How to solve ALM — first- and second-order price estimates from modified duration, Macaulay duration, and convexity questions on Exam FM?

Question

A bond is priced at $P = 5,000$ with modified duration $D_{mod} = 5.222$. If the yield decreases by $|\Delta i| = 1.386\%$, use the first-order (modified-duration) approximation to estimate the new price.

Accepted Answer

Formula
$\dfrac{\Delta P}{P}\approx-D_{\mathrm{mod}}\Delta i$; with convexity, $\dfrac{\Delta P}{P}\approx-D_{\mathrm{mod}}\Delta i+\tfrac12\mathcal C(\Delta i)^2$.

Why this formula applies
Apply the signed duration term to the yield change and include the convexity term only when requested. A yield increase normally gives a negative first-order price change.

Substitution and calculation
$P_{new} \approx \Pr(1 - D_{mod}\,\Delta i) = 5,000\,(1 - 5.222\cdot(-0.01386)) = 5,361.8846$.

Answer
The answer is 5361.8846, option (E).

How to Solve ALM — first- and second-order price estimates from modified duration, Macaulay duration, and convexity Questions on Exam FM

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