Question 10.14: Show that the duration of a 2-year bond with annual coupons ...

Denote the annual payments by C_1, C_2 and the face value by F, so that

P(y) = C_1e^{−y} + (C_2 + F)e^{−2y},
D(y)=\frac{C_1e^{−y} + 2(C_2 + F)e^{−2y}}{P(y)}

Compute the derivative of D(y) to see that it is negative:

\frac{d}{dy} D(y)=\frac{-C_1(C_2 + F)e^{−3y}}{P(y)^2}\lt 0 .