Chapter 10, Problem 16P

To determine

(a)

The joint probability distribution for savings and useful life.

Answer to Problem 16P

The following table shows the joint probability.

Savings per year	Useful life $\left(\text{years}\right)$	Joint Probability
$\$18,000$	$12$	$0.033$
$\$20,000$	$12$	$0.1169$
$\$22,000$	$12$	$0.0167$
$\$18,000$	$5$	$0.133$
$\$20,000$	$5$	$0.466$
$\$22,000$	$5$	$0.066$
$\$18,000$	$4$	$0.033$
$\$20,000$	$4$	$0.1169$
$\$22,000$	$4$	$0.0167$

Explanation of Solution

Calculation:

Calculate the joint probability.

Savings per year $\left(a\right)$	Probability $\left(b\right)$	Useful life $\left(\text{years}\right)$ $\left(c\right)$	Probability $\left(d\right)$	Joint Probability $\begin{array}{c}\left(e\right)\\ \left(b\times d\right)\end{array}$
$\$18,000$	$0.2$	$12$	$0.167$	$0.033$
$\$20,000$	$0.7$	$12$	$0.167$	$0.1169$
$\$22,000$	$0.1$	$12$	$0.167$	$0.0167$	Optimistic
$\$18,000$	$0.2$	$5$	$0.67$	$0.133$
$\$20,000$	$0.7$	$5$	$0.67$	$0.466$	Most likely
$\$22,000$	$0.1$	$5$	$0.67$	$0.066$
$\$18,000$	$0.2$	$4$	$0.167$	$0.033$	Pessimistic
$\$20,000$	$0.7$	$4$	$0.167$	$0.1169$
$\$22,000$	$0.1$	$4$	$0.167$	$0.0167$
Total				$1.00$

Conclusion:

The joint probability is shown in table.

To determine

(b)

The rate of return for optimistic, pessimistic and most likely estimates.

Answer to Problem 16P

Rate of return for optimistic scenario is $25.4\%$.

Rate of return for most likely scenario is $7.47\%$.

Rate of return for pessimistic scenario is $-4.55\%$.

Explanation of Solution

Calculation:

Write the formula to calculate the rate of return.

$\text{Cost}=\text{Savings}\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)$         ...... (I)

Here, the rate of interest is i and the time period is n.

Calculate the rate of return for optimistic scenario.

Substitute $\$81,000$ for cost, $\$22,000$ for savings and $12$ years for n in Equation (I).

$\begin{array}{l}\$81,000=\$22,000\left(\frac{{\left(1+i\right)}^{12}-1}{i{\left(1+i\right)}^{12}}\right)\\ i=25.4\%\end{array}$

Calculate the rate of return for most likely scenario.

Substitute $\$81,000$ for cost, $\$20,000$ for savings and $5$ years for n in Equation (I).

$\begin{array}{l}\$81,000=\$20,000\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ i=7.47\%\end{array}$

Calculate the rate of return for pessimistic scenario.

Substitute $\$81,000$ for cost, $\$18,000$ for savings and $4$ years for n in Equation (I).

$\begin{array}{l}\$81,000=\$18,000\left(\frac{{\left(1+i\right)}^4-1}{i{\left(1+i\right)}^4}\right)\\ i=-4.55\%\end{array}$.

Conclusion:

Thus, the rate of return for optimistic scenario is $25.4\%$.

Rate of return for most likely scenario is $7.47\%$.

Rate of return for pessimistic scenario is $-4.55\%$.