Question

What is the maximum amount you would pay for an asset that generates an income $\mathbf{\$}\mathbf{250}\mathbf{,}\mathbf{000}$at the end of each five years if the opportunity cost of using the funds is $\mathbf{8}$percent?

Short answer

The maximum amount is $\$998,178$.

Step-by-step solution

Introduction

We use the formula for the present value of a stream with five summands as the time period is 5 years:

$PV=\frac{F{V}_1}{(1+i{)}^1}+\frac{F{V}_2}{(1+i{)}^2}+\frac{F{V}_3}{(1+i{)}^3}+\frac{F{V}_4}{(1+i{)}^4}+\frac{F{V}_5}{(1+i{)}^5}$

Finding the maximum amount.

By substituting $i=8\%=0.08$into the present value formula, we get

$PV=\frac{F{V}_1}{(1+0.08{)}^1}+\frac{F{V}_2}{(1+0.08{)}^2}+\frac{F{V}_3}{(1+0.08{)}^3}+\frac{F{V}_4}{(1+0.08{)}^4}+\frac{F{V}_5}{(1+0.08{)}^5}$

$PV=\frac{F{V}_1}{(1.08{)}^1}+\frac{F{V}_2}{(1.08{)}^2}+\frac{F{V}_3}{(1.08{)}^3}+\frac{F{V}_4}{(1.08{)}^4}+\frac{F{V}_5}{(1.08{)}^5}$

Here, $F{V}_1=F{V}_2=F{V}_3=F{V}_4=F{V}_5=\$250,000$because the annual income is the same at the end of each year. Therefore, substitute $\$250,000$in to the formula:

$PV=\frac{250,000}{(1.08{)}^1}+\frac{250,000}{(1.08{)}^2}+\frac{250,000}{(1.08{)}^3}+\frac{250,000}{(1.08{)}^4}+\frac{250,000}{(1.08{)}^5}$

$PV=998,178$

Thus, the present value of the asset in the next five years is $\$998,178$.

Therefore, the maximum amount you would pay for this asset shouldn’t be more than the present value $\mathbf{\$}\mathbf{998}\mathbf{,}\mathbf{178}$.