Chapter 3, Problem 71APQ

To determine

Calculate the equivalent annual value.

Explanation of Solution

Future value (F) is $50,000. Cash flow starts from three years and deposit five consecutive years. Rate of return (i) is 20%. Time period 1 (n1) is five years and time period (n) is eight years.

Option (a):

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

$\begin{array}{c}\text{A}=\frac{\text{F}}{\left(\frac{{\left(1+\text{i}\right)}^{\text{n1}}-1}{\text{i}}\right){\left(1+\text{i}\right)}^{\text{n}}}\\ \text{1,565}=\frac{\text{50,000}}{\left(\frac{{\left(1+\text{0}\text{.2}\right)}^{\text{5}}-1}{\text{0}\text{.2}}\right){\left(1+\text{0}\text{.2}\right)}^{\text{8}}}\\ \text{1,565}=\frac{\text{50,000}}{\left(\frac{2.48832-1}{\text{0}\text{.2}}\right)\left(4.29982\right)}\\ \text{1,565}=\frac{\text{50,000}}{\left(7.4416\right)\left(4.29982\right)}\\ \text{1,565}=\frac{\text{50,000}}{31.9975}\\ \text{1,565}\approx 1,562.62\end{array}$

The calculated equivalent annual value is equal to given value. Thus, option ‘a’ is correct.

Option (b):

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

$\begin{array}{c}\text{A}=\frac{\text{F}}{\left(\frac{{\left(1+\text{i}\right)}^{\text{n1}}-1}{\text{i}}\right){\left(1+\text{i}\right)}^{\text{n}}}\\ \text{1,759}=\frac{\text{50,000}}{\left(\frac{{\left(1+\text{0}\text{.2}\right)}^{\text{5}}-1}{\text{0}\text{.2}}\right){\left(1+\text{0}\text{.2}\right)}^{\text{8}}}\\ \text{1,759}=\frac{\text{50,000}}{\left(\frac{2.48832-1}{\text{0}\text{.2}}\right)\left(4.29982\right)}\\ \text{1,759}=\frac{\text{50,000}}{\left(7.4416\right)\left(4.29982\right)}\\ \text{1,759}=\frac{\text{50,000}}{31.9975}\\ \text{1,759}>1,562.62\end{array}$

The calculated equivalent annual value is less than given value. Thus, option ‘b’ is incorrect.

Option (c):

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

$\begin{array}{c}\text{A}=\frac{\text{F}}{\left(\frac{{\left(1+\text{i}\right)}^{\text{n1}}-1}{\text{i}}\right){\left(1+\text{i}\right)}^{\text{n}}}\\ \text{1,893}=\frac{\text{50,000}}{\left(\frac{{\left(1+\text{0}\text{.2}\right)}^{\text{5}}-1}{\text{0}\text{.2}}\right){\left(1+\text{0}\text{.2}\right)}^{\text{8}}}\\ \text{1,893}=\frac{\text{50,000}}{\left(\frac{2.48832-1}{\text{0}\text{.2}}\right)\left(4.29982\right)}\\ \text{1,893}=\frac{\text{50,000}}{\left(7.4416\right)\left(4.29982\right)}\\ \text{1,893}=\frac{\text{50,000}}{31.9975}\\ \text{1,893}>1,562.62\end{array}$

The calculated equivalent annual value is less than given value. Thus, option ‘c’ is incorrect.

Option (d):

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

$\begin{array}{c}\text{A}=\frac{\text{F}}{\left(\frac{{\left(1+\text{i}\right)}^{\text{n1}}-1}{\text{i}}\right){\left(1+\text{i}\right)}^{\text{n}}}\\ \text{2,093}=\frac{\text{50,000}}{\left(\frac{{\left(1+\text{0}\text{.2}\right)}^{\text{5}}-1}{\text{0}\text{.2}}\right){\left(1+\text{0}\text{.2}\right)}^{\text{8}}}\\ \text{2,093}=\frac{\text{50,000}}{\left(\frac{2.48832-1}{\text{0}\text{.2}}\right)\left(4.29982\right)}\\ \text{2,093}=\frac{\text{50,000}}{\left(7.4416\right)\left(4.29982\right)}\\ \text{2,093}=\frac{\text{50,000}}{31.9975}\\ \text{2,093}>1,562.62\end{array}$

The calculated equivalent annual value is less than given value. Thus, option ‘d’ is incorrect.