The current price of a bond having annual coupons is $1312$. The derivative of the price function of the bond with respect to the yield to maturity is -$7443.81$ when evaluated at the current annual yield, which is 7%. Calculate the Macaulay duration and the modified duration D(.07,1) of the bond.
This problem comes is # 2 from section 9.2 from the Mathematical Interest Theory Second Edition textbook. The provided answers are Macaulay: D(.07, infinity)=6.07079, and Modified: D(.07,1)=5.67364 I have been struggling with Macaulay and Modified Durations, any help would be appreciated, thanks!
The Macaulay duration is given by $$ D(i,\infty)=-\frac{\mathrm d P}{\mathrm d \delta}\cdot\frac{1}{P}=-(1+i)\cdot\frac{\mathrm d P}{\mathrm d i}\cdot\frac{1}{P}=(1+i)\cdot D(i,1) $$ where $\delta=\ln(1+i)$, $i=\mathrm e^\delta-1$, $\frac{\mathrm d P}{\mathrm d \delta}=\frac{\mathrm d P}{\mathrm d i}\cdot\frac{\mathrm d i}{\mathrm d \delta}=\frac{\mathrm d P}{\mathrm d i}\cdot (1+i)$ and $D(i,1)=-\frac{P'(i)}{P(i)}$ is the modified duration.
Thus $$ D(7\%,1)=-\left.\frac{\mathrm d P}{\mathrm d i}\right|_{i=7\%}\times \frac{1}{P(7\%)}=-\frac{P'(7\%)}{P(7\%)}=-\frac{-7443.81}{1312}=5.67364 $$ and $$ D(7\%,\infty)=(1+7\%)\times D(7\%,1)=1.07\times 5.67364=6.07079 $$