Let € be the coupon rate per period and y be the yield per period. There are m periods per year (say; 4 for quarterly coupon payments), and let n be the number of periods remaining until maturity: Show that the duration D is given by 1+y/my - 1+y +n(c - y) / mc[(1 +y)n - 1] + my. Here, the yield per year is given by m . y:
(2) Let T denote the time to maturity and be fixed. Show that , as T -> infinity, we obtain D -> (1 + y) / my. On the other hand, with fixed T but taking m -> infinity (corresponding to continuous coupon rate), show that D =