Independent solution

How to solve this Duration and Convexity question

Setup

Setup

Write the zero-coupon price as the redemption amount discounted for n years.

P(i)=F(1+i)−nP(i)=F(1+i)^{-n}

Model

Model

Modified duration is negative price sensitivity divided by price.

Dmod=−P′(i)P(i)D_{\mathrm{mod}}=-\frac{P'(i)}{P(i)}

Compute

Compute

Differentiating the zero-coupon price cancels the redemption amount and the discount power.

P′(i)=−nF(1+i)−n−1P'(i)=-nF(1+i)^{-n-1}
Dmod=n1+iD_{\mathrm{mod}}=\frac{n}{1+i}

Answer

Answer

The resulting expression is listed as choice C.

Dmod=n1+i(C)\boxed{D_{\mathrm{mod}}=\frac{n}{1+i}\quad\text{(C)}}