1st-Order Linear Approximation

Study guide: Exam FM-2

The 1st-order linear approximation (or first-order Taylor approximation) estimates the change in bond price for a small change in Yield Rate $\Delta j$ , using Modified Duration:

$\Delta P \approx -D_{Mod} \cdot P \cdot \Delta j$

$\frac{\Delta P}{P} \approx -D_{Mod} \cdot \Delta j = -\frac{D_{Mac}}{1+j} \cdot \Delta j$

This approximation is accurate for small $\Delta j$ but underestimates price increases and overestimates price decreases for large shifts, because the price-yield relationship is convex
The Convexity term corrects for this curvature

Estimating Price Change A bond has price $ $950$ , modified duration $8.5$ years. Yields fall by 25 basis points ( $\Delta j = -0.0025$ ).

Answer $\Delta P \approx -8.5 \times 950 \times (-0.0025) = 8.5 \times 950 \times 0.0025 = 20.19$ The bond price increases by approximately $ $20.19$ .