

5a (2) + 105v9.5 = 34.4512 + 55.2133 = 89.6645

At 7%:

[2]

9.5

Interpolating:

i-7
85 - 89.6645
=
fi i = 7.8%
8 - 7 83.5938 - 89.6645

[2]

Solution 3.11
This question is taken from Subject 102, September 2003, Question 8.

(i)

These are bonds issued by the governments of developed countries in their domestic currency and are the most secure long-term investment available – there is virtually no default risk.
[½]
The return from these investments is fixed in monetary terms, so they are less volatile than other investments.
[½]
However, owing to the fixed nature of the returns, these bonds carry an inflation risk. If inflation turns out to be higher than expected at the time when the bond was purchased, the real rate of return achieved on the bond will be lower than expected. [1]

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Page 8

(ii)

CT1: Q&A Bank Part 3 – Solutions

The coupon payments are 8 at times 1, 2, 3, 4 and 5, and 4 at times 6, 7, 8, 9 and
10. The redemption payments are 50 at times 5 and 10. As the investor is liable to income tax at the rate of 30%, he only gets to keep 70% of each of the coupon payments. So the purchase price per £100 nominal of the bond is:

(

)

P = 0.7 4a10 + 4a5 + 50v5 + 50v10 @ 6%

[3]

= 32.403 + 37.363 + 27.920
= £97.69

[1]
[1]

Solution 3.12
Tax will involve payments of £2.40 per £100 nominal payable from 1 August 2001 to
1 August 2013 inclusive.
[1]
So the price per £100 nominal can be found from the equation of value:
(2)
P = 6a13| + 100v13 - 2.4v 7 /12 a13| @ 5%

= 6 ¥ 9.5096 + 100 ¥ 0.53032 - 2.4 ¥ 0.97194 ¥ 9.3936 = £88.18
So the price paid for the £10,000 holding was £8,818.

[3]
[1]

Solution 3.13
The amount of the redemption payment will be:
10, 000 ×

I (1.7.07)
193
= 10, 000 ×
= £13, 041
148
I (1.7.02)

[1]

So the money rate of return is found from the equation:
10, 250(1 + i )3 = 13, 041



i = 8.36%

[1]

Working in terms of 1.1.05 prices, the real rate of return is found from the equation:
10, 250(1 + i′)3 = 13, 041×

⇒ i′ = 3.47%

© IFE: 2009 Examinations

I (1.1.05)
175
= 13, 041×
= 11,354
I (1.1.08)
201

[2]

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CT1: Q&A Bank Part 3 – Solutions

Page 9

Solution 3.14
(i)

Arbitrage is a risk free trading profit.

[1]

An arbitrage opportunity exists if either:
(a)

(b)

(ii)

an investor can make a deal that would give her or him an immediate profit, with no risk of future loss, or
[1]
an investor can make a deal that has zero initial cost, no risk of future loss, and a non-zero probability of a future profit.
[1]

Hedging describes the use of financial instruments (including stocks, bonds, forward contracts and options) to reduce or eliminate a future risk of loss.
[1]

Solution 3.15
The forward price of the contract is:
K = 10 × 83e5×0.03 = £964.32

[1]

Therefore, on 1 February 2008, the investor pays £964.32 for £1,000 nominal of stock that has a price of £920. The investor therefore makes a loss of:
964.32 − 920 = £44.32

[1]

Solution 3.16
The risk free force of interest is log1.06 = 5.83% .
Assume that dividends are paid continuously, the dividend yield is constant and ignore tax, so that the forward price equals:
7.5e5¥(0.0583-0.03) = £8.64 per share

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[2]

© IFE: 2009 Examinations

Page 10

CT1: Q&A Bank Part 3 – Solutions

Solution 3.17


Government bonds are issued by governments. Unsecured loan stocks are issued by companies. Eurobonds may be issued either by governments,
[1]
companies or supra-national organisations (eg the World Bank).



Security will depend on the issuer. Government bonds will usually be regarded as the most secure type of debt (especially if issued by governments of developed countries).
[1]



Eurobond issues have a greater variety of individual features than issues of government bonds or unsecured loan stock.
[1]



The yield on a government bonds will usually be lower than the yield on a corresponding unsecured loan stock or Eurobond.
[1]



Government bonds and unsecured loan stock will usually pay a coupon twice per year. Eurobond coupon payments are usually made annually.
[1]



Marketability will usually be greater for government bonds than for unsecured loan stocks or Eurobonds.
[1]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 3 – Solutions

Page 11

Solution 3.18
This question is taken from Subject 102, April 2001, Question 4.

(i)

A forward contract is an agreement made at some time t = 0 , say, between two parties under which one agrees to buy from the other a specified amount of an asset (denoted S) at a specified price on a specified future date.
[1]
The investor agreeing to sell the asset is said to hold a short forward position in the asset, and the buyer is said to hold a long forward position.
[2]

(ii)

The seller of the asset has to sell one share on 1 May 2008. The holding that he requires on 1 February 2008 in order to have one share on 1 May 2008 (allowing for the growth in holding implied by the dividend yield) is: e 3
−0.03×12

[1]

In order to work out what the special dividend is, we first need to work out the holding on 1 April 2008. We do this by applying the dividend yield: e 3
−0.03×12

×e

2
0.03×12

=e

1
−0.03×12

This means that the special dividend is:
30e

1
−0.03×12

By 1 May 2008, this will have accumulated to:
30e

1
1
−0.03×12 0.05×12

e

= 30e

1
0.02×12

[1]

We then need to subtract this from the accumulated value of the current price, taking into account the growth in the holding, giving a forward price of:
150e

3
3
−0.03×12 0.05×12

e

− 30e

1
0.02×12

= £120.70

[1]

Please note that the solution given in the Examiners’ Report assumed that the special dividend was paid per shareholder rather than per share. This gives a solution of:

150e

3
3
−0.03×12 0.05×12

e

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− 30e

1
0.05×12

= £120.63

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Page 12

CT1: Q&A Bank Part 3 – Solutions

Solution 3.19
(i)

The cashflow timeline is as follows: d 2

d(1 + g) d(1 + g)
3
4

1
4

0

1

payment

1
4

time

The equation of value is given by:

P = dv¼ + d (1 + g )v ¾ + d (1 + g ) 2 v1¼ +
= dv¼ È1 + (1 + g )v½ + (1 + g ) 2 v +
Î

˘
˚

[1]

The expression in the bracket is an infinite geometric series:
P = dv¼ ¥

1
1 - (1 + g )v½

=

dv¼

[1]

1 - (1 + g )v½

Dividing the fraction through by v½ :
P=

(ii)

dv -¼ v -½ - (1 + g )

=

d (1 + i )¼

[1]

(1 + i )½ - (1 + g )

Substituting the values into the above equation gives:
3.60 =

0.12(1 + i )¼
(1 + i )½ - 1.02

fi 3.60(1 + i )½ - 3.672 = 0.12(1 + i )¼ fi 3.60(1 + i )½ - 0.12(1 + i )¼ - 3.672 = 0
Either

solving

this

as

a

[1] quadratic in

(1 + i )¼ ,

ie

0.12 ± 0.122 - 4 ¥ 3.6 ¥ -3.672
(1 + i ) = or using trial and improvement gives
2 ¥ 3.6
[2]
a rate of return of 11.1% pa.
¼

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CT1: Q&A Bank Part 3 – Solutions

Page 13

Solution 3.20
(i)

We first check whether there is a capital gain on redemption. To do this,we compare: g (1 - t1 ) = 0.6 ¥

7
= 0.042
100

with: i (2) @ 6% = 0.059126

As g (1 - t1 ) < i (2) @ 6% , there is a capital gain.

[2]

The stock is redeemable at the option of the investor. The investor will want to make the capital gain as soon as possible, so we assume that he or she will choose the earliest possible redemption date and the stock will be redeemed after
5 years.
[1]
Let P denote the maximum price that the investor can pay in order to achieve a net yield of 6% pa. The equation of value (allowing for tax) is:
P = 0.6 ¥ 7 a (2) + 100v5 - 0.25(100 - P )v5

[1]

0.25 ˆ
75
Ê fi P Á1 ˜ = 4.2 ¥ 1.014782 ¥ 4.2124 +
Ë 1.065 ¯
1.065

[1]

fi P = £91.00

[1]

5

(ii)

The investor’s net annual running yield is:

7
¥ 0.6 = 0.0462 = 4.62%
91

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[1]

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Page 14

CT1: Q&A Bank Part 3 – Solutions

Solution 3.21
The timeline for the bond (ignoring capital gains tax) is:

1/1/00

30/6/00

1/4/01
31/12/00

3

30/6/01

3
-0.23x6

1/4/02
31/12/01

3

3
-0.23x6

1/4/11
31/12/10

105+3
-0.23x6

We first need to find out if there is a capital gain:

i

(2)

1
Ê
ˆ
2 - 1˜ = 4.94%
@ 5% = 2 Á1.05
Á
˜
Ë
¯

Alternatively, you could look this value up in the Tables.

g (1 - t1 ) =

6
(1 - 0.23) = 4.4%
105

Since i (2) > g (1 - t1 ) , there is a capital gain.

[1]

Let the price paid be P. The equation of value is then:
P = 6a (2) + 105v11 - 0.4(105 - P )v11.25 - 0.23 ¥ 6a11 v 0.25
11

[3]

Substituting i = 5% :
P = 50.4539 + 61.3913 - 24.2588 + 0.231036 P - 11.3239 fi 0.768964 P = 76.2625 fi P = 99.18

So the price of the bond is £99.18%.

© IFE: 2009 Examinations

[1]
[1]

[1]

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CT1: Q&A Bank Part 3 – Solutions

Page 15

Solution 3.22
(i)

We first need to find the stock price, S0 . For £100 nominal we have:
2

S0 = 4a (4) v 12 + 50v


3
412

=

4 d (4)

2

v 12 + 50v

3
412

[1]

This needs to be evaluated at i = 4.5% . Calculating d (4) , we get: d (4) = 4(1 - 1.045-0.25 ) = 0.043776
So the price is:
S0 = 90.70725 + 41.46921 = 132.1765

[1]

The forward price is given by K 0 = ( S0 - I )(1 + i )5 , where I is the present value of income from the stock during the term and i = 0.05 . necessary values:

Calculating the

d (4) = 4(1 - 1.05-0.25 ) = 0.048494
I = 50v

3
412

2

+ 4a (4) v 12 = 40.63642 + 17.71118 = 58.3476
5

[1]

So the forward price is:
K 0 = (132.1765 − 58.3476)(1.05)5 = 94.226

[1]

Since this is for £100 nominal, the price for the stock in the question is:
4 × 94.226 = 376.91

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Page 16

(ii)

CT1: Q&A Bank Part 3 – Solutions

The value of the forward contract is given by ( K t - K 0 )(1 + i )

8
-212

, where Kt is

8 the forward price for a contract set up on 1 September 2008 with a term of 2 12

years:
K t = (140 - I ¢)(1 + i )

8
212

where I ¢ is the present value (ie the value at 1 September 2008) of the income received during the term of this new forward contract, calculated at 5%:
111

1

I ′ = 50v 12 + 4a (4) v 12 = 45.5362 + 10.3149 = 55.851
2.75

[1]

Therefore:
K t = (140 − 55.851)(1.05)

8
212

= 95.841

[1]

Finally the value of the forward contract is:
(95.841 − 94.226)(1.05)

8
−212

= 1.418

[1]

Since this is for £100 nominal, the value for the stock in the question is:
4 × 1.418 = 5.67

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[1]

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CT1: Q&A Bank Part 3 – Solutions

Page 17

Solution 3.23
(i)

The price of the stock per £100 nominal is given by:
(2)
S0 = 9a15| + 106v15 = 9 ¥ 8.0607 ¥ 1.022015 + 106 ¥ 0.27454 = £103.24%

(ii)

[2]

The forward price is calculated from the equation K = ( S0 - I )(1 + i )T , where S0 is the price of the security at time 0 and I is the present value of the fixed income payments due during the term of the forward contract.
Care must be taken since the question gives the effective rate not the force of interest. Therefore:
(2)
K 0 = (103.24 - 9a6| )(1 + i )6
(2)
= 103.24(1 + i )6 - 9 s6|

@ 6%

= 103.24 ¥ 1.41852 - 9 ¥ 6.9753 ¥ 1.014782 = £82.74%
(iii)

[3]

After three years, the price of the security is:
(2)
S3 = 9a12| + 106v12 = 9 ¥ 7.1607 ¥ 1.022015 + 106 ¥ 0.35553 = £103.55%

[1]

The value of the forward contract is:

( K3 - K0 ) v(T -t )

[1]

where T is the length of the original forward contract and t is the length of the new forward contract. Now:
K3 = ( S3 - 9a (2) )1.063 = £94.26%
3

[2]

So the value of the forward contract is:
(94.25 - 82.74)1.06 -3 = £9.67%
(iv)

[2]

The yield obtained is undefined (infinite) because the investor agreed the original forward contract at price 0 and sold it at time 3 for £9.67%.
[2]

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Page 18

CT1: Q&A Bank Part 3 – Solutions

Solution 3.24
This question is taken from Subject 102, April 2004, Question 8.
(i)

The first payment of income tax is on 1/4/04. The tax is payable on that date for both coupons in the previous 12 months. The amount of income tax payable on that date is 0.25 × 800 = 200 . The last payment of income tax is on 1/4/28.
The present value of the income tax payments on 1/4/03 is: 200a25

[1]

The present value of the income tax payments on 1/1/03 is: 200a25 v 0.25

[1]

The capital gains tax is payable on 1/4/28. The amount of capital gains tax
[1]
payable is: 0.3 (11, 000 − 9,900 ) = 330
The equation of value is:
9,900 = 800a (2) + 11, 000v 25 − 330v 25.25 − 200a25 v 0.25
25

(Equation 1) [1]

The net coupon is 6% (= 8% × (1 − 0.25)) , but there is a capital gain on redemption, so the interest rate must be higher than this. Therefore, try 7% as a first guess.
Trying values in equation 1:
At 7%, RHS = 9,158.59
At 6.5%, RHS = 9, 724.14
At 6%, RHS = 10,345.36

[3]

Interpolating between 6% and 6.5%, we get:
9,900 − 10,345.36 i−6 =
⇒ i = 6.36%
6.5 − 6 9, 724.14 − 10,345.36
So the real rate of interest is:
(ii)

0.0636 − 0.03
= 3.26%
1.03

[1]

[1]

The tax payment is being deferred, so we would expect a higher interest rate. [2]

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CT1: Q&A Bank Part 3 – Solutions

Page 19

Solution 3.25
(i)

No arbitrage means that arbitrage opportunities do not exist. Arbitrage is a riskfree trading profit, which occurs when an investor can make a deal that gives them: [1]


an immediate profit, with no risk of future loss;

[½ ]

or:



no risk of future loss, and



(ii)

no initial cost, a non-zero probability of future profit.

[½ ]

Let S0 denote the market price per £100 nominal of the security on 1 April
2007. Since the next coupon is due in 3 months’ time, and coupons are payable half-yearly thereafter, it is easier for us to set up the equation of value on 1
January 2007. This is:

S0 v¼ = 4a (2) + 110v5½

[1]



evaluated using i = 0.05 . Now:

(1 − 1.05 ) = 4.76526
@ 5% =
2 ( 1.05 − 1)
−5½

a (2)


S0 = 1.05¼ 4 × 4.76526 +

110
1.055½

[1]

= 104.4379

So the market price of the security on 1 April 2007 is £104.44 per £100 nominal.
[1]

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Page 20

(iii)

CT1: Q&A Bank Part 3 – Solutions

Let K 0 denote the price of the forward contract per £100 nominal of the security. Then:
K 0 = ( S0 - I ) (1 + i )

2

[1]

where: i = 0.035 and: I = 1.035¼ ¥ 4a (2) = 1.035¼ ¥ 4 ¥ 1.91617 = 7.7309
2

[2]

Hence:
K 0 = (104.4379 - 7.7309) ¥ 1.0352 = 103.5950
So the forward price is £103.59 per £100 nominal.
(iv)

[1]

The market price is the value of the forward contract.
Let K1 denote the price of a forward contract per £100 nominal of the security taken out on 1 July 2008 and expiring on 1 April 2009. Then:

(

)

K1 = 105 - 2v½ (1 + i ) @ 4% = 106.1148
¾

[2]

So the value on 1 July 2008 of the original forward contract is:

(106.1148 - 103.5950) ¥ 1.04-¾ ie £2.45 per £100 nominal.

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= 2.45
[1]

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CT1: Q&A Bank Part 3 – Solutions

Page 21

Solution 3.26
This question is taken from Subject A1, September 1997, Question 15.

(i)(a) The price for £100 nominal calculated using an effective nominal yield (ie not adjusted for inflation) of i per half year is (working in half years):

⎛ 206
206 × 1.07½ 2
P = 2×⎜ v+ v +
⎜ 200
200

= 2×

206
(v + 1.07½ v 2 +
200

+

206 × 1.0714½ 30 ⎞
206
v ⎟ + 100 ×
×1.0714½ v30

200
200


+ 1.0714½ v30 ) + 100 ×

206
×1.0714½ v30
200

[4]

The relationship between the effective half-yearly real yield ir and the effective half yearly nominal yield i is:

1 + ir =
(b)

1+ i
½

1.07

ie 1.015 =

1+ i
½

1.07

⇒v=

1

[2]

1.015 ×1.07½

The expression for the price (working in half years) can be written as:

P = 2×

206
1
×
200 1.07½

1
⎛ 1
+
⎜ 1.015 +
1.0152


+

1
30

1.015

206
1
1

×
⎟ + 100 × 200 ×
½
1.07
1.01530


=

206
1
×
(2a30| + 100v30 ) @1½%
200 1.07½

[3]

=

206
1
×
(2 × 24.0158 + 100 × 0.63976)
200 1.07½

[2]

= £111.53
(ii)

[1]

This investor (who is assumed to purchase the bond on the issue date) will make a real capital gain per £100 nominal of:

100 ×

206
×1.0514½ − 111.53 × 1.0515 = 208.97 − 231.86 < 0
200

ie there will be no real capital gain.

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[2]

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Page 22

CT1: Q&A Bank Part 3 – Solutions

The equation of value for this investor will therefore be:
111.53 =

206
1
×
(2a30| + 100v30 ) @ ir
½
200 1.05

[1]

where ir denotes the effective half-yearly real yield.
This is similar to the equation above whose solution was ir = 0.015 . So the solution here will be slightly greater than 0.015. Trying 0.015 and 0.016, we get
112.59 and 110.04.
[2]
So by interpolation, we find that ir = 0.0154 , ie the investor’s real annual yield is 3.08% per annum convertible half-yearly.

[2]

Solution 3.27
This question is taken from Subject A1, April 1997, Question 16.
(i)

First we need to find out if there is a capital gain:

g (1 − t1 ) =

(1 − 0.4) × 8
= 4.57%
105

i (4) @ 6% = 4(1.060.25 − 1) = 5.8695%
Since i (4) > g (1 − t1 ) , the investor makes a capital gain and the investor’s yield will decrease if the term to redemption is extended.

[2]

So the equation of value in this case should assume a term of 20 years (the maximum) and should allow for capital gains tax.
[1]
This leads to the following equation for the price per £100 nominal (with all functions calculated at 6%):
(4)
P = 0.6 × 8 a20| + [105 − 0.3(105 − P)]v 20

[2]

= 0.6 × 8 ×11.7248 + [105 − 0.3(105 − P )] × 0.311805

[1]

= 56.279 + 22.918 + 0.09354P

[1]

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CT1: Q&A Bank Part 3 – Solutions

Page 23

Rearranging to find P gives:
P = 79.197 / 0.90646 = £87.370

[1]

So for a nominal holding of £100,000 the price to ensure a yield of at least 6% would be not greater than £87,370.
[1]
(ii)

Now, find out if there is a capital gain for the second investor:

g (1 − t1 ) =

(1 − 0.2) × 8
= 6.10%
105

i (4) @ 6% = 4(1.060.25 − 1) = 5.8695%
Since i (4) < g (1 − t1 ) there will be no capital gain and CGT can be ignored.

[1]

If the full term of the loan is n years, the price for this investor can be found from the equation:
(4)
P = 0.8 × 8(1 + i )1/12 an−¾| + 105v n−10 /12

[2]

The (1 + i )1/12 term is required because the next coupon payment is due in 2 months. The first payment of the term a (4) | is in three months. n−¾ In the two cases this works out to be:
(a)
(b)

(iii)

When n = 15 : P = 6.4 × 1.00487 × 9.6106 + 105 × 0.43803 = £107.80%
When n = 20 : P = 6.4 × 1.00487 × 11.4876 + 105 × 0.32732 = £108.25%
[1½ marks each]

If redemption turns out to be after 15 years, this investor must pay no more than
£107.80% to achieve the required yield of 6%. If redemption turns out to be after 20 years, he/she must pay no more than £108.25% to achieve the required yield of 6%. To be sure of obtaining the required yield both of these conditions must be satisfied ie the price must not exceed the lower figure ie £107.80%. [2]

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CT1: Q&A Bank Part 4 – Questions

Page 1

Part 4 – Questions
Question 4.1
In the bond market of a certain country, zero-coupon bonds (all redeemable at par) are available with the following prices:
Term
4 years
5 years
6 years
7 years

Price (£%)
£79
£74
£69
£64

Calculate (to the nearest ¼%) the one-year forward rate of interest starting in 5 years’ time implied by these prices.
[2]

Question 4.2
An economist’s model of interest rates indicates that the n -year spot rate of interest is
0.1(1 + e -0.1n ) -1 . According to this model, what is the price of a 10-year zero-coupon bond redeemable at par?
[2]

Question 4.3
Consider a fixed interest security that pays coupons of 10% at the end of each year and is redeemable at par at the end of the third year.
Calculate (using an effective interest rate of 8% pa) the:
(i)

volatility of the cashflows

[2]

(ii)

discounted mean term of the cashflows

[1]

(iii)

convexity of the cashflows.

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[1]
[Total 4]

© IFE: 2009 Examinations

Page 2

CT1: Q&A Bank Part 4 – Questions

Question 4.4
The 1-year forward rates for transactions beginning at times t = 0, 1, 2 are ft where: f 0 = 0.06,

f1 = 0.065,

f 2 = 0.07

Calculate the par yield for a 3-year bond.

[3]

Question 4.5
A stochastic interest rate model assumes that the interest rates in different years are independent and identically distributed normal random variables with mean 8% and standard deviation 2%. Calculate the mean and standard deviation of the accumulated value, at time 2, of an initial investment of £10,000.
[4]

Question 4.6
A stochastic interest rate model assumes that the interest rates in different years are independent and conform to the probability distribution:

Ï 0.055
Ô
i = Ì 0.075
Ô 0.095
Ó

with probability 0.3 with probability 0.5 with probability 0.2

Calculate the standard deviation of the annual interest rate.

[3]

Question 4.7
An insurance company has a continuous payment stream of liabilities to meet over the coming 20 years. The payment stream will be at a rate of £10 million per annum throughout the period.
Calculate the duration of the continuous payment stream at a rate of interest of 4% per annum effective.
[4]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Questions

Page 3

Question 4.8
£1,000 is invested for 10 years. In any year the yield on the investment will be 4% with probability 0.4, 6% with probability 0.2 and 8% with probability 0.4 and is independent of the yield in any other year.
(i)

Calculate the mean accumulation at the end of 10 years.

[2]

(ii)

Calculate the standard deviation of the accumulation at the end of 10 years.

[5]

(iii)

Without carrying out any further calculations, explain how your answers to (i) and (ii) would change (if at all) if:
(a)

the yields had been 5%, 6% and 7% instead of 4%, 6% and 8% per annum, respectively; or

(b)

the investment had been made for 12 years instead of 10 years.

[4]
[Total 11]

Question 4.9
Yields on a fund in different years are assumed to be independently and identically distributed, and (1 + i ) follows a lognormal distribution. The mean of the interest rate in any year is 0.05 and the standard deviation is 0.012. Calculate the probability that an investment of £150 accumulates to more than £230 in 8 years’ time.
[5]

Question 4.10
A stochastic interest rate model assumes that the annual interest rate during the next year will be 7½% and that the interest rate in subsequent years will be at a fixed but unknown level with probabilities in accordance with the following probability distribution: Ï 5½%
Ô
i = Ì 7½%
Ô 9½%
Ó

with probability 0.3 with probability 0.5 with probability 0.2

What is the expected accumulated amount by the end of the fifth year of an initial investment of £20,000?
[3]

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Page 4

CT1: Q&A Bank Part 4 – Questions

Question 4.11
You are given the following information about the term structure of interest rates:
The 1-year spot rate is 4.1% pa.
The 2-year spot rate is 4.0% pa.
The 4-year spot rate is 4.4% pa.
The 5-year spot rate is 4.5% pa.
(i)

Calculate the theoretical price of a 3-year zero-coupon bond that is issued at time
2 and is redeemable at par.
[2]

(ii)

A 1-year zero-coupon bond is issued at time 3 and has a theoretical price of
£95.50 per £100 nominal. Given that the bond is redeemable at par, calculate:
(a)
(b)

(iii)

the 3-year spot rate, and the 1-year forward rate starting at time 2.

[3]

Calculate the gross redemption yield on a bond that pays annual coupons of
6% pa (at times 1, 2, 3, 4 and 5) and is redeemable at 110% in 5 years’ time. [4]
[Total 9]

Question 4.12
An individual purchases £100,000 nominal of a bond on 1 January 2003 which is redeemable at 105% in four years time and pays coupons of 4% per annum at the end of each year.
The investment manager wishes to invest the coupon payments on deposit until the bond is redeemed. It is assumed that the rate of interest at which the coupon payments can be invested is a random variable and the rate of interest in any one year is independent of that in any other year.
Deriving the necessary formulae, calculate the mean value of the total accumulated investment on 31 December 2006 if the annual effective rate of interest has an expected value of 5½% in 2004, 6% in 2005 and 4½ % in 2006.
[5]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Questions

Page 5

Question 4.13
Calculate, using an interest rate of 10%, the discounted mean term of a 20-year annuity certain payable annually in arrear under which the first payment is £1,000 if:
(i)

the payments remain level throughout the term

[2]

(ii)

the payments increase by £50 each year

[3]

(iii)

the payments increase by 5% (compound) each year

[4]

(iv)

the payments increase by 10% (compound) each year.
20

You are given that

 t 2 ¥ 1.1-t

[2]
[Total 11]

= 718.027 .

t =1

Question 4.14
(i)

Derive and simplify expressions in terms of the interest rate i for the present value and the discounted mean term of the rental income from a property if the income is expected to make payments of D (1 + j )t at time t years
(t = 1, 2,3,…) . Assume that the rental income continues indefinitely.
[5]

(ii)

Evaluate the expressions in (i) for the case where D = £5, 000 , i = 8% and j = 3% .
[2]
[Total 7]

Question 4.15
Outline the similarities and differences between deterministic and stochastic interest rate models. [3]

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© IFE: 2009 Examinations

Page 6

CT1: Q&A Bank Part 4 – Questions

Question 4.16
The expected annual effective rate of return from an insurance company’s investments is 6% and the standard deviation of annual returns is 8%. The annual effective returns are independent and (1 + it ) is lognormally distributed, where it is the return in the t th year. (a)

Calculate the expected value of an investment of £1 million after ten years.

(b)

Calculate the probability that the accumulation of the investment will be less than 90% of the expected value.
[8]

Question 4.17
Three bonds paying annual coupons in arrears of 7% and redeemable at 105 per £100 nominal reach their redemption dates in exactly one, two and three years time, respectively. The price of each of the bonds is £98 per £100 nominal.
(i)

Determine the gross redemption yield of the 3-year bond.

(ii)

Calculate all possible spot rates implied by the information given.

[3]
[5]
[Total 8]

Question 4.18
An insurance company has liabilities of £10 million due in 10 years time and £20 million due in 15 years time, and assets consisting of two zero-coupon bonds, one paying £7.404 million in 2 years time and the other paying £31.834 million in 25 years time. The current interest rate is 7% per annum effective.
(i)

Show that Redington’s first two conditions for immunisation against small changes in the rate of interest are satisfied for this insurance company.
[5]

(ii)

Determine the profit or loss, expressed as a present value, that the insurance company will make if the interest rate increases immediately to 7.5% per annum effective. [2]

(iii)

Explain how you might have anticipated, before making the calculation in (ii), whether the result would be a profit or loss.
[2]
[Total 9]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Questions

Page 7

Question 4.19
(i)

Prove that ( Ia ) n =

an − nv n i .

[3]

A government bond pays a coupon half-yearly in arrears of £10 per annum. It is to be redeemed at par in exactly ten years. The gross redemption yield from the bond is 6% per annum convertible half-yearly.
(ii)

Calculate the duration of the bond in years.

[8]

(iii)

Explain why the duration of the bond would be longer if the coupon rate were £8 per annum instead of £10 per annum.
[2]
[Total 13]

Question 4.20
An investment analyst wishes to model the annual rate of growth i of an investment fund using a probability distribution of the form:

3k
Ï 0.05 with probability
Ô
i = Ì0.075 with probability 1 - 4k
Ô 0.125 with probability k Ó where k is a suitable constant. Determine the maximum and minimum values that can be obtained for the mean and standard deviation of i using this family of distributions. [5]

Question 4.21
An investor intends to invest three lump sums in an investment account, one at the start of each of the next 3 years, and has asked you about the likely amount to which the combined payments will have accumulated by the end of the third year. The amounts of the lump sums are shown in the table below. Calculate the mean and variance of the accumulated amount assuming a varying interest rate model in which the mean and standard deviation of the rate of return in each year are as shown in the table:
Year
1
2
3

Lump sum invested Mean rate of return
£50,000
8%
£30,000
7%
£20,000
6%

SD of rate of return
2%
3%
4%
[8]

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© IFE: 2009 Examinations

Page 8

CT1: Q&A Bank Part 4 – Questions

Question 4.22
The n -year spot rate of interest sn is given by the formula:

sn = 0.050 -

n
500

for n = 1 , 2 and 3

(i)

Calculate the implied one-year forward rates applicable at times t = 1 and t = 2 .
[3]

(ii)

An investor purchases a 3-year bond that provides coupons of 6% pa payable annually in arrears and is redeemable at par. Show that the fair price for this bond is £104.36 per £100 nominal.
[1]

(iii)

Calculate the investor’s gross redemption yield.

(iv)

Calculate the par yield of the bond.

[3]
[3]
[Total 10]

Question 4.23
In any year, the rate of interest on funds invested with a particular company has mean value j and standard deviation s, and is independent of the rates of interest in all previous years.
(i)

Derive formulae for the mean and the variance of the accumulated value after n years of a single investment of 1 at time 0.
[5]

(ii)

Let it be the rate of interest earned in the t th year. Each year the value of
(1 + it ) is lognormally distributed, with parameters m = 0.04 and s 2 = 0.09 .
(a)

Show that n, the number of years that must elapse before the accumulation of a lump sum invested at time 0 has a 75% probability of at least doubling in size, satisfies:
0.04n - 0.2024 n - ln 2 = 0

(b)

Hence, or otherwise, calculate the value of n.

© IFE: 2009 Examinations

[8]
[Total 13]

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CT1: Q&A Bank Part 4 – Questions

Page 9

Question 4.24
A company incurs a liability to pay £1, 000(1 + 0.4t ) at the end of year t , for t equal to
5, 10, 15, 20 and 25. It values these liabilities assuming that in the future there will be a constant effective interest rate of 7% per annum. An amount equal to the total present value of the liabilities is immediately invested in two stocks:
Stock A pays coupons of 5% per annum annually in arrears and is redeemable in 26 years at par.
Stock B pays coupons of 4% per annum annually in arrears and is redeemable in 32 years at par.
The gross redemption yield on both stocks is the same as that used to value the liabilities. (i)

Calculate the present value of the liabilities.

[3]

(ii)

Calculate the discounted mean term of the liabilities.

[3]

(iii)

If the discounted mean term of the assets is the same as the discounted mean term of the liabilities, calculate the nominal amount of each stock which should be purchased.
[9]
[Total 15]

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© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Questions

Question 4.25
A stochastic interest rate model assumes that the annual growth factors for each future year are independently distributed lognormal random variables with parameters

m = 0.05 and s 2 = 0.01 .
(i)

Calculate the mean and standard deviation of the annual rate of return.

[3]

(ii)

Calculate the median and the upper and lower quartiles of the distribution of the annual rates of return.
[4]

(iii)

Comment on the relative sizes of the mean, median and mode of the annual rate of return.
[2]

(iv)

A single payment of £10,000 must be made at the end of 5 years. Calculate the probability that a single initial investment of £7,250 will be sufficient to meet the liability for this payment.
[3]

(v)

State, with reasons, whether the probability calculated in (iv) would be greater or smaller if the interest rate model assumed an unknown interest rate that was constant in all future years (with the same basic distribution), rather than assuming that the rates for each year are independent.
[3]
[Total 15]

You are given that the mode of a log-normal distribution is at x = e m -s .
2

Question 4.26
The annual rates of return on an investment fund are assumed to be independent and identically distributed. Each year the distribution of 1 + i is lognormal with parameters

m = 0.07 and s 2 = 0.006 , where i is the annual yield on the fund.
Calculate the amount that should be invested in the fund immediately to ensure an accumulated value of at least £500,000 in ten years’ time with a probability of 0.99. [7]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Questions

Page 11

Question 4.27
(i)

Explain what is meant by the “expectations theory” explanation for the shape of the yield curve.

(b)

(ii)

(a)

Explain how expectations theory can be modified by both “liquidity preference” and “market segmentation” theories.
[6]

Short-term, one-year annual effective interest rates are currently 10%; they are expected to be 9% in one year’s time, 8% in two years’ time, 7% in three years’ time and to remain at that level thereafter indefinitely.
(a)

If bond yields over all terms to maturity are assumed only to reflect expectations of future short-term interest rates, calculate the gross redemption yields from 1-year, 3-year, 5-year and 10-year zero coupon bonds. (b)

Draw a rough plot of the yield curve for zero coupon bonds using the data from part (ii)(a). (Graph paper is not required.)

(c)

Explain why the gross redemption yield curve for coupon paying bonds will slope down with a less steep gradient than the zero coupon bond yield curve.
[8]
[Total 14]

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CT1: Q&A Bank Part 4 – Questions

Question 4.28
An investor has to pay a lump sum of £20,000 at the end of fifteen years from now and an annuity certain of £5,000 per annum payable half-yearly in advance for twenty-five years, starting in ten years’ time.
The investor currently holds an amount of cash equal to the present value of these two liabilities valued at an effective rate of interest of 7% per annum.
The investor wishes to immunise her fund against small movements in the rate of interest by investing the cash in two zero coupon bonds, Bond X and Bond Y . The market prices of both bonds are calculated at an effective rate of interest of 7% per annum. The investor has decided to invest an amount in Bond X sufficient to provide a capital sum of £25,000 when Bond X is redeemed in ten years’ time. The remainder of the cash is invested in Bond Y .
In order to immunise her holdings:
(i)

calculate the amount of money invested in Bond Y , and

(ii)

determine the term needed for Bond Y and the redemption amount payable on the maturity date.
[10]

(iii)

Without doing any further calculations, state which other condition needs to be satisfied for immunisation to be achieved successfully.
[2]
[Total 17]

© IFE: 2009 Examinations

[5]

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CT1: Q&A Bank Part 4 – Solutions

Page 1

Part 4 – Solutions
Solution 4.1
The price of the 5-year bond implies that the accumulation factor for the next 5 years is
A(0,5) = 100 / 74 = 1.3514 , while the price of the 6-year bond implies that the accumulation factor for the next 6 years is A(0, 6) = 100 / 69 = 1.4493 . So the implied accumulation factor for the sixth year is:

A(5, 6) = A(0, 6) / A(0,5) = 1.4493 /1.3514 = 1.0725
Therefore the required forward rate is 7.25%.

[2]

Solution 4.2
Using the economist’s formula, the 10 year spot rate of interest is: i = 0.1(1 + e -0.1¥10 ) -1 = 0.07311

[1]

So the price per £100 nominal of a 10 year zero coupon bond is:
100(1 + i ) -10 = 100 /1.0731110 = £49.38

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[1]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Solutions

Solution 4.3
(i)

The present value of the cashflows for £100 nominal of this stock is:
P (i ) = 10v + 10v 2 + 110v3 = £105.15

[1]

So the volatility is:

(ii)

P ¢(i ) 10v 2 + 20v3 + 330v 4 267.01
=
=
= 2.54
P(i )
105.15
105.15

The discounted mean term is the volatility multiplied by 1 + i :

2.54 ¥ 1.08 = 2.74 years
(iii)

[1]

[1]

The convexity is:

P ¢¢(i ) 20v3 + 60v 4 + 1320v5 958.35
=
=
= 9.11
P(i )
105.15
105.15

[1]

Solution 4.4
This question is taken from Subject A1, September 1997, Question 3.
The par yield , g ,is the annual coupon that gives a price equal to the par value based on the current term structure of interest rates.
The par yield is therefore found from the equation:

1
1
1
⎛ 1

1= g⎜
+
+
⎟+
⎝ 1.06 1.06 ×1.065 1.06 ×1.065 ×1.07 ⎠ 1.06 ×1.065 ×1.07 ie [2]

1 = g (0.94340 + 0.88582 + 0.82787) + 0.82787 ⇒ g = 0.06478

The par yield is therefore 6.478%.

© IFE: 2009 Examinations

[1]

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CT1: Q&A Bank Part 4 – Solutions

Page 3

Solution 4.5
The mean accumulated amount at the end of 2 years is:

E ( S 2 ) = 10, 000 ¥ 1.082 = £11, 664

[1]

The variance of the accumulated amount at the end of 2 years is:

var(10, 000S 2 ) = 10, 0002 var( S 2 )
= 10, 0002 [((1 + j ) 2 + s 2 ) 2 - (1 + j ) 4 ]
= 10, 0002 [(1.082 + 0.022 ) 2 - (1.08) 4 ]
= 93,328

[2]

So the standard deviation is £305.

[1]

Solution 4.6
The mean interest rate is:

E (i ) = 0.3 ¥ 0.055 + 0.5 ¥ 0.075 + 0.2 ¥ 0.095 = 0.073

[1]

The variance of the interest rate is:

var(i ) = E (i 2 ) - [ E (i )]

2

= 0.3 ¥ 0.0552 + 0.5 ¥ 0.0752 + 0.2 ¥ 0.0952 - 0.0732
= 0.005525 - 0.005329
= 0.000196

[1]

So the standard deviation is:
0.000196 = 0.014

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[1]

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CT1: Q&A Bank Part 4 – Solutions

Solution 4.7
This question is taken from Subject 102, September 2004, Question 3.
Let the present value of the liabilities be P . We have:
P = 10a20

1 − v 20
1 − (1 + i ) −20
= 10
= 10 ln(1 + i ) ln(1 + i )

Evaluating this @ 4% gives P = 138.6035 .

[1]

We calculate the duration (or discounted mean term) as:
20

10 ∫ tvt dt
0
20

10 ∫ vt dt

20

=

10 ∫ te −δ t dt
0

10a20

=

( Ia )20 a20 =

120.6663
= 8.71
13.8604

[3]

0

Alternatively, find the duration by using the volatility. The definition of volatility is:



1 dP
×
P di

Differentiating the expression for the present value using the quotient rule, we get:

(

)

⎧ ln(1 + i ) × 20(1 + i ) −21 − 1 − (1 + i ) −20 × 1 ⎫ dP 1+i ⎪

= 10 ⎨

2 di [ln(1 + i )]




Evaluating this at 4% interest gives

dP
= 1,160.253 , and so the volatility is: di 1,160.253
= 8.371
138.6035
Finally, the duration is the volatility multiplied by (1 + i ) so we get a duration of:
1.04 × 8.371 = 8.71

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CT1: Q&A Bank Part 4 – Solutions

Page 5

Solution 4.8
This question is taken from Subject 102, April 2003, Question 9.
(i)

First of all, we need to find the mean, j , of the rate of return in each year:

j = (0.04 × 0.4) + (0.06 × 0.2) + (0.08 × 0.4) = 0.06

[1]

Let S10 represent the accumulated amount at time 10 of an investment of 1 at time 0. So:
E[1, 000 S10 ] = 1, 000 (1 + j )

10

(ii)

= 1, 000 (1.06 )

10

= £1, 790.85

[1]

To calculate the variance of the accumulated amount, we first need to calculate the variance, s 2 , of the rate of return in each year. s 2 = (0.042 × 0.4) + (0.062 × 0.2) + (0.082 × 0.4) − 0.062 = 0.00032

[1]

So: var (1, 000 S10 ) = 1, 0002 var[ S10 ]

(

)

+ s ) − ( (1 + j ) ) ⎥


2
= 1, 0002 E[ S10 ] − ( E[ S10 ])

(

10

= 1, 0002 ⎢ (1 + j )2 2


= 1, 0002 ⎢ 1.062 + 0.00032

= 9,145.60

(

So the standard deviation is £95.63.

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2

10 2

)

10

20 ⎤
− (1.06 ) ⎥


[2]
[1]
[1]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Solutions

(iii)(a) The average yield in any year, j , will remain unchanged because the random variable is still symmetrically distributed about 6%. The values of the yield are more closely packed around the mean, so s 2 will be smaller.
The mean accumulated value will be unchanged, but the standard deviation of this value will decrease.
[2]
(iii)(b) The values of j and s 2 are unchanged. However, the accumulation is over a longer period so both the mean and the standard deviation of the accumulated value will increase.
[2]

Solution 4.9
We have (1 + i ) ~ log N ( m ,s 2 ) . Since we are given the mean and variance of i , we need to find the mean and variance of 1 + i and hence find m and s 2 . The mean and variance of 1 + i are 1.05 and 0.0122 respectively. To find m and s 2 we must solve the simultaneous equations:

1 ˘
È
exp Í m + s 2 ˙ = 1.05
2 ˚
Î

{

}

exp È 2 m + s 2 ˘ exp Ès 2 ˘ - 1 = 0.0122
Î
˚
Î ˚

This gives:

exp Ès 2 ˘ - 1 =
Î ˚

0.0122
1.05

2

fi s 2 = 0.000130604

Substituting this back into the first equation, we get m = 0.048724862 .

[2]

We want the probability:
8
Ê
ˆ
Ê 8
230 ˆ
P Á150’ (1 + i j ) > 230˜ = P Á ’ (1 + i j ) >
150 ˜
Ë
¯
Ë i =1
¯
i =1

From the properties of the lognormal distribution, we know that:
8

’ (1 + i j ) ~ log N (8 m ,8s 2 ) = log N (0.3897989, 0.0010448)

[1]

i =1

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Page 7

So the probability becomes:

Ê 8
Ê 8
230 ˆ
230 ˆ
= P Á ln ’ (1 + i j ) > ln
P Á ’ (1 + i j ) >
˜
150 ¯
150 ˜
Ë i =1
Ë i =1
¯
230
Ê
ˆ ln - 0.3897989
Á
˜
= P Á Z > 150
˜
0.0010448
Á
˜
Ë
¯
= P ( Z > 1.165)
= 1 - 0.87799 = 0.12201

[2]

Solution 4.10
The probability distribution of the accumulated amount at the end of 5 years is:

Ï20, 000 ¥ 1.075 ¥ 1.0554 = £26, 635 with probability 0.3
Ô
Ô
S5 = Ì20, 000 ¥ 1.075 ¥ 1.0754 = £28, 713 with probability 0.5
Ô
Ô20, 000 ¥ 1.075 ¥ 1.0954 = £30,910 with probability 0.2
Ó

[2]

So the expected accumulated amount is:
E ( S5 ) = 0.3 ¥ 26, 635 + 0.5 ¥ 28, 713 + 0.2 ¥ 30,910 = £28,529

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CT1: Q&A Bank Part 4 – Solutions

Solution 4.11
(i)

Let P denote the theoretical price of the bond per £100 nominal. Discounting the cashflows back to time 0, we have:
P
1.0402

=

100
1.0455

fi P = £86.79

[2]

(ii)(a) Let y3 denote the 3-year spot rate. Again, discounting the cashflows back to time 0, we have:

95.50

(1 + y3 )3

=

100
1.0444

fi (1 + y3 ) =
3

95.50 ¥ 1.0444
= 1.134502
100

fi y3 = 0.042962

[2]

(ii)(b) Let f 2,1 denote the 1-year forward rate starting at time 2. Then:

(1 + y3 )3
1 + f 2,1 =
(1 + y2 )2
(iii)

fi f 2,1 =

1.134502
1.0402

- 1 = 0.048911

[1]

The theoretical price of the bond per £100 nominal is:
P=

6
6
6
6
116
+
+
+
+
= £114.73
2
3
4
1.041 1.040
1.042962
1.044
1.0455

[1]

The gross redemption yield is the effective annual rate of interest that satisfies the equation 114.73 = 6a5 + 110v5 .

[1]

Ê 1 - 1.044 -5 ˆ
110
= 115.11
If i = 4.4% , then RHS = 6 Á
˜+
Ë 0.044 ¯ 1.0445

[½ ]

This is too high. We can reduce the present value by increasing the interest rate.

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CT1: Q&A Bank Part 4 – Solutions

Page 9

Ê 1 - 1.045-5 ˆ
110
= 114.61 .
RHS = 6 Á
If i = 4.5% , then
˜+
Ë 0.045 ¯ 1.0455

[½ ]

This is too low, so the GRY must lie between 4.4% and 4.5%. Interpolating we get: Ê 114.73 - 115.11ˆ
4.4 + Á
¥ 0.1 = 4.48%
Ë 114.61 - 115.11 ˜
¯

[1]

Solution 4.12
This question is taken from Subject 102, April 2004, Question 1.
The coupon is 4% of 100,000 which is £4,000. The time line for this bond is:

4,000

1/1/04

1/1/03

4,000
1/1/05

i1

i2

105,000
4,000

1/1/06 i3 Amount

1/1/07

4,000

Date

i4

The accumulated amount of the investment is:
A = 4, 000(1 + i2 )(1 + i3 )(1 + i4 ) + 4, 000(1 + i3 )(1 + i4 ) + 4, 000(1 + i4 ) + 109, 000

[2]

The expected value of this is:
E [ 4, 000(1 + i2 )(1 + i3 )(1 + i4 ) + 4, 000(1 + i3 )(1 + i4 ) + 4, 000(1 + i4 ) + 4, 000] + 105, 000
= 4, 000 { E [ (1 + i2 )(1 + i3 )(1 + i4 ) ] + E [ (1 + i3 )(1 + i4 ) ] + E [ (1 + i4 ) ] + 1} + 105, 000

[1]

Since the interest rates are independent, this becomes:

4, 000 { E [1 + i2 ] E [1 + i3 ] E [1 + i4 ] + E [1 + i3 ] E [1 + i4 ] + E [1 + i4 ] + 1} + 105, 000
= 4, 000 {1.055 × 1.06 × 1.045 + 1.06 × 1.045 + 1.045 + 1} + 105, 000
= 122, 285.294
So the mean accumulated amount is £122,285.

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[2]

© IFE: 2009 Examinations

Page 10

CT1: Q&A Bank Part 4 – Solutions

Solution 4.13
(i)

Here we need to find (after cancelling a factor of 1,000):
20

20

DMT = Â tvt

 vt

t =1

[1]

t =1

The numerator is ( Ia ) 20| = 63.9205 and the denominator is a20| = 8.5136 .
So the discounted mean term is 63.9205 8.5136 = 7.51 years .
(ii)

[1]

Here we need to find:
20

DMT = Â t (950 + 50t )v t =1

t

20

 (950 + 50t )vt

[1]

t =1

The numerator is (using the figure given at the end of the question):
20

950( Ia ) 20| + 50Â t 2vt = 950 ¥ 63.9205 + 50 ¥ 718.027 = 96, 626

[½ ]

t =1

and the denominator is:
950a20| + 50( Ia ) 20| = 950 ¥ 8.5136 + 50 ¥ 63.9205 = 11, 284
So the discounted mean term is 96, 626 11, 284 = 8.56 years .

© IFE: 2009 Examinations

[½ ]
[1]

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CT1: Q&A Bank Part 4 – Solutions

(iii)

Page 11

Here we need to find (after cancelling a factor of 1, 000 1.05 ):
20

20

DMT = Â tvt ¥ 1.05t

 vt ¥ 1.05t

t =1

[1]

t =1

The denominator is just a geometric series:
20

v ¥ 1.05 - v 21 ¥ 1.0521
= 12.7177
1 - v ¥ 1.05

 vt ¥ 1.05t = t =1

[1]

The numerator can be evaluated by multiplying by 1.1 1.05 and subtracting:
20

19

t =1

t =0

(1.1 1.05 - 1) Â tvt ¥ 1.05t = Â vt ¥ 1.05t - 20v20 ¥ 1.0520

[1]

The sum on the RHS can be calculated as a20| = 13.3233 at i = 1.1 1.05 - 1 (or as the sum of a geometric series). fi 20

 tvt ¥ 1.05t = t =1

13.3233 - 20v 20 ¥ 1.0520
= 114.143
0.047619

(An alternative method of evaluating the numerator is to calculate
( Ia ) 20| @ 4.7619% .)
So the discounted mean term is 114.143 12.7177 = 8.98 years .
(iv)

[1]

Here we need to find (after cancelling a factor of 1, 000 1.1 ):
20

20

DMT =  tv ¥ 1.1 t  vt ¥ 1.1t

t

t =1

[1]

t =1

But since i = 0.1 , this is just:
20

DMT = Â t t =1

20

1

Â1 = 2 ¥ 20 ¥ 21 20 = 10.5 years

[1]

t =1

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© IFE: 2009 Examinations

Page 12

CT1: Q&A Bank Part 4 – Solutions

Solution 4.14
(i)

The present value at interest rate i can be found by summing a geometric series:


PV (i ) = Â D(1 + j )t vt = t =1

Ê1+
D(1 + j )v
D(1 + j )
=
= DÁ
1 - (1 + j )v (1 + i ) - (1 + j )
Ëi-

jˆ j˜ ¯

[2]

The discounted mean term can be found from the relationship:
DMT = (1 + i) ¥ Volatility = (1 + i) ¥ -

PV ¢(i )
PV (i )

[1]

Using the formula just derived for the present value, this gives:

DMT = (1 + i ) ¥ D

(ii)

(1 + j )
(i - j )

2

D

(1 + j ) 1 + i
=
(i - j ) i - j

[2]

With the value given:

PV = 5, 000 ¥

1.03
= £103, 000
0.08 - 0.03

[1]

and:

DMT =

1.08
= 21.6 years
0.08 - 0.03

© IFE: 2009 Examinations

[1]

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CT1: Q&A Bank Part 4 – Solutions

Page 13

Solution 4.15
Purpose of models
Both types are used to project the accumulated value of flows of money.

[½ ]

Assumptions
Deterministic models assume that future rates of return are fixed. Stochastic models assume that future rates of return are random variables.
[½ ]

Results obtained
For a given assumed set of future rates of return, a deterministic model will give a single definite answer. For a given assumption about the statistical distribution of future rates of return, a stochastic model will give a statistical distribution describing a range of possible answers. [1]

Risk and uncertainty
Stochastic interest rate models make allowance for uncertainty by enabling the probability that the actual value will lie in a given range to be calculated. Deterministic models make allowance for uncertainty by carrying out calculations based on different sets of assumptions (eg by including contingency margins in the assumptions).
[1]

Solution 4.16
This question is taken from Subject 102, September 2004, Question 8.
(a)

The mean accumulated amount after n years of a single investment of X at time 0 is X (1 + j ) n , where j is the mean rate of interest in any year.
So, the expected value of the investment is:
1m × 1.0610 = £1.7908m

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[1]

© IFE: 2009 Examinations

Page 14

(b)

CT1: Q&A Bank Part 4 – Solutions

Let S n be the accumulated value at time n of an investment of 1 at time 0. If ik is the interest rate in year k , then: n S n = ∏ (1 + ik ) k =1

Since we are told that (1 + ik ) ~ log N ( µ , σ 2 ) , by using the properties of the lognormal distribution, we know that:
S n ~ log N (nµ , nσ 2 )

[1]

We need to find the values of the parameters, µ and σ 2 . Setting up simultaneous equations for the mean and variance of the lognormal distribution:

exp ⎡ µ + 1 σ 2 ⎤ = 1.06
2



(

(

)

and exp ⎡ 2µ + σ 2 ⎤ exp ⎡σ 2 ⎤ − 1 = 0.082


⎣ ⎦

)

⇒ 1.062 exp ⎡σ 2 ⎤ − 1 = 0.082
⎣ ⎦

⎛ 0.082 ⎞
⇒ σ 2 = ln ⎜
+ 1⎟ = 0.005680
⎜ 1.062




which gives us that µ = 0.05543 .

[2]

This means that S10 ~ log N (0.5543, 0.05680) .

[1]

We can now find the probability. We require:
P ( S10 < 0.9 × 1.7908 ) = P ( S10 < 1.6118 )

[1]

Let Z ~ N (0,1) , then: ln1.6118 − 0.5543 ⎞

P ( ln S10 < ln1.6118 ) = P ⎜ Z <

0.05680


= P( Z < −0.323)

= 1 − P( Z < 0.323)
= 1 − 0.627 = 0.373

© IFE: 2009 Examinations

[1]

[1]

The Actuarial Education Company

CT1: Q&A Bank Part 4 – Solutions

Page 15

Solution 4.17
This question is taken from Subject 102, September 2002, Question 5.

(i)

The equation of value is:
98 = 7v + 7v 2 + 112v3

[1]

This equation needs to be solved by interpolation. As a first guess we can expand the right-hand side of the equation using the binomial expansion:

98 = 7(1 + i ) −1 + 7(1 + i ) −2 + 112(1 + i ) −3
⇒ i

7(1 − i ) + 7(1 − 2i ) + 112(1 − 3i )

8%

Trying 8% in the equation of value, we get the left-hand side to be 101. At 9% the left-hand side is 98.798 and at 9.5% the left-hand side is 97.536.
[1]
Interpolating between 9% and 9.5% we get the gross redemption yield to be:

⎛ 98 − 98.798 ⎞ i = 9 + 0.5 ⎜
⎟ = 9.32%
⎝ 97.536 − 98.798 ⎠
(ii)

[1]

The equation of value for the 1-year spot rate is:

98 = 112 ×

1
1 + i1

Solving this gives a 1-year spot rate of 14.3%.

[1]

The equation of value for the 2-year spot rate is:
98 = 7 ×

1
1
+ 112 ×
1 + i1
(1 + i2 ) 2

Solving this gives a 2-year spot rate of 10.4%.

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[2]

© IFE: 2009 Examinations

Page 16

CT1: Q&A Bank Part 4 – Solutions

The equation of value for the 3-year spot rate is:
98 = 7 ×

1
1
1
+ 7×
+ 112 ×
2
1 + i1
(1 + i2 )
(1 + i3 )3

Solving this gives a 3-year spot rate of 9.15%.

[2]

Solution 4.18
This question is taken from Subject 102, April 2004, Question 5.
(i)

Using an interest rate of 7%, the present value of the liabilities is:
10v10 + 20v15 = 12.332

[½ ]

The present value of the assets is:
7.404v 2 + 31.834v 25 = 12.332

[½ ]

This means that the first condition is met: the present values of the assets and liabilities are the same.
[1]
The discounted mean term of the liabilities is:

10 ×10v10 + 15 × 20v15
= 12.9
12.332

[1]

The discounted mean term of the assets is:

2 × 7.404v 2 + 25 × 31.834v 25
= 12.9
12.332

[1]

This means that the second condition is met: the discounted mean terms of the assets and liabilities are the same.
[1]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Solutions

(ii)

Page 17

We need to calculate the present values using 7.5% interest:
Liabilities:
10v10 + 20v15 = 11.611

[½ ]

7.404v 2 + 31.834v 25 = 11.627

[½ ]

Assets:

The profit is:
PVA − PVL = £0.016 million
(iii)

[1]

Redington’s first two conditions are satisfied. Furthermore, we can see that the spread of the assets is greater than the spread of the liabilities. This implies that the convexity of the assets is greater than the convexity of the liabilities. So all of Redington’s conditions are satisfied, meaning the fund is immunised.
[1]
This means that if the interest rate moves slightly from the 7% assumed, the fund moves into profit.
[1]

Solution 4.19
This question is taken from Subject 102, April 2002, Question 8.
(i)

The payments for an increasing annuity are as follows:
1
0

2

3

n – 1

n

payment

1

2

3

n –1

n

time

So the present value for these payments, ( Ia ) n , is:
( Ia ) n| = v + 2v 2 + 3v3 +

+ nv n

(1)

[1]

(2)

[1]

Multiplying equation (1) by (1 + i ) gives:
(1 + i )( Ia ) n| = 1 + 2v + 3v 2 +

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+ nv n−1

© IFE: 2009 Examinations

Page 18

CT1: Q&A Bank Part 4 – Solutions

Subtracting equation (1) from equation (2), we obtain:
+ v n −1 − nv n

i ( Ia ) n| = 1 + v + v 2 + v3 +
= an| − nv n
So:

( Ia ) n| =
(ii)

an| − nv n

[1]

i

The timeline below shows when cashflows are received from this bond:

5

5

0

5

1

5

100
5

5

2

cashflows

10

time

Now, to calculate the duration, we use the formula:

DMT (duration) =



t × PV

all cashflows



PV

all cashflows

DMT =

½ × 5v½ + 1× 5v + 1½ × 5v1½ +

+ 10 × 5v10 + 10 × 100v10

10a (2) + 100v10
10

Working in half-years:

DMT =

=

1× 5v + 2 × 5v 2 + 3 × 5v3 +

+ 20 × 5v 20 + 20 ×100v 20

5a20 + 100v 20
5( Ia ) 20 + 2000v 20

© IFE: 2009 Examinations

5a20 + 100v 20

[2]

[1]

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CT1: Q&A Bank Part 4 – Solutions

Page 19

The effective half-yearly interest rate is

a20 =
( Ia ) 20 a20 =

1 − 1.03−20
1 − 1.03−1

i (2)
= 3% .
2

[1]

= 15.324

15.324 − 20 × 1.03−20
=
= 141.676
0.03
1 − 1.03−20
= 14.877
0.03

So:

DMT =

708.381 + 1107.352 1815.732
=
= 13.994
74.387 + 55.368
129.755

[3]

As we’re working in half years we need to convert back to full years. So, the
13.994
duration of the bond is
[1]
= 7.00 years .
2
(iii)

The duration of 7 years is so late because of the big cashflow of £105 at time 10 compared to the smaller £5 cashflows made elsewhere. If the cashflows elsewhere are reduced, this will make the effect of that payment at time 10 all the more pronounced as proportionally less of the total payments are made earlier. Hence, the duration (ie the average time of the cashflows weighted by the present values) will be longer.
[2]

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© IFE: 2009 Examinations

Page 20

CT1: Q&A Bank Part 4 – Solutions

Solution 4.20
In order for this to be a sensible probability distribution, all the probabilities must be
[1]
between 0 and 1. So k must lie in the range 0 £ k £ ¼ .
The mean growth rate is:

E (i ) = 3k ¥ 0.05 + (1 - 4k ) ¥ 0.075 + k ¥ 0.125 = 0.075 - 0.025k

[1]

So the minimum value (corresponding to k = ¼ ) is 6.875% and the maximum value
(corresponding to k = 0 ) is 7.5%.
[1]
The variance of the growth rate is: var(i ) = E (i 2 ) - [ E (i )]2

= 3k ¥ 0.052 + (1 - 4k ) ¥ 0.0752 + k ¥ 0.1252 - (0.075 - 0.025k ) 2

[1]

= 0.004375k - 0.000625k 2
By
calculus, the only turning point of this function is at k = 0.004375 /(2 ¥ 0.000625) = 3.5 which is outside the permissible range of values of k . So this function must be monotonic over the range of interest. The minimum standard deviation (corresponding to k = 0 ) is 0% and the maximum standard deviation
(corresponding to k = ¼ ) is 3.25%.
[1]

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Solutions

Page 21

Solution 4.21
To find the mean of A3 , the accumulated amount at the end of the third year, we need to use the recursive relationships:
E ( A0 ) = 0
E ( Ak ) = (1 + jk )[ Pk + E ( Ak -1 )]

( k = 1, 2,3 )

[1]

where Pk denotes the lump sum invested at the start of year k and jk is the mean rate of return for year k .
This gives:
E ( A1 ) = 1.08(50, 000 + 0) = 54, 000

E ( A2 ) = 1.07(30, 000 + 54, 000) = 89,880
E ( A3 ) = 1.06(20, 000 + 89,880) = 116, 473

[2]

To find the variance of A3 , we need to use the recursive relationships:
2
E ( A0 ) = 0
2
2
2
E ( Ak ) = [(1 + jk ) 2 + sk ] [ Pk2 + 2 Pk E ( Ak -1 ) + E ( Ak -1 )]

(k = 1, 2,3)

[1]

where sk is the standard deviation of the rate of return for year k .
This gives:
2
E ( A1 ) = (1.082 + 0.022 )[50, 0002 + 2(50, 000)(0) + 0] = 2,917, 000, 000
2
E ( A2 ) = (1.07 2 + 0.032 )[30, 0002 + 2(30, 000)(54, 000) + 2,917, 000, 000]

= 8, 085,910, 600
2
E ( A3 ) = (1.062 + 0.042 )[20, 0002 + 2(20, 000)(89,880) + 8, 085,910, 600]

= 13,593, 665, 650

[2]

We then find that:
2
var( A3 ) = E ( A3 ) - [ E ( A3 )]2 = 13,593, 665, 650 - 116, 4732 = (£5, 264)2

[1]

So the mean of the accumulated amount is £116,473 and the standard deviation is
£5,264.
[1]

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© IFE: 2009 Examinations

Page 22

CT1: Q&A Bank Part 4 – Solutions

Solution 4.22
(i)

The spot rates are: s1 = 0.048

s2 = 0.046

s3 = 0.044

[1]

Let ft denote the one-year forward rate applicable at time t . Then:
(1 + s2 ) 2
1.0462
fi f1 =
- 1 = 0.04400
1 + f1 =
1 + s1
1.048
1 + f2 =
(ii)

(1 + s3 )3
(1 + s2 ) 2

fi f1 =

1.0462

- 1 = 0.04001

[1]

The fair price for £100 nominal of the bond is:
P = 6a3 + 100v3 =

(iii)

1.0443

[1]

6
6
106
+
+
= £104.36
2
1.048 1.046
1.0443

[1]

The gross redemption yield is the constant annual interest rate that satisfies the equation 104.36 = 6a3 + 100v3 . i = 4% fi RHS = 105.55

[1] i = 4.5% fi RHS = 104.12

[1]

Interpolating gives:

Ê 104.36 - 105.55 ˆ i ª 4 + 0.5 Á
= 4.4%
Ë 104.12 - 105.55 ˜
¯
(iv)

[1]

The par yield is the annual coupon g that satisfies the equation of value, assuming the given spot rates:
1
1 ˆ
100
Ê 1
100 = g Á
+
+
+
2

Ë 1.048 1.046
1.044 ¯ 1.0443
Solving this gives g = 4.41%

© IFE: 2009 Examinations

[2]

[1]

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CT1: Q&A Bank Part 4 – Solutions

Page 23

Solution 4.23
(i)

Let S n be the accumulated value, after n years, of a single investment of 1 at time 0. Let it be the rate of interest earned in the t th year. Then:

E [ Sn ] = E [(1 + i1 )(1 + i2 )… (1 + in )]
= E [1 + i1 ] E [1 + i2 ]… E [1 + in ]
= (1 + j ) n

[1]

2
2
2
2
E È Sn ˘ = E È(1 + i1 ) (1 + i2 ) … (1 + in ) ˘
Í
˙
Î ˚
Î
˚
2
2
2
= E È(1 + i1 ) ˘ E È(1 + i2 ) ˘ … E È (1 + in ) ˘
Í
˙ Í
˙
Í
˙
Î
˚ Î
˚
Î
˚

[1]

Now, since E[ X 2 ] = var[ X ] + E 2 [ X ] :
E[(1 + ik ) 2 ] = E 2 [1 + ik ] + var[1 + ik ] = (1 + E[ik ]) 2 + var[ik ] = (1 + j ) 2 + s 2
[1]
So:
2
E È Sn ˘ = È (1 + j ) 2 + s 2 ˘
Î ˚ Î
˚

n

[1]

Hence:

var [ Sn ] = È (1 + j ) 2 + s 2 ˘ - (1 + j )2n
Î
˚ n The Actuarial Education Company

[1]

© IFE: 2009 Examinations

Page 24

CT1: Q&A Bank Part 4 – Solutions

(ii)(a) Now:
S n = (1 + i1 )(1 + i2 )… (1 + in )
So:

ln Sn = ln È(1 + i1 )(1 + i2 )… (1 + in )˘
Î
˚
= ln (1 + i1 ) + ln (1 + i2 ) +

+ ln (1 + in )

[1]

Since ln (1 + it ) ~ N (0.04, 0.09) using the additive property of independent normal distributions means that ln S n ~ N (0.04n, 0.09n) .

[2]

The probability of at least doubling is:
P ( S n ≥ 2) = 0.75 fi P ( Sn < 2) = 0.25

[1]

Standardising: z= ln 2 - 0.04n
= -0.6745
0.09n

[1]

Rearranging:

ln 2 - 0.04n = -0.6745 ¥ 0.3 n
0.04n - 0.2024 n - ln 2 = 0

[1]

(ii)(b) Letting x = n , the equation becomes:
0.04 x 2 - 0.2024 x - ln 2 = 0
Solving the quadratic equation:

0.2024 ± 0.20242 + 0.16 ln 2 x= = -2.3413, 7.4013
0.08

[1]

n = 7.40132 = 54.8 years

[1]

So:

© IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Solutions

Page 25

Solution 4.24
This question is taken from Subject 102, September 2002, Question 10.
The timeline for the liabilities is:
3,000

(i)

7,000

9,000

11,000

Payment

5

0

5,000

10

15

20

25

Time (yrs)

The present value of the liabilities is:
3, 000v5 + 5, 000v10 + 7, 000v15 + 9, 000v 20 + 11, 000v 25
= 2,138.9585 + 2,541.7465 + 2,537.1221 + 2,325.7710 + 2, 026.7410
= 11,570.3391

(ii)

[1]
[1]
[1]

The discounted mean term is given by:
15, 000v5 + 50, 000v10 + 105, 000v15 + 180, 000v 20 + 275, 000v 25
[1]
11,570.3391
10, 694.7927 + 25, 417.4646 + 38, 056.8321 + 46,515.4205 + 50, 668.5238
=
11,570.3391

171,353.0337
11,570.3391
= 14.810
=

(iii)

[2]

Assuming that A nominal of stock A and B nominal of stock B is purchased, the present value of the assets is:
Av 26 + 0.05 Aa26 + Bv32 + 0.04 Ba32 = 0.7634844 A + 0.6206033B

[1]

Since the present values of the liabilities and assets are the same, we have:

0.7634844 A + 0.6206033B = 11,570.3391

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Equation (1)
[1]

© IFE: 2009 Examinations

Page 26

CT1: Q&A Bank Part 4 – Solutions

To calculate the discounted mean term, we need the sum of the product of the time of each cashflow with the present value of each cashflow. For an annuity payable annually in arrears, this is: v + 2v 2 + 3v3 +

+ nv n = ( Ia ) n

The present values of the liabilities and the assets are the same. The discounted mean term of the assets is then:
26 Av 26 + 0.05 A( Ia )26 + 32 Bv32 + 0.04 B ( Ia)32
11,570.3391

[2]

Now:

( Ia) 26 = 116.807148
( Ia)32 = 140.858544
So the discounted mean term is:
10.31744 A + 9.306058 B
11,570.3391

[1]

Since this must be equal to the discounted mean term of the liabilities, we have:

10.31744 A + 9.306058B = 14.810 ×11,570.3391
⇒ 10.31744 A + 9.306058B = 171,353.0337

Equation (2)
[1]
Rearranging Equation (1) and substituting it into Equation (2), we get:

10.31744(15,154.65 − 0.812857 B ) + 9.306058B = 171,353.0337
⇒ 156,357 − 8.38660 B + 9.306058 B = 171,353.0337
14,995.8
⇒ B=
= 16,309
0.91946
Substituting this value back into Equation (1), we get A = 1,897 .

[2]
[1]

So £1,897 nominal of Stock A and £16,309 nominal of Stock B should be purchased. © IFE: 2009 Examinations

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CT1: Q&A Bank Part 4 – Solutions

Page 27

Solution 4.25
(i)

The annual growth factor 1 + i has a log N ( m , s 2 ) distribution.
So, using the formulae for the mean and variance of a lognormal distribution:

E (1 + i ) = e m +½s = e0.05+½(0.01) = 1.0565
2

[1]

and: var(1 + i ) = e 2 m +s

2

(e - 1) = e s2 2(0.05) + 0.01

(e

0.01

)

- 1 = 0.01122

[1]

The mean and variance of the annual rate of return are:

E (i ) = E (1 + i ) - 1 = 0.0565 and: var(i ) = var(1 + i ) = 0.01122 = 0.1062

[1]

The mean and standard deviation of the annual return are 5.65% and 10.6%.
(ii)

Since the annual growth factors have a lognormal distribution:

ln(1 + i ) - 0.05
~ N (0,1)
0.1
The median ( im ) and the upper and lower quartiles ( iu and il ) correspond to the
50%, 25% and 75% points of the standard normal distribution:

ln(1 + im ) - 0.05
=0

0.1
ln(1 + iu ) - 0.05
= 0.6745
0.1
ln(1 + il ) - 0.05
= -0.6745
0.1

[1]

im = e0.05 - 1 = 0.051

[1]



[1]



iu = e0.11745 - 1 = 0.125 il = e -0.01745 - 1 = -0.017

[1]

So the median and upper and lower quartiles are 5.1%, 12.5% and –1.7%.

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Page 28

(iii)

CT1: Q&A Bank Part 4 – Solutions

Using the formula given, the mode of the distribution of 1 + i is at:
1 + i = e m -s = e0.05-0.01 = 1.0408
2



i = 4.1%

[1]

The mean is greater than the median, which is greater than the mode. (This is true for a positively skewed distribution. The lognormal distribution is a positively skewed distribution, so this should not be a surprise).
[1]
(iv)

The probability that a payment of £7,250 will be sufficient to meet the liability is:
10, 000 ˆ
Ê
P (7, 250 S5 > 10, 000) = P Á S5 >
= P( S5 > 1.379)
Ë
¯
7, 250 ˜

[1]

where S5 is the accumulation factor for the 5-year period and has a log N (5 m ,5s 2 ) distribution.
This probability can be evaluated using the normal distribution:
Ê ln S5 - 5 m ln1.379 - 5 ¥ 0.05 ˆ
P ( S5 > 1.379) = P Á
>
˜
Ë
¯
5s
5 ¥ 0.1
= 1 - F(0.320)
= 1 - 0.626
= 0.374
(v)

[2]

If the interest rate was constant in all future years, the variance in the accumulation factor would be increased. In fact, under the fixed rate model, S5 would have a log N (5 m ,52s 2 ) distribution. So the probability would become:
Ê ln S5 - 5 m ln1.379 - 5 ¥ 0.05 ˆ
>
P ( S5 > 1.379) = P Á
˜
Ë
¯
5s
5 ¥ 0.1
= 1 - F(0.143)
= 1 - 0.557
= 0.443
The probability is greater since, if the variance is higher, extreme values are more likely. [3]

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CT1: Q&A Bank Part 4 – Solutions

Page 29

Solution 4.26
Let X be the amount invested at time 0, and let ik denote the yield obtained in year k ,

k = 1, 2,...,10 . The accumulated value of the fund after 10 years is then:
X (1 + i1 )(1 + i2 ) ... (1 + i10 ) = XS10
This will be at least £500,000 if and only if:

S10 ≥

500, 000
X

[1]

Since the yields are independent, it follows that:

S10 ~ log N (0.7, 0.06)

[1]

Hence:
500, 000 ˘
È
P Í S10 ≥
˙
X
Î
˚
È
Ê 500, 000 ˆ ˘
= P Í ln S10 ≥ ln Á
˜
Ë
X ¯˙
Î
˚
È
˘
Ê 500, 000 ˆ ln Á
˜ - 0.7 ˙
Í
Ë
X ¯
˙
= P ÍZ ≥
0.06
Í
˙
Í
˙
Î
˚

Since this probability is 0.99,

[2]

Ê 500, 000 ˆ ln Á
˜ - 0.7
Ë
X ¯
0.06

must be the lower 1% point of the

standard normal distribution, ie: ln 500, 000 - ln X - 0.7
0.06

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= -2.3263

[1]

© IFE: 2009 Examinations

Page 30

CT1: Q&A Bank Part 4 – Solutions

Solving for X gives:

ln X = ln 500, 000 - 0.7 + 2.3263 0.06 = 12.992188 fi X = 438,970.80
So the amount to be invested (rounded to the nearest £1) is £438,971.

[2]

Solution 4.27
This question is taken from Subject 102, September 2003, Question 12.
(i)(a) Expectations theory states that the shape of the yield curve can be explained by the relative attractiveness of short and long-term investments, and how this changes according to expectations of future movements in interest rates.
[1]
An expectation of a fall in interest rates will make short-term investments less attractive and longer-term investments more attractive. This will make yields on short-term investments rise, and yields on long-term investments fall. An expectation of a rise in interest rates will have the opposite effect.
[1]
(i)(b) The liquidity preference theory states that investors prefer short-term investments to long-term investments. (This is because longer-term bonds are more sensitive to interest rate movements than short-term bonds. Investors require compensation for this extra risk.) This preference for short-term investments helps to increase the yield on longer-term bonds.
[2]
The market segmentation theory says that bonds of different terms are attractive to different types of investors. For example, banks prefer very short-term investments, as their liabilities are very short-term. Pension funds have liabilities that are very long-term. So the longest-term bonds are attractive investments for pension funds. The demand for bonds will therefore differ for bonds of different terms. Bonds that are less attractive overall will have higher yields than those that are more attractive.
[2]

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CT1: Q&A Bank Part 4 – Solutions

Page 31

(ii)(a) For a 1-year zero-coupon bond, the GRY is 10%.

[1]

For a 3-year zero-coupon bond, the GRY, s3 , satisfies the equation:

(1 + s3 )3 = 1.1×1.09 ×1.08 = 1.29492
So:

s3 = (1.29492 )

1/ 3

− 1 = 8.997%

[1]

For a 5-year zero-coupon bond, the GRY, s5 , satisfies the equation:

(1 + s5 )5 = 1.1×1.09 ×1.08 ×1.072 = 1.48255
So:

s5 = (1.48255 )

1/ 5

− 1 = 8.194%

[1]

For a 10-year zero-coupon bond, the GRY, s10 , satisfies the equation:

(1 + s10 )10 = 1.1×1.09 ×1.08 ×1.077 = 2.07936
So:

s10 = ( 2.07936 )

1/10

− 1 = 7.595%

[1]

(ii)(b) The calculations in (ii)(a) show that the yield curve is downward sloping. As the term of the bond tends to infinity, the yield will tend to 7%. So the graph has an asymptote at 7%.
Yield

7%

Term to maturity

[2]

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Page 32

CT1: Q&A Bank Part 4 – Solutions

(ii)(c) The gross redemption yield of a coupon-paying bond is a weighted average of the yields obtained on the interest and redemption payments. As the interest payments are received earlier than the redemption payment, the yield achieved on this part of the return will be higher (under a downward-sloping yield curve).
The net result is that the yield curve will go down at a lower rate for couponpaying bonds than for zero-coupon bonds (where the return comes as a single redemption payment).
[2]

Solution 4.28
This question is taken from Subject A1, April 1998, Question 15.
(i)

The present value of the liabilities is:
(2)
PVL = 20, 000v15 + 5, 000v10 a25|

[1]

@ 7%

= 20, 000 × 1.07 −15 + 5, 000 × 1.07 −10 ×

1 − 1.07 −25
2(1 − 1.07 −½ )

= £38, 416

[1]

The present value of the amount invested in Bond X is:
PVX = 25, 000v10 = £12, 709 @ 7%

[1]

For immunisation, the present value of the liabilities must be equal to the present value of the assets.
[1]
So the amount invested in Bond Y (ie the present value) is:

38, 416 − 12, 709 = £25, 707
(ii)

[1]

If the liabilities are to be immunised then the discounted mean term of the assets must equal the discounted mean term of the liabilities.
[1]

DMTL =

15 × 20, 000v15 + 2,500v10 (10 + 10.5v½ + 11v +
PVL

+ 34.5v 24½ )

@ 7%
[2]

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CT1: Q&A Bank Part 4 – Solutions

Page 33

Working in half-yearly time periods, the expression in brackets can be evaluated as: 9.5a50| + 0.5( Ia )50|

@ 3.4408043%

[2]

= 9.5 × 24.5239 + 0.5 × 460.307
= 463.1306

So:

DMTL =

[1]

108, 733.8 + 1, 270.873 × 463.1306 697,314
=
PVL
PVL

[1]

If n is the term of Bond Y then:

DMTA =

12, 708.7 × 10 + 25, 707 n 127, 087 + 25, 707 n
=
PVA
PVA

[1]

Since PVA = PVL and DMTA = DMTL , then:

127, 087 + 25, 707 n = 697,314

[1]

⇒ n = 22.182 years
Thus, the redemption proceeds are: 25, 707 ×1.07 22.182 = £115,302
(iii)

[1]

The third condition for immunisation is that the convexity of the assets must be greater than the convexity of the liabilities where:
PV ′′ n
= ∑ Ck tk (tk + 1)vtk + 2
Convexity =
PV k =1

where Ck is the cashflow at time tk .

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n

∑ Ck v t

k

k =1

[2]

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CT1: Q&A Bank Part 5 – Revision Questions

Page 1

Part 5 – Revision Questions
This part contains 100 marks of questions testing the material from the whole course.
You may like to try these questions under exam conditions as a mock exam.

Question 5.1
List the reasons why the running yield from property investments will normally be higher than that for ordinary shares.
[2]

Question 5.2
Show whether or not:

Ê i iˆ a ( |p ) = Á ( p ) + ˜ an| n p¯
Ëi

[2]

Question 5.3
State the main features of certificates of deposit.

[3]

Question 5.4
The 1, 2, and 3-year spot rates are 3.5%, 4% and 3.7% respectively. The 2-year forward rate from time 3 is 5% and the 1-year forward rate from time 4 is 4.9%. Calculate:
(i)

the 3-year forward rate from time 1

(ii)

the present value at time 0 of payments of £100 at times 3, 4 and 5.

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[2]
[2]
[Total 4]

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CT1: Q&A Bank Part 5 – Revision Questions

Question 5.5
A loan of £10,000 is repaid over a five-year period by level monthly repayments in arrears of £250. Calculate:
(i)

the flat rate of interest per annum

(ii)

[2]

the APR on the transaction.

[2]
[Total 4]

Question 5.6
An investor purchases a bond, redeemable at par, which pays half-yearly coupons at a rate of 8% per annum. There are 8 days until the next coupon payment and the bond is ex-dividend. The bond has 7 years to maturity after the next coupon payment.
Calculate the purchase price to provide a yield to maturity of 6% per annum effective.
[4]

Question 5.7
A continuous cashflow is to be paid at a rate r (t ) = 10 + 2t , 0 £ t £ 10 . T