Independent solution

How to solve this Duration and Convexity question

Setup

Setup

Discount the two equal payments at times one and two under the constant force.

P=e−δ+e−2δP=e^{-\delta}+e^{-2\delta}

Model

Model

Multiply each present value by its payment time to form the duration numerator.

M=e−δ+2e−2δM=e^{-\delta}+2e^{-2\delta}

Compute

Compute

Cancel the common first-year discount factor from numerator and denominator.

DM=MP=1+2e−δ1+e−δD_M=\frac{M}{P}=\frac{1+2e^{-\delta}}{1+e^{-\delta}}

Answer

Answer

The simplified duration is the expression in choice A.

DM=1+2e−δ1+e−δ(A)\boxed{D_M=\frac{1+2e^{-\delta}}{1+e^{-\delta}}\quad\text{(A)}}