Chapter 10, Problem 17P

To determine

(a)

The joint probability of saving and useful life.

Answer to Problem 17P

The joint probability of saving and useful life is shown in the following table.

Yearly saving	Joint probability
$\$15000$	$0.18$
$\$30000$	$0.30$
$\$45000$	$0.12$
$\$15000$	$0.12$
$\$30000$	$0.20$
$\$45000$	$0.08$

Explanation of Solution

Given:

The first cost is $\$150000$.

The salvage value is zero.

The interest rate is $20\%$.

Calculations:

The table for joint probability of saving and useful life is given below.

Yearly saving	Probability of saving $\left(A\right)$	Useful life	Probability of useful life $\left(B\right)$	Joint probability $\left(A\times B\right)$
$\$15000$	$0.3$	$3$	$0.6$	$0.18$
$\$30000$	$0.5$	$3$	$0.6$	$0.30$
$\$45000$	$0.2$	$3$	$0.6$	$0.12$
$\$15000$	$0.3$	$5$	$0.4$	$0.12$
$\$30000$	$0.5$	$5$	$0.4$	$0.20$
$\$45000$	$0.2$	$5$	$0.4$	$0.08$

Table (1).

Conclusion:

The joint probability of saving and useful life is

Yearly saving	Joint probability
$\$15000$	$0.18$
$\$30000$	$0.30$
$\$45000$	$0.12$
$\$15000$	$0.12$
$\$30000$	$0.20$
$\$45000$	$0.08$

Table (2).

To determine

(b)

The present worth of optimistic, most likely and pessimistic scenario.

Answer to Problem 17P

The present worth for optimistic estimates is $-\$15405$.

The present worth for most likely estimates is $-\$72330$.

The present worth for optimistic estimates is $-\$118440$.

Explanation of Solution

Given:

The first cost is $\$150000$ and zero salvage value. The interest rate is $20\%$ and useful life is of $4$ years.

Concept used:

The estimates which under any circumstance can give the best possible results are called as optimistic estimates. These estimates are estimated at the best future events.

The estimates which are most probable or likely to happen are called as most likely estimates. These estimates occur at normal circumstances.

The estimates which under any circumstance can give the worst possible results are called as pessimistic estimates. These estimates are estimated at the abnormal circumstances.

Write the expression for present worth.

$PW=-O+A\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]$ ...... (I)

Here, the outflow is $O$, the annual inflow is $A$, the interest rate is $i$ and useful life is $n$.

Calculations:

The optimistic estimates are saving of $\$45000$ per year with project life of $5$ years.

The most likely estimates are saving of $\$30000$ per year with project life of $4$ years.

The pessimistic estimates are saving of $\$15000$ per year with project life of $3$ years.

Calculate the present worth for optimistic estimates.

Substitute $\$150000$ for $O$, $\$45000$ for $A$, $20\%$ for $i$ and $5$ for $n$.

$\begin{array}{c}PW=-\$150000+\$45000\left[\frac{{\left(1+0.2\right)}^5-1}{0.2{\left(1+0.2\right)}^5}\right]\\ =-\$150000+\$45000\left[\frac{{\left(1.2\right)}^5-1}{0.2{\left(1.2\right)}^5}\right]\\ =-\$150000+\$45000\left(2.991\right)\\ =-\$15405\end{array}$

The present worth for optimistic estimates is $-\$15405$.

Calculate the present worth for most likely estimates.

Substitute $\$150000$ for $O$, $\$30000$ for $A$, $20\%$ for $i$ and $4$ for $n$.

$\begin{array}{c}PW=-\$150000+\$30000\left[\frac{{\left(1+0.2\right)}^4-1}{0.2{\left(1+0.2\right)}^4}\right]\\ =-\$150000+\$30000\left[\frac{{\left(1.2\right)}^4-1}{0.2{\left(1.2\right)}^4}\right]\\ =-\$150000+\$30000\left(2.589\right)\\ =-\$72330\end{array}$

The present worth for most likely estimates is $-\$72330$.

Calculate the present worth for optimistic estimates.

Substitute $\$150000$ for $O$, $\$15000$ for $A$, $20\%$ for $i$ and $3$ for $n$.

$\begin{array}{c}PW=-\$150000+\$15000\left[\frac{{\left(1+0.2\right)}^3-1}{0.2{\left(1+0.2\right)}^3}\right]\\ =-\$150000+\$15000\left[\frac{{\left(1.2\right)}^3-1}{0.2{\left(1.2\right)}^3}\right]\\ =-\$150000+\$15000\left(2.106\right)\\ =-\$118440\end{array}$

The present worth for optimistic estimates is $-\$118440$.

Conclusion:

Thus, the present worth for optimistic estimates is $-\$15405$.

The present worth for most likely estimates is $-\$72330$.

The present worth for optimistic estimates is $-\$118440$.