Chapter 9, Problem 36P

(a):

To determine

Calculate the incremental benefit–cost ratio of the mutual alternative.

Explanation of Solution

Table-1 shows the cash flow of two different projects.

Table-1

Item	1	2
Initial cost (C)	60	500
Benefit (B) per month	1.25	8
Time period (n)	60	60

The discount rate (i) is 0.5%.

The incremental benefit–cost ratio (BC) between the initial alternate do nothing and the new alternate 1 can be calculated as follows:

$\begin{array}{c}\text{BC}=\frac{\text{B1}}{\text{C1}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)}\\ =\frac{\text{1}\text{.25}}{\text{60}\left(\frac{\text{0}\text{.005}{\left(1+\text{0}\text{.005}\right)}^{\text{60}}}{{\left(1+\text{0}\text{.005}\right)}^{\text{60}}-1}\right)}\\ =\frac{\text{1}\text{.25}}{\text{60}\left(\frac{\text{0}\text{.005}\left(1.34885\right)}{\left(1.34885\right)-1}\right)}\\ =\frac{\text{1}\text{.25}}{\text{60}\left(\frac{0.00674}{\left(0.34885\right)}\right)}\\ =\frac{\text{1}\text{.25}}{\text{60}\left(0.01932\right)}\\ =\frac{\text{1}\text{.25}}{1.1592}\\ =1.08\end{array}$

The incremental benefit–cost ratio is 1.08. Since the incremental benefit–cost ratio is greater than 1, select the new alternate 1 and eliminate the initial alternate do nothing.

The incremental benefit–cost ratio (BC) between the initial alternate 1 and new alternate 2 can be calculated as follows:

$\begin{array}{c}\text{BC}=\frac{\text{B2}-\text{B1}}{\left(\text{C2}-\text{C1}\right)\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)}\\ =\frac{\text{8}-\text{1}\text{.25}}{\left(\text{500}-\text{60}\right)\left(\frac{\text{0}\text{.005}{\left(1+\text{0}\text{.005}\right)}^{\text{60}}}{{\left(1+\text{0}\text{.005}\right)}^{\text{60}}-1}\right)}\\ =\frac{\text{6}\text{.75}}{\text{440}\left(\frac{\text{0}\text{.005}\left(1.34885\right)}{\left(1.34885\right)-1}\right)}\\ =\frac{\text{6}\text{.75}}{\text{440}\left(\frac{0.00674}{\left(0.34885\right)}\right)}\\ =\frac{\text{6}\text{.75}}{\text{440}\left(0.01932\right)}\\ =\frac{\text{6}\text{.75}}{8.5008}\\ =0.79\end{array}$

The incremental benefit cost ratio is 0.79. Since the incremental benefit cost ratio is less than 1, select the initial alternate 1 and eliminate the new alternate 2.

(b):

To determine

Calculate the incremental benefit–cost ratio of the independent alternative.

Explanation of Solution

The benefit–cost ratio (BC) of alternate 1 can be calculated as follows:

$\begin{array}{c}\text{BC}=\frac{\text{B1}}{\text{C1}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)}\\ =\frac{\text{1}\text{.25}}{\text{60}\left(\frac{\text{0}\text{.005}{\left(1+\text{0}\text{.005}\right)}^{\text{60}}}{{\left(1+\text{0}\text{.005}\right)}^{\text{60}}-1}\right)}\\ =\frac{\text{1}\text{.25}}{\text{60}\left(\frac{\text{0}\text{.005}\left(1.34885\right)}{\left(1.34885\right)-1}\right)}\\ =\frac{\text{1}\text{.25}}{\text{60}\left(\frac{0.00674}{\left(0.34885\right)}\right)}\\ =\frac{\text{1}\text{.25}}{\text{60}\left(0.01932\right)}\\ =\frac{\text{1}\text{.25}}{1.1592}\\ =1.08\end{array}$

The benefit–cost ratio is 1.08. Since the benefit–cost ratio is greater than 1, select the alternate 1.

The benefit–cost ratio (BC) of alternate 2 can be calculated as follows:

$\begin{array}{c}\text{BC}=\frac{\text{B2}}{\text{C2}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)}\\ =\frac{\text{8}}{\text{500}\left(\frac{\text{0}\text{.005}{\left(1+\text{0}\text{.005}\right)}^{\text{60}}}{{\left(1+\text{0}\text{.005}\right)}^{\text{60}}-1}\right)}\\ =\frac{\text{8}}{\text{500}\left(\frac{\text{0}\text{.005}\left(1.34885\right)}{\left(1.34885\right)-1}\right)}\\ =\frac{\text{8}}{\text{500}\left(\frac{0.00674}{\left(0.34885\right)}\right)}\\ =\frac{\text{8}}{\text{500}\left(0.01932\right)}\\ =\frac{\text{8}}{9.66}\\ =0.83\end{array}$

The incremental benefit–cost ratio is 0.83. Since the benefit–cost ratio is less than 1, eliminate the new alternate 2.