Independent solution

How to solve this First-Order Duration Approximation question

Setup

Setup

Immediately before a payment, the level perpetuity is a perpetuity-due. Compute its initial price and modified duration.

P0=1.06250.0625=17P_0=\frac{1.0625}{0.0625}=17
Dmod=10.0625(1.0625)=15.058824D_{\mathrm{mod}}=\frac{1}{0.0625(1.0625)}=15.058824

Model

Model

The yield change is negative one-half percent. Apply the first-order modified-duration estimate to the initial price.

Δi=0.0575−0.0625=−0.005\Delta i=0.0575-0.0625=-0.005
X=P0(1−DmodΔi)X=P_0(1-D_{\mathrm{mod}}\Delta i)

Compute

Compute

The estimate is 18.28. The exact due-perpetuity value at the new rate is 18.391304.

X=18.280000X=18.280000
Y=1.05750.0575=18.391304Y=\frac{1.0575}{0.0575}=18.391304
X−YY=−0.0060520\frac{X-Y}{Y}=-0.0060520

Answer

Answer

The estimate is low by about 0.61%, which is choice C.

X−YY≈−0.61%(C)\boxed{\frac{X-Y}{Y}\approx-0.61\%\quad\text{(C)}}