1. Let be the coupon rate per period and y be the yield per period. There are m periods per year (say, m = 4 for quarterly coupon payments), and let n be the number of periods remaining until maturity. (1) Show that the duration D is given by D= 1+ y 1+y+n(c-y) my m[(1 + y)" - 1] + my Here, the yield per year A is given by m.y. (2) Let T denote the time to maturity and m be fixed. Show that, as T +, we obtain D+ T (3) On the other hand, with fixed T but taking m + corresponding to continuous coupon rate), show that 1 1 m D Remark The above analytic results are revealed in the following numerical example. Consider the duration calculated for various bonds as shown in the following table, where 1 = 0.05 and m= 2. We obtain D+} + oos = 20.5, as T + . The duration for large yields tends to be relatively short. Duration of a Bond Yielding 5% as Function of Maturity and Coupon Rate Coupon rate Years to maturity 1% 2% 10% 1 0.997 0.995 0.988 0.977 2 1.984 1.969 1.928 1.868 4.875 4.763 4.485 4.156 10 9.116 8.950 7.989 7.107 25 20.161 17.715 14.536 12.751 26.666 22.281 18.765 17.381 100 22.572 21.200 20.363 20.067 20.500 20.500 20.500 20.500 The table shows that duration does not increase appreciably with maturity. In fact, with a fixed yield, duration increases only to a finite limit as maturity is increased. 1. Let be the coupon rate per period and y be the yield per period. There are m periods per year (say, m = 4 for quarterly coupon payments), and let n be the number of periods remaining until maturity. (1) Show that the duration D is given by D= 1+ y 1+y+n(c-y) my m[(1 + y)" - 1] + my Here, the yield per year A is given by m.y. (2) Let T denote the time to maturity and m be fixed. Show that, as T +, we obtain D+ T (3) On the other hand, with fixed T but taking m + corresponding to continuous coupon rate), show that 1 1 m D Remark The above analytic results are revealed in the following numerical example. Consider the duration calculated for various bonds as shown in the following table, where 1 = 0.05 and m= 2. We obtain D+} + oos = 20.5, as T + . The duration for large yields tends to be relatively short. Duration of a Bond Yielding 5% as Function of Maturity and Coupon Rate Coupon rate Years to maturity 1% 2% 10% 1 0.997 0.995 0.988 0.977 2 1.984 1.969 1.928 1.868 4.875 4.763 4.485 4.156 10 9.116 8.950 7.989 7.107 25 20.161 17.715 14.536 12.751 26.666 22.281 18.765 17.381 100 22.572 21.200 20.363 20.067 20.500 20.500 20.500 20.500 The table shows that duration does not increase appreciably with maturity. In fact, with a fixed yield, duration increases only to a finite limit as maturity is increased
