Independent solution

How to solve this Duration and Convexity question

Setup

Setup

The supplied estimated value was obtained with the Macaulay yield-factor approximation, so first recover its exponent.

121212=123000(1.051.054)DM121212=123000\left(\frac{1.05}{1.054}\right)^{D_M}

Model

Model

Taking logarithms of the price ratio and accumulation-factor ratio isolates Macaulay duration.

DM=ln⁡(121212/123000)ln⁡(1.05/1.054)=3.8512D_M=\frac{\ln(121212/123000)}{\ln(1.05/1.054)}=3.8512

Compute

Compute

That duration is 3.8512; modified duration at the original 5% yield is 3.8512 divided by 1.05.

Dmod=DM/1.05=3.6678D_{mod}=D_M/1.05=3.6678

Answer

Answer

The requested modified duration is 3.67 years, choice A.

Dmod=3.67(A)\boxed{D_{mod}=3.67\quad\text{(A)}}