How to solve ALM — Macaulay duration, modified duration, and convexity questions on Exam FM?

Question

A series of cash flows pays $1000$ at time $t=1$; $1000$ at time $t=2$; $800$ at time $t=4$; $100$ at time $t=7$ (times in years). At an effective annual interest rate of $i = 0.0456$, calculate the convexity of the cash-flow stream.

Accepted Answer

Formula
$D_{\mathrm{Mac}}=\dfrac{\sum_t t\,CF_t v^t}{P}$, $D_{\mathrm{mod}}=D_{\mathrm{Mac}}/(1+i)$, and $\mathcal C=\dfrac{\sum_t t(t+1)CF_t(1+i)^{-t-2}}{P}$.

Why this formula applies
Treat bond price as a yield-dependent present value. Duration measures its first-order sensitivity and convexity its second-order curvature, with the appropriate adjustment between Macaulay and modified duration.

Substitution and calculation
With $v = 1/(1+i) = 0.95639$, the price is $P = \sum v^t CF_t = 2613.5667$, and the Macaulay duration is $D_{mac} = \sum t\,v^t CF_t / P = 2.28627$. $C = \sum t(t+1) v^{t+2} CF_t / P = 8.70930$.

Answer
The answer is 8.7093, option (A).

How to Solve ALM — Macaulay duration, modified duration, and convexity Questions on Exam FM

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