Chapter 2, Problem 18PS

Perpetuities You have just read an advertisement stating, “Pay us $100 a year for 10 years and we will pay you $100 a year thereafter in perpetuity.” If this is a fair deal, what is the rate of interest?

Summary Introduction

To determine: The rate of interest.

Answer to Problem 18PS

The rate of interest is 7.18%.

Explanation of Solution

Determine the rate of interest

The PV of $100 a year for 10 year is calculated as,

$\begin{array}{l}PV=\left[C\times \left(\left(\frac{1}{r}\right)-\left(\frac{1}{\left(r\times {\left(1+r\right)}^t\right)}\right)\right)\right]\\ PV=\left[\$100\times \left(\left(\frac{1}{r}\right)-\left(\frac{1}{\left(r\times {\left(1+r\right)}^{100}\right)}\right)\right)\right]\\ \cdots \left(1\right)\end{array}$

The PV of $100 a year forever is calculated as,

$\begin{array}{l}PV=\left[\frac{\left(\frac{C}{r}\right)}{{\left(1+r\right)}^n}\right]\\ PV=\left[\frac{\left(\frac{\$100}{r}\right)}{{\left(1+r\right)}^{10}}\right]\\ \cdots \left(2\right)\end{array}$

Combine Equation (1) and (2),

$\$100\times \left[\$100\times \left(\left(\frac{1}{r}\right)-\left(\frac{1}{\left(r\times {\left(1+r\right)}^{100}\right)}\right)\right)\right]=\left[\frac{\left(\frac{\$100}{r}\right)}{{\left(1+r\right)}^{10}}\right]$

By solving for r using trial and error method, we get r as 7.18%.

Therefore the rate of interest is 7.18%.