Chapter 9, Problem 29P

To determine

Calculate incremental benefit cost ratio.

Explanation of Solution

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

Table -1

Item	R	C
Initial cost (C)	880,000	2,900,000
Annual maintenance cost (AM)	92,000	30,000
Reduces flood damage (R)	200,000	600,000

Discount rate (i) is 8%. Time period is 20 years. Damage is prevented in the year (n1) 3, (n2) 9 and (n3) 18.

Incremental benefit cost ratio (BC) between initial alternate R and new alternate C can be calculated as follows.

$\begin{array}{c}\text{BC}=\frac{\left(\text{RC}-\text{RR}\right)\left(\frac{1}{{\left(1+\text{i}\right)}^{\text{n1}}}+\frac{1}{{\left(1+\text{i}\right)}^{\text{n2}}}+\frac{1}{{\left(1+\text{i}\right)}^{\text{n3}}}\right)}{\left(\begin{array}{l}\left(\text{CC}+\text{AMC}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)\right)\\ -\left(\text{CR}+\text{AMR}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)\right)\end{array}\right)}\\ =\frac{\left(600,000-200,000\right)\left(\frac{1}{{\left(1+\text{0}\text{.08}\right)}^{\text{3}}}+\frac{1}{{\left(1+\text{0}\text{.08}\right)}^{\text{9}}}+\frac{1}{{\left(1+\text{0}\text{.08}\right)}^{\text{18}}}\right)}{\left(\begin{array}{l}\left(\text{2,900,000}+\text{30,000}\left(\frac{{\left(1+\text{0}\text{.08}\right)}^{\text{20}}-1}{\text{0}\text{.08}{\left(1+\text{0}\text{.08}\right)}^{\text{20}}}\right)\right)\\ -\left(\text{880,000}+\text{92,000}\left(\frac{{\left(1+\text{0}\text{.08}\right)}^{\text{20}}-1}{\text{0}\text{.08}{\left(1+\text{0}\text{.08}\right)}^{\text{20}}}\right)\right)\end{array}\right)}\\ =\frac{400,000\left(\frac{1}{1.259712}+\frac{1}{1.999005}+\frac{1}{3.996019}\right)}{\left(\begin{array}{l}\left(\text{2,900,000}+\text{30,000}\left(\frac{4.660957-1}{\text{0}\text{.08}\left(4.660957\right)}\right)\right)\\ -\left(\text{880,000}+\text{92,000}\left(\frac{4.660957-1}{\text{0}\text{.08}\left(4.660957\right)}\right)\right)\end{array}\right)}\\ =\frac{400,000\left(0.793832+0.500249+0.250249\right)}{\left(\begin{array}{l}\left(\text{2,900,000}+\text{30,000}\left(\frac{3.660957}{0.372877}\right)\right)\\ -\left(\text{880,000}+\text{92,000}\left(\frac{3.660957}{0.372877}\right)\right)\end{array}\right)}\\ =\frac{400,000\left(1.54433\right)}{\left(\begin{array}{l}\left(\text{2,900,000}+\text{30,000}\left(9.818136\right)\right)\\ -\left(\text{880,000}+\text{92,000}\left(9.818136\right)\right)\end{array}\right)}\\ =\frac{617,732}{\left(\left(\text{2,900,000}+294,544.08\right)-\left(\text{880,000}+903,268.51\right)\right)}\\ =\frac{617,732}{\left(3,194,544.08-\left(1,783268.51\right)\right)}\\ =\frac{617,732}{1,411,275.57}\\ =0.44\end{array}$

Benefit cost ratio is 0.44. Since the incremental cost benefit analysis between the initial alternative R and new alternative C is less than 1, retain the initial alternate R.