managingbondportfolios(编辑修改稿)

managingbondportfolios(编辑修改稿)内容摘要：

zerocoupon bond, each with 2 years to maturity. We assume that the yield to maturity on each bond is 10%, or 5% per halfyear. The present value of each payment is discounted at 5% per period for the number of (semiannual) periods shown in column B. The weight associated with each payment time (column F) is the present value of the payment for that period (column E) divided by the bond price (the sum of the present values in column E). Spreadsheet Calculating the duration of two bonds Column sums subject to rounding error. Please visit us at p. 513 Spreadsheet Spreadsheet formulas for calculating duration Please visit us at The numbers in column G are the products of time to payment and payment weight. Each of these products corresponds to one of the terms in Equation . According to that equation, we can calculate the duration of each bond by adding the numbers in column G. The duration of the zerocoupon bond is exactly equal to its time to maturity, 2 years. This makes sense, because with only one payment, the average time until payment must be the bond39。 s maturity. In contrast, the 2year coupon bond has a shorter duration of years. Spreadsheet shows the spreadsheet formulas used to produce the entries in Spreadsheet . The inputs in the spreadsheet— specifying the cash flows the bond will pay— are given in columns B– D. In column E we calculate the present value of each cash flow using the assumed yield to maturity, in column F we calculate the weights for Equation , and in column G we pute the product of time to payment and payment weight. Each of these terms corresponds to one of the values that is summed in Equation . The sums puted in cells G8 and G14 are therefore the durations of each bond. Using the spreadsheet, you can easily answer several “what if” questions such as the one in Concept Check 1. CONCEPT CHECK 1 Suppose the interest rate decreases to 9% as an annual percentage rate. What will happen to the prices and durations of the two bonds in Spreadsheet ? Duration is a key concept in fixedine portfolio management for at least three reasons. First, as we have noted, it is a simple summary statistic of the effective average maturity of the portfolio. Second, it turns out to be an essential tool in immunizing portfolios from interest rate risk. We explore this application in Section . Third, duration is a measure of the interest rate sensitivity of a portfolio, which we explore here. p. 514 We have seen that longterm bonds are more sensitive to interest rate movements than are shortterm bonds. The duration measure enables us to quantify this relationship. Specifically, it can be shown that when interest rates change, the proportional change in a bond39。 s price can be related to the change in its yield to maturity, y, according to the rule The proportional price change equals the proportional change in 1 plus the bond39。 s yield times the bond39。 s duration. Practitioners monly use Equation in a slightly different form. They define modified duration Macaulay39。 s duration divided by 1 + yield to maturity. Measures interest rate sensitivity of bonds. as D* = D/(1 + y), note that Δ(1 + y) = Δ y, and rewrite Equation as The percentage change in bond price is just the product of modified duration and the change in the bond39。 s yield to maturity. Because the percentage change in the bond price is proportional to modified duration, modified duration is a natural measure of the bond39。 s exposure to changes in interest rates. Actually, as we will see below, Equation , or equivalently , is only approximately valid for large changes in the bond39。 s yield. The approximation bees exact as one considers smaller, or localized, changes in Example Duration Consider the 2year maturity, 8% coupon bond in Spreadsheet making semiannual coupon payments and selling at a price of $, for a yield to maturity of 10%. The duration of this bond is years. For parison, we will also consider a zerocoupon bond with maturity and duration of years. As we found in Spreadsheet , because the coupon bond makes payments semiannually, it is best to treat one “period” as a half year. So the duration of each bond is 2 = 4 (semiannual) periods, with a per period interest rate of 5%. The modified duration of each bond is therefore Suppose the semiannual interest rate increases from 5% to %. According to Equation , the bond prices should fall by Now pute the price change of each bond directly. The coupon bond, which initially sells at $, falls to $ when its yield increases to %, which is a percentage decline of .0359%. The zerocoupon bond initially sells for $1,000/ = . At the higher yield, it sells for $1,000/ = . This price also falls by .0359%. We conclude that bonds with equal durations do in fact have equal interest rate sensitivity and that (at least for small changes in yields) the percentage price change is the modified duration times the change in yield. p. 515 CONCEPT CHECK 2 a. In Concept Check 1, you calculated the price and duration of a 2year maturity, 8% coupon bond making semiannual coupon payments when the market interest rate is 9%. Now suppose the interest rate increases to %. Calculate the new value of the bond and the percentage change in the bond39。 s price. b. Calculate the percentage change in the bond39。 s price predicted by the duration formula in Equation or . Compare this value to your answer for (a). What Determines Duration? Malkiel39。 s bond price relations, which we laid out in the previous section, characterize the determinants of interest rate sensitivity. Duration allows us to quantify tha。