Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

Index the immediate payment of one at time zero and subsequent payments two, three, and so on.

X=∑k=0∞(k+1)vkX=\sum_{k=0}^{\infty}(k+1)v^k

Model

Model

The generating function for this increasing geometric-weighted sequence is one divided by one minus v squared.

X=1(1−v)2X=\frac{1}{(1-v)^2}

Compute

Compute

Replace one minus v by i divided by one plus i.

X=(1+i)2i2=1.0520.052=441X=\frac{(1+i)^2}{i^2}=\frac{1.05^2}{0.05^2}=441

Answer

Answer

The present value is 441, corresponding to choice D.

X=441(D)\boxed{X=441\quad\text{(D)}}