Chapter 7, Problem 18P

(a)

To determine

Calculate the rate of return.

Explanation of Solution

Installation cost is (IN) $4,970,000. Saving per year (A) is $1,300,000. Time period is 10.

Rate of return (i) can be calculated as follows:

$\begin{array}{c}\text{S}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)=\text{IN}\\ \text{1,300,000}\left(\frac{{\left(1+\text{i}\right)}^{\text{10}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{10}}}\right)=\text{4,970,000}\\ \left(\frac{{\left(1+\text{i}\right)}^{\text{10}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{10}}}\right)=\frac{\text{4,970,000}}{\text{1,300,000}}\\ \left(\frac{{\left(1+\text{i}\right)}^{\text{10}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{10}}}\right)=\text{3}\text{.8231}\end{array}$

Substitute the rate of return as 22% by trial and error method in the above Equation.

$\begin{array}{c}\left(\frac{{\left(1+\text{0}\text{.22}\right)}^{\text{10}}-1}{\text{0}\text{.22}{\left(1+\text{0}\text{.22}\right)}^{\text{10}}}\right)=\text{3}\text{.8231}\\ \left(\frac{7.304631-1}{\text{0}\text{.22}\left(7.304631\right)}\right)=\text{3}\text{.8231}\\ \left(\frac{6.304631}{1.607019}\right)=\text{3}\text{.8231}\\ 3.9231>\text{3}\text{.8231}\end{array}$

When the rate of return is substituted as 22%, the calculated value is greater than the 3.8231. Thus, the rate of return increases to 22.8%.

$\begin{array}{c}\left(\frac{{\left(1+\text{0}\text{.228}\right)}^{\text{10}}-1}{\text{0}\text{.228}{\left(1+\text{0}\text{.228}\right)}^{\text{10}}}\right)=\text{3}\text{.8231}\\ \left(\frac{7.798008-1}{\text{0}\text{.228}\left(7.798008\right)}\right)=\text{3}\text{.8231}\\ \left(\frac{6.798008}{1.777946}\right)=\text{3}\text{.8231}\\ 3.8235\approx \text{3}\text{.8231}\end{array}$

The calculated value is nearly equal to initial cost with rate of return 22.8%. Thus, it is confirmed that the rate of return is 22.8%.

The interest rate can be calculated through spreadsheet function that given below:

=RATE(10,1300000,-4970000)

The above functions gives the value of 22.8%.

(b)

To determine

Calculate the rate of return.

Explanation of Solution

Length of the G (G) is 113. Cost of G (PG) per unit is $72,000. Saving per year (A) is $1,100,000. Time period is 10.

Rate of return (i) can be calculated as follows:

$\begin{array}{c}\text{S}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)=\text{G}\times \text{PG}\\ \text{1,100,000}\left(\frac{{\left(1+\text{i}\right)}^{\text{10}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{10}}}\right)=\text{113}\times \text{72,000}\\ \left(\frac{{\left(1+\text{i}\right)}^{\text{10}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{10}}}\right)=\frac{\text{8,136,000}}{\text{1,100,000}}\\ \left(\frac{{\left(1+\text{i}\right)}^{\text{10}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{10}}}\right)=\text{7}\text{.3964}\end{array}$

Substitute the rate of return as 6% by trial and error method in the above Equation.

$\begin{array}{c}\left(\frac{{\left(1+\text{0}\text{.06}\right)}^{\text{10}}-1}{\text{0}\text{.06}{\left(1+\text{0}\text{.06}\right)}^{\text{10}}}\right)=\text{7}\text{.3964}\\ \left(\frac{1.790848-1}{\text{0}\text{.06}\left(1.790848\right)}\right)=\text{7}\text{.3964}\\ \left(\frac{0.790848}{0.107451}\right)=\text{7}\text{.3964}\\ 7.3601<\text{7}\text{.3964}\end{array}$

When the rate of return is substituted as 6%, the calculated value is less than the 7.3964. Thus, the rate of return decreases to 5.896%.

$\begin{array}{c}\left(\frac{{\left(1+\text{0}\text{.05896}\right)}^{\text{10}}-1}{\text{0}\text{.05896}{\left(1+\text{0}\text{.05896}\right)}^{\text{10}}}\right)=\text{7}\text{.3964}\\ \left(\frac{1.773354-1}{\text{0}\text{.05896}\left(1.773354\right)}\right)=\text{7}\text{.3964}\\ \left(\frac{0.773354}{0.104557}\right)=\text{7}\text{.3964}\\ 7.3965<\text{7}\text{.3964}\end{array}$

The calculated value is nearly equal to the value 7.3964t with rate of return 5.896%. Thus, it is confirmed that the rate of return is 5.896%.

The interest rate can be calculated through spreadsheet function that given below:

= RATE(10,1100000,-72000*113)

The above functions gives the value of 5.896%.