Convexity Adjustments

A bond investor is assessing the impact of a 50-basis-point interest rate increase. His bond portfolio has eight years of average maturity, 6.8 Macaulay duration, 138.45 convexity, and 7.5% yield-to-maturity. Which of the following is _closest_ to the percentage change in the value of his bond portfolio?

Incorrect. The answer choice appears to have been obtained by plugging in the value of Macaulay duration into the formula. It should be modified duration.

Incorrect. This answer choice multiplies the modified duration by the 50-basis-point change to arrive at –3.16%. It ignores the effect of convexity on the percentage change in bond price due to an increase in interest rate.

Correct! The formula for calculating the change in price is: $$\displaystyle \% \Delta PV^{Full} \approx -(\text{AnnModDur} \times \Delta \text{Yield}) + \left[ \frac{1}{2} \times \text{AnnConvexity} \times (\Delta \text{Yield})^2 \right] $$ $$\displaystyle \text{AnnModDur} = \frac{\text{AnnMacDur}}{(1+r)} = \frac{6.8}{(1+0.075)} = 6.3256$$ Therefore: $$\displaystyle \% \Delta PV^{Full} \approx (-6.3256 \times 0.0050) $$ $$\displaystyle + \left (\frac{1}{2} \times 138.45 \times 0.0050^{2}\right ) = -0.0299 = -2.99 \% $$

–3.23%

–3.16%

–2.99%