Chapter 2, Problem 73APQ

To determine

Calculate the interest rate.

Explanation of Solution

First year operating cost (FC₁) is $26,000 per year. Operation cost increases (G) by $1,500 per year. Time period (n) is 5 years. Interest rate (i) is 8%.

Option (b):

Equivalent annual value (A) can be calculated as follows:

$\begin{array}{c}\text{A}={\text{CF}}_1+\text{G}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{28,770}=\text{26,000}+\text{1,500}\left(\frac{1}{\text{0}\text{.08}}-\frac{\text{5}}{{\left(1+\text{0}\text{.08}\right)}^{\text{5}}-1}\right)\\ \text{28,770}=\text{26,000}+\text{1,500}\left(12.5-\frac{\text{5}}{1.4693-1}\right)\\ \text{28,770}=\text{26,000}+\text{1,500}\left(12.5-10.6542\right)\\ \text{28,770}=\text{26,000}+\text{1,500}\left(1.8458\right)\\ \text{28,770}=\text{26,000}+2,768.7\\ 28,770\approx 28,768.7\end{array}$

Since the calculated equivalent annual value is nearly equal to actual equivalent annual value, option ‘b’ is correct.

Option (a):

Equivalent annual value (A) can be calculated as follows:

$\begin{array}{c}\text{A}={\text{CF}}_1+\text{G}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{30,850}=\text{26,000}+\text{1,500}\left(\frac{1}{\text{0}\text{.08}}-\frac{\text{5}}{{\left(1+\text{0}\text{.08}\right)}^{\text{5}}-1}\right)\\ \text{30,850}=\text{26,000}+\text{1,500}\left(12.5-\frac{\text{5}}{1.4693-1}\right)\\ \text{30,850}=\text{26,000}+\text{1,500}\left(12.5-10.6542\right)\\ \text{30,850}=\text{26,000}+\text{1,500}\left(1.8458\right)\\ \text{30,850}=\text{26,000}+2,768.7\\ \text{30,850}>28,768.7\end{array}$

Since the calculated equivalent annual value is less than the actual equivalent annual value, option ‘a’ is incorrect.

Option (c):

Equivalent annual value (A) can be calculated as follows:

$\begin{array}{c}\text{A}={\text{CF}}_1+\text{G}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{26,930}=\text{26,000}+\text{1,500}\left(\frac{1}{\text{0}\text{.08}}-\frac{\text{5}}{{\left(1+\text{0}\text{.08}\right)}^{\text{5}}-1}\right)\\ \text{26,930}=\text{26,000}+\text{1,500}\left(12.5-\frac{\text{5}}{1.4693-1}\right)\\ \text{26,930}=\text{26,000}+\text{1,500}\left(12.5-10.6542\right)\\ \text{26,930}=\text{26,000}+\text{1,500}\left(1.8458\right)\\ \text{26,930}=\text{26,000}+2,768.7\\ \text{26,930}<28,768.7\end{array}$

Since the calculated equivalent annual value is greater than the actual equivalent annual value, option ‘c’ is incorrect.

Option (d):

Equivalent annual value (A) can be calculated as follows:

$\begin{array}{c}\text{A}={\text{CF}}_1+\text{G}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{23,670}=\text{26,000}+\text{1,500}\left(\frac{1}{\text{0}\text{.08}}-\frac{\text{5}}{{\left(1+\text{0}\text{.08}\right)}^{\text{5}}-1}\right)\\ \text{23,670}=\text{26,000}+\text{1,500}\left(12.5-\frac{\text{5}}{1.4693-1}\right)\\ \text{23,670}=\text{26,000}+\text{1,500}\left(12.5-10.6542\right)\\ \text{23,670}=\text{26,000}+\text{1,500}\left(1.8458\right)\\ \text{23,670}=\text{26,000}+2,768.7\\ \text{23,670}<28,768.7\end{array}$

Since the calculated equivalent annual value is greater than the actual equivalent annual value, option ‘d’ is incorrect.