Chapter 2, Problem 76APQ

To determine

Calculate the equivalent annual value.

Explanation of Solution

First year cash flow (CF₁) is $800. Cash flow decreases (G) by 100 per year per year. Interest rate (i) is 8%. Time period (n) is 6 years.

Option (c):

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

$\begin{array}{c}\text{A}={\text{CF}}_1-\text{G}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{572}=800-\text{100}\left(\frac{1}{\text{0}\text{.08}}-\frac{\text{6}}{{\left(1+\text{0}\text{.08}\right)}^{\text{6}}-1}\right)\\ \text{572}=800-\text{100}\left(12.5-\frac{\text{6}}{1.5869-1}\right)\\ \text{572}=800-\text{100}\left(12.5-10.2232\right)\\ \text{572}=800-\text{100}\left(2.2768\right)\\ \text{572}=800-227.68\\ \text{572}\approx 572.32\end{array}$

Since the calculated equivalent annual value is nearly equal to the given value, Option (c) is correct.

Option (a):

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

$\begin{array}{c}\text{A}={\text{CF}}_1-\text{G}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{302}=800-\text{100}\left(\frac{1}{\text{0}\text{.08}}-\frac{\text{6}}{{\left(1+\text{0}\text{.08}\right)}^{\text{6}}-1}\right)\\ \text{302}=800-\text{100}\left(12.5-\frac{\text{6}}{1.5869-1}\right)\\ \text{302}=800-\text{100}\left(12.5-10.2232\right)\\ \text{302}=800-\text{100}\left(2.2768\right)\\ \text{302}=800-227.68\\ \text{302}<572.32\end{array}$

Since the calculated equivalent annual value is greater than the given value, option ‘a’ is incorrect.

Option (b):

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

$\begin{array}{c}\text{A}={\text{CF}}_1-\text{G}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{421}=800-\text{100}\left(\frac{1}{\text{0}\text{.08}}-\frac{\text{6}}{{\left(1+\text{0}\text{.08}\right)}^{\text{6}}-1}\right)\\ \text{421}=800-\text{100}\left(12.5-\frac{\text{6}}{1.5869-1}\right)\\ \text{421}=800-\text{100}\left(12.5-10.2232\right)\\ \text{421}=800-\text{100}\left(2.2768\right)\\ \text{421}=800-227.68\\ \text{421}<572.32\end{array}$

Since the calculated equivalent annual value is greater than the given value, option ‘b’ is incorrect.

Option (d):

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

$\begin{array}{c}\text{A}={\text{CF}}_1-\text{G}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{824}=800-\text{100}\left(\frac{1}{\text{0}\text{.08}}-\frac{\text{6}}{{\left(1+\text{0}\text{.08}\right)}^{\text{6}}-1}\right)\\ \text{824}=800-\text{100}\left(12.5-\frac{\text{6}}{1.5869-1}\right)\\ \text{824}=800-\text{100}\left(12.5-10.2232\right)\\ \text{824}=800-\text{100}\left(2.2768\right)\\ \text{824}=800-227.68\\ \text{824}>572.32\end{array}$

Since the calculated equivalent annual value is less than the given value, option ‘c’ is incorrect.