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Duration Basics

Introduction

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

...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...Assignment Print View http://ezto.mheducation.com/hm_finance.tpx 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 http://ezto.mheducation.com/hm_finance.tpx 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...

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