Chapter 15, Problem 23P

To determine

(a)

The projects to be funded and the opportunity cost of capital.

Answer to Problem 23P

Projects $4,5\text{and}6$ should be funded.

The opportunity cost of the capital is $19.4\%$.

Explanation of Solution

Given:

Project	First cost	Annual benefits	Salvage Value	Life(years)
$1$	$\$20,000$	$\$4,000$		$20$
$2$	$\$20,000$	$\$3200$	$\$20,000$	$20$
$3$	$\$20,000$	$\$3300$	$\$10,000$	$30$
$4$	$\$20,000$	$\$4500$		$15$
$5$	$\$20,000$	$\$4500$	$-\$20,000$	$25$
$6$	$\$20,000$	$\$5800$		$10$
$7$	$\$20,000$	$\$4000$	$\$10,000$	$15$

The budget is $\$60,000$.

Concept used:

Write the formula to calculate the present worth factor.

$\left(\frac{P}{A},i,n\right)=\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}$

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

Calculate the Internal rate of return (IRR).

To determine the IRR, equate the present worth of project to zero.

$\text{PW}\text{of}\text{project}=\left[\begin{array}{l}-\left(\text{First}\text{cost}\right)+\left(\text{Benefit}\right)\left(\frac{P}{A},i,n\right)\\ +\left(\text{Salvage}\text{value}\right)\left(\frac{P}{F},i,n\right)\end{array}\right]$ ...... (I)

For project $1$.

Substitute $0$ for PW, $\$20,000$ for First cost, $\$4,000$ for Benefit, $0$ for Salvage value and $20$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$20,000+\$4,000\left(\frac{{\left(1+i\right)}^{20}-1}{i{\left(1+i\right)}^{20}}\right)\\ 5=\left(\frac{{\left(1+i\right)}^{20}-1}{i{\left(1+i\right)}^{20}}\right)\\ i=19.4\%\end{array}$

For project $2$.

Substitute $0$ for PW, $\$20,000$ for First cost, $\$3200$ for Benefit, $\$20,000$ for Salvage value and $20$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$20,000+\$3200\left(\frac{{\left(1+i\right)}^{20}-1}{i{\left(1+i\right)}^{20}}\right)+\$20,000\left(\frac{1}{{\left(1+i\right)}^{20}}\right)\\ 25=4\left(\frac{{\left(1+i\right)}^{20}-1}{i{\left(1+i\right)}^{20}}\right)+25\left(\frac{1}{{\left(1+i\right)}^{20}}\right)\\ i=16\%\end{array}$

For project $3$.

Substitute $0$ for PW, $\$20,000$ for First cost, $\$3300$ for Benefit, $\$10,000$ for Salvage value and $30$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$20,000+\$3300\left(\frac{{\left(1+i\right)}^{30}-1}{i{\left(1+i\right)}^{30}}\right)+\$10,000\left(\frac{1}{{\left(1+i\right)}^{30}}\right)\\ 200=33\left(\frac{{\left(1+i\right)}^{30}-1}{i{\left(1+i\right)}^{30}}\right)+10\left(\frac{1}{{\left(1+i\right)}^{30}}\right)\\ i=16.4\%\end{array}$

For project $4$.

Substitute $0$ for PW, $\$20,000$ for First cost, $\$4500$ for Benefit, $0$ for Salvage value and $15$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$20,000+\$4,500\left(\frac{{\left(1+i\right)}^{15}-1}{i{\left(1+i\right)}^{15}}\right)\\ 4.44=\left(\frac{{\left(1+i\right)}^{15}-1}{i{\left(1+i\right)}^{15}}\right)\\ i=21.2\%\end{array}$

For project $5$.

Substitute $0$ for PW, $\$20,000$ for First cost, $\$4500$ for Benefit, $-\$20,000$ for Salvage value and $25$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$20,000+\$4500\left(\frac{{\left(1+i\right)}^{25}-1}{i{\left(1+i\right)}^{25}}\right)-\$20,000\left(\frac{1}{{\left(1+i\right)}^{25}}\right)\\ 40=9\left(\frac{{\left(1+i\right)}^{25}-1}{i{\left(1+i\right)}^{25}}\right)-40\left(\frac{1}{{\left(1+i\right)}^{25}}\right)\\ i=22.2\%\end{array}$

For project $6$.

Substitute $0$ for PW, $\$20,000$ for First cost, $\$5800$ for Benefit, $0$ for Salvage value and $10$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$20,000+\$5800\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ 3.45=\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ i=26.16\%\end{array}$

For project $7$.

Substitute $0$ for PW, $\$20,000$ for First cost, $\$4000$ for Benefit, $\$10,000$ for Salvage value and $15$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$20,000+\$4000\left(\frac{{\left(1+i\right)}^{15}-1}{i{\left(1+i\right)}^{15}}\right)+\$10,000\left(\frac{1}{{\left(1+i\right)}^{15}}\right)\\ 5=\left(\frac{{\left(1+i\right)}^{15}-1}{i{\left(1+i\right)}^{15}}\right)+2.5\left(\frac{1}{{\left(1+i\right)}^{15}}\right)\\ i=19.26\%\end{array}$

Tabulate the results.

Project	First cost	IRR
$1$	$\$20,000$	$19.4\%$
$2$	$\$20,000$	$16\%$
$3$	$\$20,000$	$16.4\%$
$4$	$\$20,000$	$21.2\%$
$5$	$\$20,000$	$22.2\%$
$6$	$\$20,000$	$26.16\%$
$7$	$\$20,000$	$19.26\%$

Projects $4,5\text{and}6$ should be chosen as they have the highest IRR and a combined initial cost of $\$60,000$.

Thus, the opportunity cost of the capital is $19.4\%$.

Conclusion:

Projects $4,5\text{and}6$ should be funded.

The opportunity cost of the capital is $19.4\%$.

To determine

(b)

The projects to be funded and the opportunity cost of capital.

Answer to Problem 23P

Projects $1,3,4,5,6\text{and}7$ should be funded.

The opportunity cost of the capital is $16\%$.

Explanation of Solution

Given:

The budget is $\$120,000$.

Calculation:

Projects $1,3,4,5,6\text{and}7$ should be chosen as they have the highest IRR and a combined initial cost of $\$120,000$.

Thus, the opportunity cost of the capital is $16\%$.

Conclusion:

Projects $1,3,4,5,6\text{and}7$ should be funded.

The opportunity cost of the capital is $16\%$.

To determine

(c)

The projects to be funded and the opportunity cost of capital.

Answer to Problem 23P

Projects $1,5\text{and}6$ should be funded.

The opportunity cost of the capital is $19.26\%$.

Explanation of Solution

Given:

Project $4$ has an external cost of $\$1000$ annually.

Calculation:

Calculate the internal rate of return for Project $4$.

New annual benefit. $\begin{array}{l}=4500-1000\\ =3500\end{array}$

Substitute $0$ for PW, $\$20,000$ for First cost, $\$3500$ for Benefit, $0$ for Salvage value and $15$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$20,000+\$3,500\left(\frac{{\left(1+i\right)}^{15}-1}{i{\left(1+i\right)}^{15}}\right)\\ 5.714=\left(\frac{{\left(1+i\right)}^{15}-1}{i{\left(1+i\right)}^{15}}\right)\\ i=15.4\%\end{array}$

Thus, the new IRR.

Project	First cost	IRR
$1$	$\$20,000$	$19.4\%$
$2$	$\$20,000$	$16\%$
$3$	$\$20,000$	$16.4\%$
$4$	$\$20,000$	$15.4\%$
$5$	$\$20,000$	$22.2\%$
$6$	$\$20,000$	$26.16\%$
$7$	$\$20,000$	$19.26\%$

Projects $1,5\text{and}6$ should be funded.

The opportunity cost of the capital is $19.26\%$.

Conclusion:

Projects $1,5\text{and}6$ should be funded.

The opportunity cost of the capital is $19.26\%$.

To determine

(d)

The projects to be funded and the opportunity cost of capital.

Explanation of Solution

Projects $1,2,3,5,6\text{and}7$ should be funded.

The opportunity cost of the capital is $15.4\%$.

Calculation:

Projects $1,2,3,5,6\text{and}7$ should be chosen as they have the highest IRR and a combined initial cost of $\$120,000$.

The opportunity cost of the capital is $15.4\%$.

Conclusion:

Projects $1,2,3,5,6\text{and}7$ should be funded.

The opportunity cost of the capital is $15.4\%$.