Bond Duration

10 Key excerpts on "Bond Duration"

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  eBook - PDF

  Corporate Bonds and Structured Financial Products

  • Moorad Choudhry(Author)
  • 2004(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Bond analysts use duration to measure this property (it is sometimes known as Macaulay's duration , after its inventor, who first introduced it in 1938). 5 Duration is the weighted average time until the receipt of cash flows from a bond, where the weights are the present values of the cash flows, measured in years. At the time that he introduced the concept, Macaulay used the duration measure as an alternative for the length of time that a bond investment had remaining to maturity. 1.10.1 Duration Recall that the price/yield formula for a plain vanilla bond is as given at (1.17) below, assuming complete years to maturity paying annual coupons, and with no accrued interest at the calculation date. The yield to maturity reverts to the symbol r in this section. P ˆ C 1 ‡ r † ‡ C 1 ‡ r † 2 ‡ C 1 ‡ r † 3 ‡ ‡ C 1 ‡ r † n ‡ M 1 ‡ r † n 1 : 17 † If we take the first derivative of this expression we obtain (1.18). dP dr ˆ 1 C 1 ‡ r † 2 ‡ 2 C 1 ‡ r † 3 ‡ ‡ n † C 1 ‡ r † n ‡ 1 ‡ n † M 1 ‡ r † n ‡ 1 1 : 18 † If we re-arrange (1.18) we will obtain the expression at (1.19), which is our equation to calculate the approximate change in price for a small change in yield. dP dr ˆ 1 1 ‡ r † 1 C 1 ‡ r † ‡ 2 C 1 ‡ r † 2 ‡ ‡ nC 1 ‡ r † n ‡ nM 1 ‡ r † n # 1 : 19 † 5 Macaulay, F., Some theoretical problems suggested by the movements of interest rates, bond yields and stock prices in the United States since 1865 , National Bureau of Economic Research, NY 1938. This remains a fascinating read and is available from Risk Classics publishing, under the title Interest rates, bond yields and stock prices in the United States since 1856 . Chapter 1: A Primer on Bond Basics 29 Readers may feel a sense of familiarity regarding the expression in brackets in equation (1.19) as this is the weighted average time to maturity of the cash flows from a bond, where the weights are the present values of each cash flow.

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  Bond and Money Markets

  Strategy, Trading, Analysis

  • Moorad Choudhry(Author)
  • 2003(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Let us summarise here the main interpretations of the duration measures. In its original formulation by Macaulay duration was defined as the weighted-average time until receipt of a financial instrument’s cash flows. The original formula for duration given as (7.32) can be re-written as (7.33). ( ) 1 1 . 1 N n n n nC D P r = = + ∑ (7.32) ( ) 1 . 1 N n n n C P D n r =      = ×        + ∑ (7.33) So expression (7.33) defines duration as the time-weighted average of a bond instrument’s discounted cash flows as a proportion of the bond’s price. The duration concept was later developed as a measure of a bond’s price elasticity with respect to changes in its yield to maturity. This allowed us to view duration as a measure of a bond’s price volatility. To derive this measure, we needed to obtain the first derivative of the bond’s price with respect to its yield shown at (7.34) ( ) 1 1 d d 1 N n n n nC P r r + = − = + ∑ (7.34) which can be rearranged using algebra 6 to give (7.35): d . d 1 P DP r r = − + (7.35) This enables us to interpret duration now as the bond yield elasticity of the bond price, as shown in (7.36). ( ) d Percentage change in bond price . d 1 Percentage change in bond yield P P D r r = = − + (7.36) This further allows us to use the modified duration measure as the price volatility of a bond, if we rearrange (7.36) to give us (7.37): d d . 1 P r D P r = − + (7.37) One further interpretation of the duration measure was given by Babcock (1985). This views definition as a weighted average of two factors, shown as (7.38): ( ) ( ) ( ) 1 1 Ann rc rc D N PV r ,N r r r = − + × + (7.38) where rc is the bond’s running yield 6 The process is illustrated at Appendix 7.1.

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  Streetwise

  The Best of The Journal of Portfolio Management

  • Peter L. Bernstein, Frank J. Fabozzi, Peter L. Bernstein, Frank J. Fabozzi(Authors)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  Moreover, it is evident from this example that the same probability of default prior to maturity is more serious with the deep discount bond because a larger proportion of the investor's outlay is riding on the more distant cash receipts. Therefore, we might expect to find the low coupon bonds displaying more than average price sensitivity to changes in perceived risk. HOW TO CALCULATE DURATION Macaulay proposed the concept of duration as a measure of a bond's life that explicitly considers the timing of the return of value. In essence, duration is simply a weighted average maturity stated in present value terms; the number of years into the future when a cash flow is received is weighted by the proportion that flow contributes to the total present value or price of the bond. For ease of exposition, let us assume annual compounding and one bond coupon payment per year. Then, algebraically, duration (D) is defined as D = C, (1 + r)< C, (1 + r)< (1) where n is the life of the bond in years, Q is the cash receipt at the end of year t — equal to the annual coupon except for the last year, when it is equal to the annual coupon plus the maturity value — and r is the yield to maturity. 3 The numerator of the expression within the brackets is the present value of a sing'° year's cash receipt; the denominator is the sum of all these present values, which is equal to the total pres-ent value or price of the bond. Therefore, the entire expression within the brackets is the weight given to the t th receipt. The t outside the brackets is simply the number of years from the present when the cash is received, equal to 1, 2, 3, and so on. The number of years into the future of a receipt, multiplied by its weight and summed over all receipts, is the bond's du-ration. Table 2 contains an example that should make this measure considerably more comprehensible. It shows the computation of the duration of an 8% coupon bond with five years to maturity priced at par.

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  Fixed Income Mathematics

  • Robert Zipf(Author)
  • 2003(Publication Date)
  • Academic Press

    (Publisher)

  Duration may not be appropriate for cash flows that are sensitive to changes in interest rates. These include bonds with embedded options, such as a call option, which may be exercised at any time, and especially include mortgage backed securities (MBS). These securities require careful analysis if you want to use the duration concept. We’ll discuss MBS in Chapter 21. CHAPTER SUMMARY We started with duration as a measure of the length of time a bond (or a series of cash flows) was outstanding. Duration measures both time and risk, in our presentation. Duration’s importance is as a measure of bond volatility. Macaulay duration is a measure of the weighted average life of bond, including coupons and a present value factor. Negative Duration 201 Dollar duration is a measure of the dollar price change of the bond for a given yield change. It is also sometimes called the price value of one basis point change. Portfolio modified duration is approximately the weighted average modi-fied durations of the bonds in the portfolio. It may also be calculated by com-puting the duration of the entire portfolio’s flow of funds, using the internal rate of return that sets the present value of the entire flow equal to the market value of the portfolio. C OMPUTER P ROJECTS 1. Use your favorite computer program to compute duration. Compute the duration for the 4% bond, due in 3 years, at a 4% yield, as we did earlier. Now change the yield to 5% and to 6%. What do you expect to happen to the duration? Will it increase or decrease? By how much? A relatively large amount or a relatively small amount? Why? 2. What happens to the duration as yields become very large and very small? Why? 3. Now change Problem 1 to use a 4% bond due in 10 years, but callable at par in 3 years. Price the bond to the par call date when the yield becomes equal to or less than 3%. What happens to the duration as the yield crosses 3%? Why? T OPICS FOR C LASS D ISCUSSION 1.

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  Fixed Income Analysis

  • (Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  • Bond Duration, in general, measures the sensitivity of the full price (including accrued inter- est) to a change in interest rates. • Yield duration statistics measuring the sensitivity of a bond’s full price to the bond’s own yield-to-maturity include the Macaulay duration, modified duration, money duration, and price value of a basis point. • Curve duration statistics measuring the sensitivity of a bond’s full price to the benchmark yield curve are usually called “effective durations.” • Macaulay duration is the weighted average of the time to receipt of coupon interest and principal payments, in which the weights are the shares of the full price corresponding to each payment. This statistic is annualized by dividing by the periodicity (number of coupon payments or compounding periods in a year). • Modified duration provides a linear estimate of the percentage price change for a bond given a change in its yield-to-maturity. 250 Fixed Income Analysis • Approximate modified duration approaches modified duration as the change in the yield-to-maturity approaches zero. • Effective duration is very similar to approximate modified duration. The difference is that approximate modified duration is a yield duration statistic that measures interest rate risk in terms of a change in the bond’s own yield-to-maturity, whereas effective duration is a curve duration statistic that measures interest rate risk assuming a parallel shift in the benchmark yield curve. • Key rate duration is a measure of a bond’s sensitivity to a change in the benchmark yield curve at specific maturity segments. Key rate durations can be used to measure a bond’s sensitivity to changes in the shape of the yield curve. • Bonds with an embedded option do not have a meaningful internal rate of return because future cash flows are contingent on interest rates. Therefore, effective duration is the appro- priate interest rate risk measure, not modified duration.

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  An Introduction to International Capital Markets

  Products, Strategies, Participants

  • Andrew M. Chisholm(Author)
  • 2009(Publication Date)
  • Wiley

    (Publisher)

  It is possible, for example, for yields to rise at the 10-year point but at the same time fall at the five-year maturity, so that the trader in the example could actually lose on both the long and the short position. This is sometimes known as curve risk. 5.17 CHAPTER SUMMARY The price of low coupon, long maturity bonds is highly sensitive to changes in market yields. This is because most of the cash flows occur far out into the future and cannot be reinvested at new levels of interest rates for a long time. The weighted average life of a bond’s cash flows is measured by Macaulay’s duration. A zero coupon bond has a Macaulay’s duration equal to its maturity. The duration of a coupon bond and its effective exposure to interest rate changes are less than its maturity because the coupons can be reinvested at the prevailing market rate. Modified duration is a related measure which can be used to estimate the change in the money value of the bond for a 0.01 % yield change. This is known as the price value of a basis point or basis point value. Duration can be used to put together a portfolio of bonds which matches expected future cash flows. This technique is called immunization. Duration can also be used to construct a hedge against the fall in the value of a bond portfolio. Duration is an approximate measure because the relationship between the price of a bond and its yield is nonlinear. Convexity is a measure of this curvature. It can be used to provide a better estimate of the actual change in the price of a bond for a substantial yield change.

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  Financial Institutions, Markets, and Money

  • David S. Kidwell, David W. Blackwell, David A. Whidbee, Richard W. Sias(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  By using duration com- bined with a strategy of “dedicating” a bond portfolio’s assets to paying a particular stream of pension fund obligations, fund managers can ensure that they are able to pay retirees the contracted amount regardless of interest rate changes. In Chapter 20, we show how manag- ers of commercial banks and thrift institutions can use duration to reduce fluctuations in net interest income (the difference between the average return of the assets minus the average cost of liabilities) resulting from interest rate changes. In addition, we show that banks can protect their net worth from interest rate risk by matching the average duration of the assets with the average duration of the liabilities. DO YOU UNDERSTAND? 1. Consider a 4‐year bond selling at par with a 7 percent annual coupon. Suppose that yields on similar bonds increase by 50 basis points. Use duration to estimate the percentage change in the bond price. Check your answer by calculating the new bond price. 2. Define price risk and reinvestment risk. Explain how the two risks offset each other. 3. What is the duration of a bond portfolio made up of two bonds: 37 percent of a bond with duration of 7.7 years and 63 percent of a bond with duration of 16.4 years? 4. How can duration be used as a way to rank bonds on their interest rate risk? 5. To eliminate interest rate risk, should you match the maturity or the duration of your bond investment to your holding period? Explain. 6. If you have a 4‐year investment horizon and you think interest rates will fall should you pick a bond with a 3‐year duration, a 4‐year duration, or a bond with a 5‐year duration? Explain. S U M M A R Y O F L E A R N I N G O B J E C T I V E S 1 Explain the time value of money and its appli- cation to the pricing of bonds.

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  Fixed Income Securities

  Concepts and Applications

  • Sunil Kumar Parameswaran(Author)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  5  Duration, Convexity, and Immunization

  The duration of a plain vanilla bond can be defined as its average life. It is very easy to define duration in the case of securities that yield a single cash flow, like a zero coupon bond. In such cases there is no difference between the average time to maturity and the actual time to maturity, for we need concern ourselves only with the terminal payment. Consequently, in such cases, the duration of the security is nothing but its stated term to maturity.

  However the definition is not so clearcut in the case of a conventional coupon paying debt security. In such cases, the asset gives rise to a series of cash flows, usually on a semiannual basis, as well as a relatively large cash flow at the end that constitutes the principal repayment. The average life of such a security can be obtained only by taking cognizance of the times to maturity of the component cash flows. Because the cash flows occur at different points in time, we also need to factor in the issue of the time value of money.

  Convexity of a bond accounts for the fact that the price-yield relationship of a bond is convex and not linear. Whereas, duration accounts for a first order approximation to the price-yield relationship, convexity factors into the fact that the relationship is indeed convex. Immunization strategies protect a bond or a bond portfolio against interest rate risk. As discussed earlier, interest rate changes impact bonds in two ways. The higher the interest rate, the more the income from the re-investment of coupons. However the higher the rate, the lower the sale price of the bond at the end of the investment horizon. See Figure 5.1

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  Investing in Mortgage Securities

  • Laurence G. Taff(Author)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  53 3 More Advanced Fixed-Income Topics Some of the major mathematical tools, including duration and convexity, used in fixed-income portfolio management are developed in this chapter. Duration and convexity can materially aid in managing interest rate-sensitive revenue and expense cash flows over time. They also play a major role in maintaining the price balance between assets and liabilities. Next negative convexity , which leads to price com-pression , is explained and its consequences outlined. Debt with an embedded call option is also treated in this chapter because almost all residential mortgages have one. I. DURATION A variety of time periods are associated with fixed-income instruments. The most obvious one is maturity, the date upon which the instrument repays its redemption amount. The next one is the term-to-maturity — the amount of time remaining until maturity is reached. A temporal concept frequently utilized in the mortgage industry is the weighted average maturity . It measures the average time to receive each dollar of principal (see Chapter 5). The first part of this chapter is concerned with two other measures of time associated with the cash flows from fixed-income securities: Macaulay duration and modified duration and their accidental relationship in some special financial circumstances. (A third kind of duration, effective duration , was invented to accommodate contingent cash flows.) A. M ACAULAY D URATION 1. Duration and Portfolio Management Macaulay duration has an important role to play in fixed-income portfolio manage-ment. Suppose that you were managing a bond fund. Then you would have cash flows in — namely interest earnings from your assets — and cash flows out — the income stream expected by the fund’s owners. The assets are the investments orig-inally made by you, the bond fund manager, and any reinvestments of additional cash flows and excess earnings. You, the manager of the bond fund, have two very different problems: 1.

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  Managing a Corporate Bond Portfolio

  • Frank J. Fabozzi, Leland E. Crabbe(Authors)
  • 2003(Publication Date)
  • Wiley

    (Publisher)

  Chapter 6 87 by 100 basis points for the duration measure to be useful. This is a critical assumption and its importance cannot be overemphasized. We return to this point later in this chapter when we discuss yield curve risk measures. Similarly, the dollar duration of a portfolio can be obtained by calculat- ing the weighted average of the dollar duration of the bonds in the portfolio. An alternative procedure for calculating the duration of a portfolio is to calculate the dollar price change for a given number of basis points for each secu- rity in the portfolio and then adding up all the price changes. Dividing the total of the price changes by the initial market value of the portfolio produces a percent- age price change that can be adjusted to obtain the portfolio’s duration. For example, consider the three-bond portfolio shown before. Suppose that we calculate the dollar price change for each bond in the portfolio based on its respective duration for a 50 basis point change in yield. We would then have: Thus, a 50 basis point change in all rates changes the market value of the three- bond portfolio by $310,663. Since the market value of the portfolio is $9,609,961, a 50 basis point change produced a change in value of 3.23% ($310,663 divided by $9,609,961). Since duration is the approximate percentage change for a 100 basis point change in rates, this means that the portfolio duration is 6.46 (found by doubling 3.23). This is the same value for the portfolio’s duration as found earlier. Contribution to Portfolio Duration Some portfolio managers look at their exposure to an issue or to a sector in terms of the percentage of that issue or sector in the portfolio. A better measure of expo- sure of an individual issue or sector to changes in interest rates is in terms of its contribution to portfolio duration. This is found by multiplying the percentage that the individual issue or sector is of the portfolio by the duration of the individ- ual issue or sector.

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