Chapter 14A.6, Problem 1MQ

To determine

The equation when machine deteriorates at the rate of d per year.

Explanation of Solution

Given, equation 14A.33 is $\left(1+r\right)\left(1+p^e\right)=1+r+p^e+r{p}^e=1+i$

If the machine deteriorates at the rate of d per year then equation 14A.33 becomes:

  $\left(1+r\right)\left(1+d\right)\approx \left(1+r+d\right)$

The present discounted value from machine ownership when depreciation rate d = 0, will be:

  $PDV=\frac{v}{1+r}+\frac{v}{{\left(1+r\right)}^2}+\mathrm{...}+\frac{v}{{\left(1+r\right)}^n}+\mathrm{...}$

Here, r = rate of return,

v = rental rate.

When machine deteriorates at the rate of d per year then for period n, the numerator value of this equation should be $\frac{v}{{\left(1+d\right)}^n}$ .

After considering the depreciation rate d, the equation for present discounted value becomes:

  $\begin{array}{l}P=PDV={\displaystyle \sum _1^n\frac{v}{{\left(1+d\right)}^i{\left(1+r\right)}^i}}\\ As,\left(1+d\right)\left(1+r\right)\approx \left(1+d+r\right)\\ P={\displaystyle \sum _1^n\frac{v}{{\left(1+d+r\right)}^i}}\end{array}$

When n is tends to infinity, the equation for yield is:

  $\begin{array}{l}P=\frac{v}{r+d}\\ or,v=P\left(r+d\right)\end{array}$

Thus, the equation for rental rate is $v=P\left(r+d\right)$ .