Premium Essay


In: Business and Management

Submitted By maxankit
Words 2749
Pages 11
California Debt and Investment Advisory Commission

Duration Basics
Duration is a term used by fixed-income investors, financial advisors, and investment advisors. It is an important measure for investors to consider, as bonds with higher durations (given equal credit, inflation and reinvestment risk) may have greater price volatility than bonds with lower durations. It is an important tool in structuring and managing a fixed-income portfolio based on selected investment objectives. Investment theory tells us that the value of a fixed-income investment is the sum of all of its cash flows discounted at an interest rate that reflects the inherent investment risk. In addition, due to the time value of money, it assumes that cash flows returned earlier are worth more than cash flows returned later. In its most basic form, duration measures the weighted average of the present value of the cash flows of a fixed-income investment. All of the components of a bond—price, coupon, maturity, and interest rates—are used in the calculation of its duration. Although a bond’s price is dependent on many variables apart from duration, duration can be used to determine how the bond’s price may react to changes in interest rates. This issue brief will provide the following information:

< A basic overview of bond math and the components of a bond that will affect its volatility.

< The different types of duration and how they are calculated. < Why duration is an important measure when comparing individual bonds and constructing bond portfolios.

< An explanation of the concept of convexity and how it is used in conjunction with the duration measure.

January 2007

issue brief

Basic Bond Math and Risk Measurement
The price of a bond, or any fixed-income investment, is determined by summing the cash flows discounted by a rate of return. The rate of

Similar Documents

Premium Essay

Convexity & Duration

...the yield increases by 1 basis point. the price of the bond will decrease to 99.95. If the yield decreases by 1 basis point. the price of the bond will increase to 100.04. What is the modified duration of the bond? a) 5.0 b) -5.0 c) 4.5 d) -4.5 Example 1-6: FRl\1 Exam 1998--Question 22 What is the price impact of a 10-basis-point increase in yield on a 10-year par bond with a modified duration of 7 and convexity of 50? a) -0.705 b) -0.700 c) -0.698 d) -0.690 Example 1-8: FRl\1 Exam 1998--Question 20 Coupon curve duration is a useful method for estimating duration from market prices of a mortgage-backed security (MBS). Assume the coupon curve of prices for Ginnie Maes in June 2001 is as follows: 6% at 92. 7% at 94. and 8% at 96.5. What is the estimated duration of the 7s? a) 2.45 b) 2.40 c) 2.33 d) 2.25 Example 1-9: FRl\'1 Exam 1998--Question 21 Coupon curve duration is a useful method for estimating convexity from market prices of an MBS. Assume the coupon curve of prices for Ginnie Maes in June 2001 is as follows: 6% at 92. 7% at 94. and 8% at 96.5. What is the estimated convexity of the 7s? a) 53 b)26 c) 13 d) -53 Example 1-10: FRM Exam 2001-Question 71 Calculate the modified duration of a bond with a Macauley duration of 13.083 years. Assume market interest rates are 11.5% and the coupon on the bond is paid semiannually. a) 13.083 b) 12.732 c) 12.459 d) 12.371 Example I-II: FRl\'1 Exam 2002-Question...

Words: 1001 - Pages: 5

Premium Essay


...sensitivity Duration Cash-flow matching Duration matching: immunization Convexity Prof. Lasse H. Pedersen 2 Interest-Rate Sensitivity First order effect: Bond prices and yields are negatively related Maturity matters: Prices of long-term bonds are more sensitive to interest-rate changes than short-term bonds Convexity: An increase in a bond’s YTM results in a smaller price decline than the price gain associated with a decrease of equal magnitude in the YTM. Prof. Lasse H. Pedersen 3 Duration The duration (D) of a bond with cashflows c(t) is defined as minus the elasticity of its price (P) with respect to 1 plus its yield (y): dP 1 + y T c(t ) D=− = ∑ f (t ) t , where f (t ) = dy P (1 + y ) t P 1 We see that the duration is equal to the average of the cash-flow times t weighted by f(t), the fraction of the present value of the bond that comes from c(t) ! The relative price-response to a yield change is therefore: ∆P ∆y D modified P ≅ −D 1+ y =− 1+ y { ∆y = − D ∆y modified duration Prof. Lasse H. Pedersen 4 Example: Duration of a Coupon Bond What is the duration of a 3-year coupon bond with a coupon rate of 8% and a YTM of 10% ? If the YTM changes to 10.1%, what would be the (relative) change in price ? If the YTM changes to 11%, what would be the (relative) change in price ? Prof. Lasse H. Pedersen 5 Duration Facts The duration of a portfolio is the weighted average of the durations: D p...

Words: 871 - Pages: 4

Premium Essay

Project Management

...In evaluating the schedule for the project of house flipping for a profit it is determined that the resources necessary for competition are to outsource utilizing local contractors who have been chosen through a competitive bid process. Quotes were sent to area contractors who specialized in the task needed to complete the various aspects of the project. Once the quotes were submitted, the contractors were selected based upon the lowest responsible bid. This process gives all area contractors the opportunity to bid on work and also benefits the home owner by ensuring that they are receiving the best possible price. After the winning contractors were selected, they were then notified as to the outcome and presented with a contractual agreement outlining the specific job and the expectations. After the contractors have been awarded the jobs, the schedule is then set up. The tasks are entered into Microsoft Project and then broke out into the subtasks such as siding, painting, and removing shingles. The tasks are assigned resources which include the specific contractors that will be performing the work. When evaluating the schedule, it is determined that because each contractor is responsible for various areas of the project, there will be an issues with allocation. What this means is that while the contractors who were hired to complete the exterior work can tackle the jobs such as siding and roofing, the contractors who were hired to complete the interior...

Words: 1131 - Pages: 5

Free Essay

Duration of Benefits

...analysis of the relationship between the marital status of an employee and the period of time taken off from work due to injuries sustained at the workplace Abstract The primary aim of this paper is to analyze the statistical relationship between the marital status of an employee and the period of time said employee takes off from work due to injuries sustained at the workplace. The analysis will be conducted on the basis of data consisting of 7,150 observations and 13 variables. This paper will aim to observe as many factors which have bearing on the duration of benefits, as is reasonably possible, with a specific focus on the role of the marital status of an employee. Such an endeavor will necessitate the observation of a variety of aspects consisting of emotional, physical and sexual factors. The overarching aim of our analysis is to draw the attention of employers towards the different factors which impact durat (the time duration of the provision of benefits) and pique the interest of other researchers to conduct further studies on the issue we have raised in our current undertaking. Introduction The primary assumption of this paper shall be that married individuals have greater tendency to prolong the time they take off due to injuries, as compared to unmarried individuals, because their spouses are also members of the active workforce. Accordingly, married individuals can afford to take more time off because they are not the sole breadwinners of their household. Such an...

Words: 2905 - Pages: 12

Premium Essay


...factor affecting asset and liability values is interest rate changes. If interest rates increase, the value of both loans (assets) and deposits and debt (liabilities) fall. If assets and liabilities are held until maturity, it does not affect the book valuation of the FI. However, if deposits or loans have to be refinanced, then market value accounting presents a better picture of the condition of the FI. The process by which changes in the economic value of assets and liabilities are accounted is called marking to market. The changes can be beneficial as well as detrimental to the total economic health of the FI. 6. Consider three Treasury bonds each of which has a 10 percent semiannual coupon and trades at par. a. Calculate the duration for a bond that has a maturity of four years, three years, and two years? Four-year Treasury Bond: Par value = $1,000 Coupon rate = 10% Semiannual payments R = 10% CF t 0.5 $50 1 $50 1.5 $50 2 $50 2.5...

Words: 5086 - Pages: 21

Premium Essay

Finance Notes and Problems - Managing Bond Portfolios

...Bond Portfolios 1. Duration can be thought of as a weighted average of the ‘maturities’ of the cash flows paid to holders of the perpetuity, where the weight for each cash flow is equal to the present value of that cash flow divided by the total present value of all cash flows. For cash flows in the distant future, present value approaches zero (i.e., the weight becomes very small) so that these distant cash flows have little impact, and eventually, virtually no impact on the weighted average. 2. A low coupon, long maturity bond will have the highest duration and will, therefore, produce the largest price change when interest rates change. 3. An intermarket spread swap should work. The trade would be to long the corporate bonds and short the treasuries. A relative gain will be realized when the rate spreads return to normal. 4. Change in Price = – (Modified Duration Change in YTM) Price = -Macaulay's Duration1+ YTM Change in YTM Price Given the current bond price is $1,050, yield to maturity is 6%, and the increase in YTM and new price, we can calculate D: $1,025 – $1,050 = – Macaulay's Duration1+ 0.06 0.0025 $1,050 D = 10.0952 5. d. None of the above. 6. The increase will be larger than the decrease in price. 7. While it is true that short-term rates are more volatile than long-term rates, the longer duration of the longer-term bonds makes their rates of return more volatile. The higher duration magnifies the sensitivity...

Words: 4938 - Pages: 20

Premium Essay


...Chapter Nine Interest Rate Risk II Chapter Outline Introduction Duration: A Simple Introduction A General Formula for Duration • The Duration of Interest Bearing Bonds • The Duration of a Zero-Coupon Bond • The Duration of a Consol Bond (Perpetuities) Features of Duration • Duration and Maturity • Duration and Yield • Duration and Coupon Interest The Economic Meaning of Duration • Semiannual Coupon Bonds Duration and Interest Rate Risk • Duration and Interest Rate Risk Management on a Single Security • Duration and Interest Rate Risk Management on the Whole Balance Sheet of an FI Immunization and Regulatory Considerations Difficulties in Applying the Duration Model • Duration Matching can be Costly • Immunization is a Dynamic Problem • Large Interest Rate Changes and Convexity Summary Appendix 9A: The Basics of Bond Valuation Appendix 9B: Incorporating Convexity into the Duration Model • The Problem of the Flat Term Structure • The Problem of Default Risk • Floating-Rate Loans and Bonds • Demand Deposits and Passbook Savings • Mortgages and Mortgage-Backed Securities • Futures, Options, Swaps, Caps, and Other Contingent Claims Solutions for End-of-Chapter Questions and Problems: Chapter Nine ***signed to the questions 2 3 16 20 1. What is the difference between book value accounting and market value accounting? How do interest rate changes...

Words: 11605 - Pages: 47

Premium Essay


...LECTURE 10: DURATION BONDS III FIN300 (Matt Marcinkowski, Fall '13) DURATION • Consider two bonds with 10 years to maturity and $1,000 face value (assume annual coupons/compounding): • Bond A: Coupon rate = 10% • Bond B: Coupon rate = 0% (discount paper) Yield Bond 8% A B $1,134.20 (+13.4%) $463.19 (+20%) 10% $1,000 $385.54 12% 887.00 (-11.3%) $321.97 (-16.5%) FIN300 (Matt Marcinkowski, Fall '13) DURATION • Now, consider two bonds with 10 percent coupon rate and $1,000 face value (assume annual coupons/compounding): • Bond C: Time to maturity = 5 years • Bond D: Time to maturity = 25 years Yield Bond 8% C D $1,079.85 (+8%) $1,213.50 (+21.4%) 10% $1,000 $1,000 12% $927.90 (-7.2%) $843.14 (-15.7%) FIN300 (Matt Marcinkowski, Fall '13) DURATION • We have observed the following: • The price of Bond A is less sensitive (in relative terms) to interest rate changes than the price of Bond B. • This is due to the fact that Bond A has a higher coupon rate (10%) than Bond B (0%) • We have also observed that: • The price of Bond C is less sensitive (in relative terms) to interest rate changes than the price of Bond D. • This is due to the fact that Bond D has a longer time to maturity than Bond C. FIN300 (Matt Marcinkowski, Fall '13) DURATION • Bond prices are more sensitive in relative terms to interest rate changes if the coupon rate is lower and if the time to maturity is longer. • To compare interest rate sensitivity of bonds with different coupon...

Words: 1459 - Pages: 6

Free Essay

Calculate Method

...CFA一级培训项目 CFA 级培训项目 前导课程 汤震宇 金程教育首席培训师 Ph.D CFA FRM CTP CAIA CMA RFP 地点: ■ 上海 □北京 □深圳 汤震宇  工作职称:博士, 金程教育首席培训师、上海交通大学继续教育学院客座教授、综合开发研究院 (中国·深圳)培训中心副教授,南京大学中国机构投资者研究中心专家、CFA(注册金融分析 师)、FRM(金融风险管理师)、CTP(国际财资管理师)、CAIA(另类投资分析师)、CMA(美 国管理会计师)、RFP(注册财务策划师)、CISI会员(英国特许证券与投资协会会员)  教育背景:中国人民大学投资系学士,复旦大学国际金融系硕士毕业,复旦大学管理学院博士  工作背景:“中国CFA第一人”,国内授课时间最长、人气最高、口碑最好的CFA金牌教师。十余 年CFA授课经验,为金程教育讲授CFA一级达二百多个班次、CFA二级六十多班次,CFA三级十个班 次,深受学员的欢迎和赞誉。行业经验丰富,先后供职于大型企业财务公司从事投资项目评估工 作, 参与成立证券营业部并任部门经理;任职于某民营公司,参与海外融资和资金管理工作。  服务客户:上海证券交易所、深圳综合开发研究院、山东省银行同业协会、对外经济贸易大学、 摩根士丹利、中国银行总行、广发证券、中国建设银行、中国工商银行总行、交通银行、招商银 行、农业银行、上海银行、太平洋保险、平安证券、富国基金等。  主编出版:《固定收益证券定价理论》、《财务报表分析技术》、《公司财务》、《衍生产品定 价理论》、《商业银行管理学》多本金融教材,备受金融学习者与从业人员好评。  新浪微博:汤震宇CFA_金程教育  联系方式: 电话:021-33926711 2-156 邮箱 100% Contribution Breeds Professionalism 前导课程大纲  CFA一级框架结构 金 金程服务平台及百题分析报告 务 台 析  计算器使用  财务前导 3-156 100% Contribution Breeds Professionalism CFA 考试知识点及其比重 TOPIC AREA LEVELⅠ LEVELⅡ LEVEL Ⅲ Ethical and Professional Standards (total) 15 10 10 Quantitative Methods 12 5-10 0 Economics 10 5-10 0 Financial St t Fi i l Statement A l i t Analysis 20 15-25 15 25 0 Corporate Finance 8 5-15 0 Investment Tools (total) 50 30 60 30-60 0 Analysis of Equity Investments 10 20-30 5--15 Analysis of Fixed Income Investments 12 5-15 ...

Words: 6750 - Pages: 27

Premium Essay

Home Work

...Assignment Print View award: 1.00 point A pension fund has an average duration of its liabilities equal to 14 years. The fund is looking at 5-year maturity zero-coupon bonds and 4% yield perpetuities to immunize its interest rate risk. How much of its portfolio should it allocate to the zero-coupon bonds to immunize if there are no other assets funding the plan? → 57.14% 42.86% 35.71% 26.00% Duration of the perpetuity = 1.04/0.04 = 26 years Duration of the zero = 1 years 14 = (wz)(5) + (1 – wz)26; wz = 57.14% Learning Objective: 11-04 Formulate fixed-income immunization strategies for various investment horizons. Multiple Choice Difficulty: 3 Hard award: 1.00 point You own a bond that has a duration of 5 years. Interest rates are currently 6%, but you believe the Fed is about to increase interest rates by 29 basis points. Your predicted price change on this bond is ________. (Select the closest answer.) +1.37% → –1.37% –4.72% +4.72% D* = 5/1.06 = 4.72 ∆P/P = –D*(∆y) = –4.72(0.29%) = –1.37% Learning Objective: 11-02 Compute the duration of bonds; and use duration to measure interest rate sensitivity. Multiple Choice Difficulty: 2 Medium 1 of 13 11/29/2014 1:56 PM Assignment Print View award: 1.00 point You have purchased a guaranteed investment contract (GIC) from an insurance firm that promises to pay you a 7% compound rate of return per...

Words: 2817 - Pages: 12

Premium Essay

Assignment 3

...FINS 2624 Portfolio Management Tutorial 4 – Group Presentation After-Tax Yield to Maturity (Yip S3) – Discussion Questions A. Define the after-tax yield to maturity of a bond The after-tax YTM is the annualised discount rate that equates the present value of all the after-tax cash flows of a bond, to its settlement price (on the assumption that the bond is held to maturity). The after-tax YTM allows the investor to compare the after-tax returns of different investments and compare the after-tax returns of bonds with different coupon rates B. An Investor whose marginal tax rate is 15% would like advice on the choice between a Low Coupon (LC) bond and a High Coupon (HC) bond with the following attributes ➢ LC: 2% Coupon, paid semi-annually, 10 years to maturity, 5% YTM ➢ HC: 15% Coupon, paid semi-annually, 10 years to maturity, 5% YTM i) Show how the After-Tax YTM is computed Low Coupon Bond [pic] High Coupon Bond [pic] ii) Which bond offers the higher after-tax yield? The Low Coupon Rate Bond offers the higher after-tax yield iii) Will investors always prefer low coupon bonds to high coupon bonds? Why? No – they may prefer to pay tax ‘as they go’, for several reasons: • A large taxable gain at maturity may push them into the next tax bracket • They anticipate their taxable income will be higher at maturity so would prefer to pay tax earlier and receive a capital loss at maturity (particularly for longer term bonds) ...

Words: 2041 - Pages: 9

Free Essay

Men Fairness Cream

...Martin Seyffert Research Associate Equity Technology Team Merrill Lynch Investment Managers 800 Scudders Mill Road Plainsboro, NJ 08536 Phone: 609-282-6632 Fax: 609-282-6597 UNDERLYING ASSUMPTION We specialize in technology equity portfolio management. As a subset of this work, we also examine the timing and relationships within software companies of the following variables: 1) R&D cycle; 2) product deployment period; 3) sales cycle; 4) contract duration; and 5) "disposable life" of software. Our assumptions may, or may not, be valid. Our assumptions are as follows: The technology industry is in a state of flux with the duration of the above listed items completely mismatched within companies. The mismatched time horizons are causing more volatile stocks, stemming in part from, less stable financial performance for software companies. We believe that faster development cycles and Internet-based distribution channels have accelerated parts of the business, while R&D cycles and contract duration have not yet been adjusted - or even recognized as an issue within many companies. Below are the specific areas we would like to have addressed in an MBA research project: POTENTIAL RESEARCH TOPIC #1  Does rapid implementation and niche software solutions lead to accelerated market penetration and faster "time to peak" revenue performance for today's young software companies? HERE IS OUR PREMISE It is our contention that investors are constantly "looking...

Words: 1120 - Pages: 5

Premium Essay


...------------------------------------------------- FINAL EXAM ------------------------------------------------- Fall 2015 ------------------------------------------------- Investments ------------------------------------------------- BUS 315 Student Name: _____________________________Student Number:_________________ DURATION: 3 HOURS No. of Students: 125 Department Name & Course Number: BUSI 315 Section: D100 Course Instructor(s): Yuriy Zabolotnyuk ------------------------------------------------- AUTHORIZED MEMORANDA Financial calculator Students MUST count the number of pages in this examination question paper before beginning to write, and report any discrepancy to a proctor. This question paper has 9 (nine) pages. This examination question paper may not be taken from the examination room. In addition to this question paper, students require: an examination booklet yes Scantron sheet yes ------------------------------------------------- Do ALL 25 multiple choice problems: 2 marks per question for a total of 50 marks. 1) Compared to investing in a single security, diversification provides investors a way to: a) Increase the expected rate of return. b) Decrease the volatility of returns. c) Increase the probability of high returns. d) All of the above 2) In a 5-year period, the annual returns on an investment are 5%, -3%, -4%, 2%, and 6%. The standard deviation of annual returns on this investment...

Words: 2309 - Pages: 10

Free Essay


...numerator of equation 4.10 is first derivative of the price w.r.t. yield using equation 4.9. Consider either equation 4.3 or the numerator of 4.9. Determine only the sign of following second derivative and mixed partial derivatives: * ∂2P/∂y2 * ∂2P/∂y∂C * ∂2P/∂y∂n (a) Does duration increase or decrease as the initial yield increases?(decrease) (b) Does duration increase or decrease as the coupon increases?(decrease) (c) Does duration increase or decrease as the maturity increases?(increase) 3. (This is questions 2 and 4 from the text.) Consider semi-annual bonds A and B. | Bond A | Bond B | Coupon | 8% | 9% | Yield to maturity | 8% | 8% | Maturity (years) | 2 | 5 | Par | $100.00 | $100.00 | Price | $100.00 | $104.055 | Produce an Excel spreadsheet to answer the following questions: (a) Compute the PVBP (aka DV01) given the initial yields show above. (b) Compute Modified Duration (D*) using Equation 4.10 (c) Use the Excel “=DURATION” formula to calculate the duration for each bond. Hint: You must use date values two years apart and five years apart for “SETTLE” and “MATURITY”. (d) Does the Excel duration...

Words: 1140 - Pages: 5

Premium Essay

Fixed Income: Asset Liability Management

...Financial Mathematics for Actuaries Chapter 8 Bond Management Learning Objectives 1. Macaulay duration and modified duration 2. Duration and interest-rate sensitivity 3. Convexity 4. Some rules for duration calculation 5. Asset-liability matching and immunization strategies 6. Target-date immunization and duration matching 7. Redington immunization and full immunization 8. Cases of nonflat term structure 2 8.1 Macaulay Duration and Modified Duration • Suppose an investor purchases a n-year semiannual coupon bond for P0 at time 0 and holds it until maturity. • As the amounts of the payments she receives are different at different times, one way to summarize the horizon is to consider the weighted average of the time of the cash flows. • We use the present values of the cash flows (not their nominal values) to compute the weights. • Consider an investment that generates cash flows of amount Ct at time t = 1, · · · , n, measured in payment periods. Suppose the rate of interest is i per payment period and the initial investment is P . 3 • We denote the present value of Ct by PV(Ct ), which is given by Ct . PV(Ct ) = t (1 + i) and we have P = n X (8.1) PV(Ct ). (8.2) t=1 • Using PV(Ct ) as the factor of proportion, we define the weighted average of the time of the cash flows, denoted by D, as D = = n X t=1 n X t " PV(Ct ) P twt , # (8.3) t=1 where PV(Ct ) wt = . P 4 (8.4) P •...

Words: 7983 - Pages: 32