Chapter 12, Problem 31P

(a):

To determine

Calculate the internal rate of return (ir).

Explanation of Solution

Table 1 shows the project cash flow:

Table 1

Alternate	X	Y	Z	A	B
Initial cost (FC)	-30,000	-15,000	-45,000	-70,000	-40,000
NCF	9,000	-4,900	11,100	19,000	10,000
Life time (n)	10	10	10	10	10

MARR (i) is 15%.

The internal rate of return can be calculated using the following formula:

$\text{FC}=\text{NCF}\left(\frac{{\left(1+\text{ir}\right)}^{\text{n}}-1}{\text{ir}{\left(1+\text{ir}\right)}^{\text{n}}}\right)$        (1)

The internal rate of return (ir) can be calculated using the following formula:

$30,000=9,000\left(\frac{{\left(1+\text{ir}\right)}^{\text{10}}-1}{\text{ir}{\left(1+\text{ir}\right)}^{\text{10}}}\right)$

Substitute ir as 0.27 by trial and error method in the above calculation.   

$\begin{array}{l}30,000=9,000\left(\frac{{\left(1+\text{0}\text{.27}\right)}^{10}-1}{\text{0}\text{.27}{\left(1+0.27\right)}^{\text{10}}}\right)\\ 30,000=9,000\left(\frac{10.9153-1}{\text{0}\text{.27}\left(10.9153\right)}\right)\\ 30,000=9,000\left(\frac{9.9153}{2.9471}\right)\\ 30,000=9,000\left(3.3644\right)\\ 30,000<30,279.6\end{array}$

The calculated value is greater than the initial cost. Thus, increase the ir value to 0.273.

$\begin{array}{l}30,000=9,000\left(\frac{{\left(1+\text{0}\text{.273}\right)}^{10}-1}{\text{0}\text{.273}{\left(1+0.273\right)}^{\text{10}}}\right)\\ 30,000=9,000\left(\frac{11.1759-1}{\text{0}\text{.273}\left(11.1759\right)}\right)\\ 30,000=9,000\left(\frac{10.1759}{3.05102}\right)\\ 30,000=9,000\left(3.3352\right)\\ 30,000\approx 30,016.8\end{array}$

Since the calculated value is equal to the initial cost, the value of ir is confirmed as 27.3%.

Table 2 shows the internal rate of return value for different alternatives that are obtained using Equation (1). The alternates are raked based on the internal rate of return as follows:

Table 2

Alternate	ir
Y	30.4%
X	27.3%
A	24%
B	21.4%
Z	21%

Since there is an investment limit, alternates Y, X, and A can be selected.

The internal rate of return can be calculated using the spreadsheet as follows:

(b):

To determine

The overall rate of return (ROR).

Explanation of Solution

After investing in alternate Y, X, and A, the remaining money (RM) is $15,000$\left(120,000-115,000\right)$. The overall rate of return can be calculated as follows:

$\begin{array}{c}\text{ROR}=\frac{\left({\text{FC}}_{\text{Y}}\times {\text{ir}}_{\text{Y}}\right)+\left({\text{FC}}_{\text{X}}\times {\text{ir}}_{\text{X}}\right)+\left({\text{FC}}_{\text{A}}\times {\text{ir}}_{\text{A}}\right)+\left(\text{RM}\times \text{i}\right)}{{\text{FC}}_{\text{Y}}+{\text{FC}}_{\text{X}}{\text{FC}}_{\text{A}}+\text{RM}}\\ =\frac{\left(\text{15,000}\times 0.304\right)+\left(\text{30,000}\times \text{0}\text{.273}\right)+\left(\text{70,000}\times 0.\text{24}\right)+\left(\text{5,000}\times 0.15\right)}{15,000+30,000+70,000+5,000}\\ =\frac{4,560+8,190+16,800+750}{120,000}\\ =0.2525\end{array}$

The overall rate of return is 25.25%.

The overall rate of return can be calculated using the spreadsheet as follows: