7. Duration - APPENDIX TO CHAPTER 4 Measuring Interest-Rate...

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In our discussion of interest-rate risk, we saw that when interest rates change, a bond with a longer term to maturity has a larger change in its price and hence more interest- rate risk than a bond with a shorter term to maturity. Although this is a useful general fact, in order to measure interest-rate risk, the manager of a financial institution needs more precise information on the actual capital gain or loss that occurs when the inter- est rate changes by a certain amount. To do this, the manager needs to make use of the concept of duration, the average lifetime of a debt security’s stream of payments. The fact that two bonds have the same term to maturity does not mean that they have the same interest-rate risk. A long-term discount bond with ten years to maturity, a so-called zero-coupon bond , makes all of its payments at the end of the ten years, whereas a 10% coupon bond with ten years to maturity makes substantial cash pay- ments before the maturity date. Since the coupon bond makes payments earlier than the zero-coupon bond, we might intuitively guess that the coupon bond’s effective matu- rity , the term to maturity that accurately measures interest-rate risk, is shorter than it is for the zero-coupon discount bond. Indeed, this is exactly what we find in example 1. APPLICATION Rate of Capital Gain Calculate the rate of capital gain or loss on a ten-year zero-coupon bond for which the interest rate has increased from 10% to 20%. The bond has a face value of $1,000. Solution The rate of capital gain or loss is 2 49.7%. g 5 where P t 1 1 5 price of the bond one year from now 5 5 $193.81 P t 5 price of the bond today 5 5 $385.54 $1,000 (1 1 0.10) 10 $1,000 (1 1 0.20) 9 P t 1 1 2 P t P t 1 APPENDIX TO CHAPTER 4 Measuring Interest-Rate Risk: Duration
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Thus: g 5 g 52 0.497 49.7% But as we have already calculated in Table 2 in Chapter 4, the capital gain on the 10% ten-year coupon bond is 2 40.3%. We see that interest-rate risk for the ten-year coupon bond is less than for the ten-year zero-coupon bond, so the effective maturity on the coupon bond (which measures interest-rate risk) is, as expected, shorter than the effective maturity on the zero-coupon bond. Calculating Duration To calculate the duration or effective maturity on any debt security, Frederick Macaulay, a researcher at the National Bureau of Economic Research, invented the concept of duration more than half a century ago. Because a zero-coupon bond makes no cash pay- ments before the bond matures, it makes sense to define its effective maturity as equal to its actual term to maturity. Macaulay then realized that he could measure the effec- tive maturity of a coupon bond by recognizing that a coupon bond is equivalent to a set of zero-coupon discount bonds. A ten-year 10% coupon bond with a face value of $1,000 has cash payments identical to the following set of zero-coupon bonds: a $100 one-year zero-coupon bond (which pays the equivalent of the $100 coupon payment made by the $1,000 ten-year 10% coupon bond at the end of one year), a $100 two-
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7. Duration - APPENDIX TO CHAPTER 4 Measuring Interest-Rate...

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