Chapter 9, Problem 70APQ

To determine

Benefit–cost ratio.

Explanation of Solution

Benefit (B) is $10,000 per year forever. Cost (C1) is $50,000 at year zero and (C2) $50,000 at the end of year (n) 2. Interest rate (i) is 10%.

Option (b):

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

$\begin{array}{l}\text{BC}=\frac{\frac{\text{B}}{\text{i}}}{\left(\text{C1}+\left(\frac{\text{C2}}{{\left(1+\text{i}\right)}^{\text{n}}}\right)\right)}\\ \text{1}\text{.1}=\frac{\frac{\text{10,000}}{\text{0}\text{.1}}}{\left(\text{50,000}+\left(\frac{\text{50,000}}{{\left(1+\text{0}\text{.1}\right)}^{\text{2}}}\right)\right)}\\ \text{1}\text{.1}=\frac{\text{100,000}}{\left(\text{50,000}+\left(\frac{\text{50,000}}{1.21}\right)\right)}\\ \text{1}\text{.1}=\frac{\text{100,000}}{\left(\text{50,000}+41,322.31\right)}\\ \text{1}\text{.1}=\frac{\text{100,000}}{\left(91,322.31\right)}\\ \text{1}\text{.1}\approx 1.095\end{array}$

The calculated BC value is nearly equal to the given value. Thus, option ‘b’ is correct.

Option (a):

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

$\begin{array}{l}\text{BC}=\frac{\frac{\text{B}}{\text{i}}}{\left(\text{C1}+\left(\frac{\text{C2}}{{\left(1+\text{i}\right)}^{\text{n}}}\right)\right)}\\ \text{0}\text{.93}=\frac{\frac{\text{10,000}}{\text{0}\text{.1}}}{\left(\text{50,000}+\left(\frac{\text{50,000}}{{\left(1+\text{0}\text{.1}\right)}^{\text{2}}}\right)\right)}\\ \text{0}\text{.93}=\frac{\text{100,000}}{\left(\text{50,000}+\left(\frac{\text{50,000}}{1.21}\right)\right)}\\ \text{0}\text{.93}=\frac{\text{100,000}}{\left(\text{50,000}+41,322.31\right)}\\ \text{0}\text{.93}=\frac{\text{100,000}}{\left(91,322.31\right)}\\ 0.93<1.095\end{array}$

The calculated BC value is greater than the given value. Thus, option ‘a’ is incorrect.

Option (c):

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

$\begin{array}{l}\text{BC}=\frac{\frac{\text{B}}{\text{i}}}{\left(\text{C1}+\left(\frac{\text{C2}}{{\left(1+\text{i}\right)}^{\text{n}}}\right)\right)}\\ \text{1}\text{.24}=\frac{\frac{\text{10,000}}{\text{0}\text{.1}}}{\left(\text{50,000}+\left(\frac{\text{50,000}}{{\left(1+\text{0}\text{.1}\right)}^{\text{2}}}\right)\right)}\\ \text{1}\text{.24}=\frac{\text{100,000}}{\left(\text{50,000}+\left(\frac{\text{50,000}}{1.21}\right)\right)}\\ 1.24=\frac{\text{100,000}}{\left(\text{50,000}+41,322.31\right)}\\ \text{1}\text{.24}=\frac{\text{100,000}}{\left(91,322.31\right)}\\ 1.24>1.095\end{array}$

The calculated BC value is less than the given value. Thus, option ‘c’ is incorrect.

Option (d):

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

$\begin{array}{l}\text{BC}=\frac{\frac{\text{B}}{\text{i}}}{\left(\text{C1}+\left(\frac{\text{C2}}{{\left(1+\text{i}\right)}^{\text{n}}}\right)\right)}\\ \text{1}\text{.73}=\frac{\frac{\text{10,000}}{\text{0}\text{.1}}}{\left(\text{50,000}+\left(\frac{\text{50,000}}{{\left(1+\text{0}\text{.1}\right)}^{\text{2}}}\right)\right)}\\ \text{1}\text{.73}=\frac{\text{100,000}}{\left(\text{50,000}+\left(\frac{\text{50,000}}{1.21}\right)\right)}\\ \text{1}\text{.73}=\frac{\text{100,000}}{\left(\text{50,000}+41,322.31\right)}\\ \text{1}\text{.73}=\frac{\text{100,000}}{\left(91,322.31\right)}\\ 1.73>1.095\end{array}$

The calculated BC value is less than the given value. Thus, option ‘d’ is incorrect.