Chapter 9, Problem 33P

To determine

Calculate the incremental benefit–cost ratio.

Explanation of Solution

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

Table-1

Item	1	2
Initial cost (C)	1,200,000	2,000,000
Annual maintenance cost (AM)	80,000	75,000
Annual benefit (B)	520,000	580,000
Annual disbenefit (DB)	90,000	140,000
Time period (n)	10	20

The interest rate (i) is 8%.

The alternatives are ranked based on the initial cost of the alternate. The first is do nothing, the second is alternate 1, and the third is alternate 2.

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

$\begin{array}{c}\text{BC}=\frac{\text{B1}-\text{DB1}}{\text{C1}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n1}}}{{\left(1+\text{i}\right)}^{\text{n1}}-1}\right)+\text{AM1}}\\ =\frac{\text{520,000}-\text{90,000}}{\text{1,200,000}\left(\frac{\text{0}\text{.08}{\left(1+\text{0}\text{.08}\right)}^{\text{10}}}{{\left(1+\text{0}\text{.08}\right)}^{\text{10}}-1}\right)+\text{80,000}}\\ =\frac{430,000}{\text{1,200,000}\left(\frac{\text{0}\text{.08}\left(2.158925\right)}{2.158925-1}\right)+\text{80,000}}\\ =\frac{430,000}{\text{1,200,000}\left(\frac{0.172714}{1.158925}\right)+\text{80,000}}\\ =\frac{430,000}{\text{1,200,000}\left(0.149029\right)+\text{80,000}}\\ =\frac{430,000}{178,834.8+\text{80,000}}\\ =\frac{430,000}{258,834.8}\\ =1.66\end{array}$

The incremental benefit–cost ratio is 1.66. 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:

$\begin{array}{c}\text{BC}=\frac{\left(\text{B2}-\text{B1}\right)-\left(\text{DB2}-\text{DB1}\right)}{\left(\text{C2}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n2}}}{{\left(1+\text{i}\right)}^{\text{n2}}-1}\right)+\text{AM2}\right)-\text{C1}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n1}}}{{\left(1+\text{i}\right)}^{\text{n1}}-1}\right)+\text{AM1}}\\ =\frac{\left(\text{580,000}-\text{520,000}\right)-\left(\text{140,000}-\text{90,000}\right)}{\left(\text{2,000,000}\left(\frac{\text{0}\text{.08}{\left(1+\text{0}\text{.08}\right)}^{\text{20}}}{{\left(1+\text{0}\text{.08}\right)}^{\text{20}}-1}\right)+\text{75,000}\right)-\left(\text{1,200,000}\left(\frac{\text{0}\text{.08}{\left(1+\text{0}\text{.08}\right)}^{\text{10}}}{{\left(1+\text{0}\text{.08}\right)}^{\text{10}}-1}\right)+\text{80,000}\right)}\\ =\frac{60,000-50,000}{\left(\text{2,000,000}\left(\frac{\text{0}\text{.08}\left(4.660957\right)}{4.660957-1}\right)+\text{75,000}\right)-\left(\text{1,200,000}\left(\frac{\text{0}\text{.08}\left(2.158925\right)}{2.158925-1}\right)+\text{80,000}\right)}\\ =\frac{10,000}{\left(\text{2,000,000}\left(\frac{0.372877}{3.660957}\right)+\text{75,000}\right)-\left(\text{1,200,000}\left(\frac{0.172714}{1.158925}\right)+\text{80,000}\right)}\\ =\frac{10,000}{\left(\text{2,000,000}\left(0.101852\right)+\text{75,000}\right)-\left(\text{1,200,000}\left(0.149029\right)+\text{80,000}\right)}\\ =\frac{10,000}{\left(203,704+\text{75,000}\right)-\left(178,834.8+\text{80,000}\right)}\\ =\frac{10,000}{2798,704-258,834.8}\\ =\frac{10,000}{2798,704-258,834.8}\\ =\frac{10,000}{19,869.2}\\ =0.503\end{array}$

The incremental benefit–cost ratio is 0.503. Since the incremental benefit–cost ratio is less than 1, select the initial alternate 1 and eliminate the new alternate 2.