Independent solution

How to solve this Duration and Convexity question

Setup

Setup

A beginning-of-year perpetuity has payments at times zero, one, two, and so on.

DM=∑n=0∞nvn∑n=0∞vn=1iD_M=\frac{\sum_{n=0}^{\infty}nv^n}{\sum_{n=0}^{\infty}v^n}=\frac{1}{i}

Model

Model

The geometric-series present value and its time-weighted derivative reduce the Macaulay duration to the reciprocal of the yield.

30=1i⟹i=13030=\frac{1}{i}\Longrightarrow i=\frac{1}{30}

Compute

Compute

The given duration therefore fixes the yield at one-thirtieth. Modified duration divides the Macaulay value by one plus that yield.

Dmod=DM1+i=301+1/30=29.0323D_{mod}=\frac{D_M}{1+i}=\frac{30}{1+1/30}=29.0323

Answer

Answer

The resulting modified duration is 29.03 years, so choice C is correct.

Dmod=29.03(C)\boxed{D_{mod}=29.03\quad\text{(C)}}