Chapter 7, Problem 17P

To determine

Calculate the rate of return.

Explanation of Solution

Initial cost (C) is $130,000. Repayment in year 1 (R1) is 78,000 and increases (I) by $1,000 per year thereafter. Equivalent annual cost (AC) is $49,000. Time period (1) is 8.

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

$\begin{array}{c}\text{C}=\left(\begin{array}{l}-\text{AC}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)+\text{R1}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)\\ +\text{I}\times \frac{1}{\text{i}}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}}\right)+\frac{\text{SV}}{{\left(1+\text{i}\right)}^{\text{n}}}\end{array}\right)\\ \text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(\frac{{\left(1+\text{i}\right)}^{\text{8}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{8}}}\right)+\text{78,000}\left(\frac{{\left(1+\text{i}\right)}^{\text{8}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{8}}}\right)\\ +\text{1,000}\times \frac{1}{\text{i}}\left(\frac{{\left(1+\text{i}\right)}^{\text{8}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{8}}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{8}}}\right)+\frac{23,000}{{\left(1+\text{i}\right)}^{\text{8}}}\end{array}\right)\end{array}$

Substitute the rate of return as 19% by trial and error method in the above Equation.

$\begin{array}{l}\text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(\frac{{\left(1+\text{0}\text{.19}\right)}^{\text{8}}-1}{\text{0}\text{.19}{\left(1+\text{0}\text{.19}\right)}^{\text{8}}}\right)+\text{78,000}\left(\frac{{\left(1+\text{0}\text{.19}\right)}^{\text{8}}-1}{\text{0}\text{.19}{\left(1+\text{0}\text{.19}\right)}^{\text{8}}}\right)\\ +\text{1,000}\times \frac{1}{0.19}\left(\frac{{\left(1+\text{0}\text{.19}\right)}^{\text{8}}-1}{\text{0}\text{.19}{\left(1+\text{0}\text{.19}\right)}^{\text{8}}}-\frac{\text{8}}{{\left(1+\text{0}\text{.19}\right)}^{\text{8}}}\right)+\frac{23,000}{{\left(1+\text{0}\text{.19}\right)}^{\text{8}}}\end{array}\right)\\ \text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(\frac{4.021385-1}{\text{0}\text{.19}\left(4.021385\right)}\right)+\text{78,000}\left(\frac{4.021385-1}{\text{0}\text{.19}\left(4.021385\right)}\right)\\ +\text{1,000}\times 5.263158\left(\frac{4.021385-1}{\text{0}\text{.19}\left(4.021385\right)}-\frac{\text{8}}{4.021385}\right)+\frac{23,000}{4.021385}\end{array}\right)\\ \text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(\frac{3.021385}{0.764063}\right)+\text{78,000}\left(\frac{3.021385}{0.764063}\right)\\ +5,268.158\left(\frac{3.021385}{0.764063}-1.989364\right)+5,719.42\end{array}\right)\\ \text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(3.954573\right)+\text{78,000}\left(3.954573\right)\\ +5,268.158\left(3.954573-1.989364\right)+5,719.42\end{array}\right)\\ \text{130,000}=\left(\begin{array}{l}-193,774.08+308,456.69\\ +\text{1,000}\times 5.263158\left(1.965209\right)+5,719.42\end{array}\right)\\ \text{130,000}=\left(-193,774.08+308,456.69+10,343.21+5,719.42\right)\\ \text{130,000}=130,745.24\end{array}$

When the rate of return is substituted as 19%, the calculated value is greater than the initial cost. Thus, the rate of return increases to 19.17%.

$\begin{array}{l}\text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(\frac{{\left(1+\text{0}\text{.1917}\right)}^{\text{8}}-1}{\text{0}\text{.1917}{\left(1+\text{0}\text{.1917}\right)}^{\text{8}}}\right)+\text{78,000}\left(\frac{{\left(1+\text{0}\text{.1917}\right)}^{\text{8}}-1}{\text{0}\text{.1917}{\left(1+\text{0}\text{.1917}\right)}^{\text{8}}}\right)\\ +\text{1,000}\times \frac{1}{0.1917}\left(\frac{{\left(1+\text{0}\text{.1917}\right)}^{\text{8}}-1}{\text{0}\text{.1917}{\left(1+\text{0}\text{.1917}\right)}^{\text{8}}}-\frac{\text{8}}{{\left(1+\text{0}\text{.1917}\right)}^{\text{8}}}\right)+\frac{23,000}{{\left(1+\text{0}\text{.1917}\right)}^{\text{8}}}\end{array}\right)\\ \text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(\frac{4.067574-1}{\text{0}\text{.1917}\left(4.067574\right)}\right)+\text{78,000}\left(\frac{4.067574-1}{\text{0}\text{.1917}\left(4.067574\right)}\right)\\ +\text{1,000}\times \frac{1}{0.1917}\left(\frac{4.067574-1}{\text{0}\text{.1917}\left(4.067574\right)}-\frac{\text{8}}{4.067574}\right)+\frac{23,000}{4.067574}\end{array}\right)\\ \text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(\frac{3.067574}{0.779754}\right)+\text{78,000}\left(\frac{3.067574}{0.779754}\right)\\ +\text{1,000}\times 5.216584\left(\frac{3.067574}{0.779754}-\frac{\text{8}}{4.067574}\right)+5,654.48\end{array}\right)\\ \text{130,000}=\left(\begin{array}{l}-\text{49,000}\left(3.934028\right)+\text{78,000}\left(3.934028\right)\\ +\text{1,000}\times 5.216584\left(3.934028-1.966734\right)+5,654.48\end{array}\right)\\ \text{130,000}=\left(-192,767.37+306,854.18+\text{1,000}\times 5.216584\left(1.967294\right)+5,654.48\right)\\ \text{130,000}=\left(-192,767.37+306,854.18+10,262.55+5,654.48\right)\\ \text{130,000}\approx \text{130,003}\text{.84}\end{array}$

The calculated value is nearly equal to initial cost with rate of return 19.17%. Thus, it is confirmed that the rate of return is 19.17%.

The spreadsheet function to calculate the rate of return is given below:

The rate of return is 19.17%.