Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

View the payment of k units as k overlapping unit perpetuities beginning at successive dates.

PV=vn−1i+vni+vn+1i+⋯PV=\frac{v^{n-1}}i+\frac{v^n}i+\frac{v^{n+1}}i+\cdots

Model

Model

The values of those deferred level perpetuities themselves form a geometric series.

PV=vn−1i11−vPV=\frac{v^{n-1}}i\frac{1}{1-v}

Compute

Compute

Summing that series and using d = iv converts the result to v to the n divided by d squared.

PV=vn−1id=vnd2PV=\frac{v^{n-1}}{id}=\frac{v^n}{d^2}

Answer

Answer

That symbolic form appears in choice D.

vnd2(D)\boxed{\frac{v^n}{d^2}\quad\text{(D)}}