Second-Order Bond Price Change via Convexity Adjustment

Scenario

Modified duration provides a first-order linear approximation to bond price changes, but exhibits substantial pricing error for large yield movements. Convexity provides the second-order quadratic correction.

Your Task

1. Formulate the Taylor series expansion of bond price $P(y + \Delta y)$ around current yield $y$ up to the second order in $\Delta y$.
2. Define the bond convexity measure $C = \frac{1}{P} \frac{d^2 P}{dy^2}$ and show that the relative price change is approximated by: $\frac{\Delta P}{P} \approx -D_{mod} \Delta y + \frac{1}{2} C (\Delta y)^2$
3. Explain why the convexity term $\frac{1}{2} C (\Delta y)^2$ is strictly positive for standard option-free bonds regardless of whether yields rise or fall.
4. Given P = \100.00$,$D_{mod} = 7.50$, and$C = 72.00$, compute the estimated percentage price change for$\Delta y = +200$bps and$\Delta y = -200$ bps.

Constraints

• Option-free bond with positive cash flows.

Examples For $\Delta y = +2\%$, linear duration predicts $-15.0\%$. The convexity adjustment $+0.5 \times 72 \times (0.02)^2 = +1.44\%$ dampens the drop to $-13.56\%$.