Chapter 16 Managing Bond Portfolios
Change in Bond Price as a Function of Change in Yield to Maturity
Interest Rate Sensitivity Inverse relationship between price and yield. An increase in a bond s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield. Long-term bonds tend to be more price sensitive than short-term bonds.
Interest Rate Sensitivity Price sensitivity is inversely related to a bond s coupon rate. Price sensitivity is inversely related to the yield to maturity at which the bond is selling.
Interest Rate Sensitivity Prices of 8% Coupon Bond (Coupons Paid Semiannually): As YTM rises, price falls. Fall in price is higher when maturity is higher As maturity increases, price sensitivity increases at a decreasing rate.
Interest Rate Sensitivity Prices of Zero-Coupon Bond (Semiannually Compounding) Because we know that long-term bonds are more sensitive to interest rate movements than are short-term bonds, this observation suggests that in some sense a zero-coupon bond represents a longer-term bond than equal-time-to-maturity coupon bond
Ambiguity with Effective Maturity Note that the times to maturity of the two bonds in this example are not perfect measures of the long-term or shortterm nature of the bonds. The 20-year 8% bond makes many coupon payments, most of which come years before the bond s maturity date. Each of these payments may be considered to have its own maturity date, and the effective maturity of the bond is therefore some sort of average of the maturities of all the cash flows paid out by the bond. The zero-coupon bond, by contrast, makes only one payment at maturity. Its time to maturity is, therefore, a well-defined concept.
Duration To deal with the ambiguity of the maturity of a bond making many payments, we need a measure of the average maturity of the bond s promised cash flows to serve as a useful summary statistic of the effective maturity of the bond. Duration is a measure of the effective maturity of a bond. Frederick Macaulay termed the effective maturity concept the duration of the bond.
Duration Macaulay s duration is computed as the weighted average of the times until each payment is received, with the weights proportional to the present value of the payment. Duration is shorter than maturity for all bonds except zero coupon bonds. Duration is equal to maturity for zero coupon bonds.
Duration: Calculation w t CF t 1 Price y t D T t1 t w t CFt Cash Flow for period t
Duration Calculation: Example 8% coupon paid semiannually; maturity: 2 years; YTM=10% Time (years) Payment PV of CF Weight C 1 C 4.5 40 38.095.0395.0197 1 40 36.281.0376.0376 1.5 40 34.553.0358.0537 2.0 1040 855.611.8871 1.7742 sum 964.540 1.000 1.8852
Duration/Price Relationship We have seen that long-term bonds are more sensitive to interest rate movements than are short-term 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 bond s price can be related to its YTM according to the rule: P/P = D 1+y 1+y D = modified duration D = D 1+y = 1.8852 / 1.10 = 1.71 years P/P = D y
Rules for Duration Rule 1 The duration of a zero-coupon bond equals its time to maturity. Rule 2 Holding maturity constant, a bond s duration is higher when the coupon rate is lower. Rule 3 Holding the coupon rate constant, a bond s duration generally increases with its time to maturity. Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond s yield to maturity is lower.
Rules for Duration (cont d) Rules 5 The duration of a level perpetuity is equal to: ( 1 y y) For example, at a 10% yield, the duration of a perpetuity that pays $100 once a year forever is 1.10 / 0.10 = 11 years, but at an 8% yield, it is 1.08 / 0.08 = 13.5 years This rule makes it obvious that maturity and duration can differ substantially. The maturity of a perpetuity is infinite, whereas the duration at a 10% yield is only 11 years
Rules for Duration (cont d) Rule 6 The duration of a level annuity is equal to: 1 y y (1 T y) T 1 For example, a 10-year annual annuity with a yield of 8% will have duration 1.08 10 10 0.08 1.08 1 4.87 years
Rules for Duration (cont d) Rule 7 The duration for a coupon bond is equal to: 1 y y (1 y) c[(1 T ( c y) y) T 1] y The duration of a 8% coupon (paid semiannually) bond with maturity of 2 years and YTM of 10% would be 1.05 1.05 4(0.04 0.05) 1.01 21 3.7645halfyears 1.88years 4 0.05 0.04[1.05 1] 0.05 0.0586 Same result that we got in slide 9
Duration and Convexity P/P = D y This rule asserts that the percentage price change is directly proportional to the change in the bond s yield. If this were exactly so, a graph of the percentage change in bond price as a function of the change in its yield would plot as a straight line, with slope equal to D. Yet, we know from the Figure presented in slide 2 that the relationship between bond prices and yields is not linear. The duration rule is a good approximation for small changes in bond yield, but it is less accurate for larger changes.
Duration and Convexity 30-Year Maturity, 8% Coupon; Initial YTM = 8%
Correction for Convexity n 1 CFt Convexity t t t (1 y) 2 P (1 y) t1 2 ( ) Correction for Convexity: P * 1 2 D y 2 [ Convexity ( y) ] P
Correction for Convexity Suppose, a bond has 30-year maturity, 8% coupon and 8% YTM. If a bond s yield increases from 8% to 10%, and D =11.26 years, without adjusting for convexity, percentage change in price P * D y P 11.26 0.02 22.52% Convexity of the bond can be calculated to be 212.4. Adjusting for convexity, percentage change in price P 1 P 2 2 11.26 0.02 [212.4 (0.02) ] 18.27%
Why Do Investors Like Convexity?
Practice Problems Chapter 16: 2, 3, 4, 5, 7, CFA Problem: 3