Chapter 15, Problem 21P

To determine

(a)

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

Answer to Problem 21P

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

The opportunity cost of the capital is $19.36\%$.

Explanation of Solution

Given:

Project	First cost	Annual benefits	Life(years)
$1$	$\$200,000$	$\$50,000$	$15$
$2$	$\$300,000$	$\$70000$	$10$
$3$	$\$100,000$	$\$40000$	$5$
$4$	$\$50,000$	$\$12,500$	$10$
$5$	$\$250,000$	$\$75,000$	$5$
$6$	$\$150,000$	$\$32,000$	$20$
$7$	$\$400,000$	$\$125,000$	$5$

The budget is $\$500,000$.

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.

Calculation:

Calculate the Internal rate of return (IRR).

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

$\text{PW}\text{of}\text{project}=-\left(\text{First}\text{cost}\right)+\text{Benefit}\left(\frac{P}{A},i,n\right)$ ...... (I)

For project $1$.

Substitute $0$ for PW, $\$200,000$ for First cost, $\$50000$ for Benefit and $5$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$200,000+\$50,000\left(\frac{{\left(1+i\right)}^{15}-1}{i{\left(1+i\right)}^{15}}\right)\\ 4=\left(\frac{{\left(1+i\right)}^{15}-1}{i{\left(1+i\right)}^{15}}\right)\\ i=24.01\%\end{array}$

For project $2$.

Substitute $0$ for PW, $\$300,000$ for First cost, $\$70000$ for Benefit and $10$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$300,000+\$70000\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ 4.286=\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ i=19.36\%\end{array}$

For project $3$.

Substitute $0$ for PW, $\$100,000$ for First cost, $\$40000$ for Benefit and $5$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$100,000+\$40000\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ 2.5=\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ i=28.65\%\end{array}$

For project $4$.

Substitute $0$ for PW, $\$50,000$ for First cost, $\$12,500$ for Benefit and $10$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$50,000+\$12,500\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ 4=\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ i=21.41\%\end{array}$

For project $5$.

Substitute $0$ for PW, $\$250,000$ for First cost, $\$75,000$ for Benefit and $5$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$250,000+\$75,000\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ 3.33=\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ i=15.24\%\end{array}$

For project $6$.

Substitute $0$ for PW, $\$150,000$ for First cost, $\$32,000$ for Benefit and $20$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$150,000+\$32,000\left(\frac{{\left(1+i\right)}^{20}-1}{i{\left(1+i\right)}^{20}}\right)\\ 4.688=\left(\frac{{\left(1+i\right)}^{20}-1}{i{\left(1+i\right)}^{20}}\right)\\ i=20.85\%\end{array}$

For project $7$.

Substitute $0$ for PW, $\$400,000$ for First cost, $\$125,000$ for Benefit and $5$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$400,000+\$125,000\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ 3.2=\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ i=16.99\%\end{array}$

Tabulate the results.

Project	First cost	IRR
$1$	$\$200,000$	$24.01\%$
$2$	$\$300,000$	$19.36\%$
$3$	$\$100,000$	$28.65\%$
$4$	$\$50,000$	$21.41\%$
$5$	$\$250,000$	$15.24\%$
$6$	$\$150,000$	$20.85\%$
$7$	$\$400,000$	$16.99\%$

Table-(1)

From Table-(1), projects $1,3,4\text{and}6$ should be chosen as they have the highest IRR and a combined initial cost of $\$500,000$.

Conclusion:

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

Thus, the opportunity cost of the capital is $19.36\%$.

To determine

(b)

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

Answer to Problem 21P

Projects $1,2,3,4,6\text{and}7$ should be funded.

The opportunity cost of the capital is $15.24\%$.

Explanation of Solution

Given:

The budget is $\$1.2\text{million}$.

Calculation:

From Table-(1), projects $1,2,3,4,6\text{and}7$ should be chosen as they have the highest IRR and a combined initial cost of $\$1.2\text{million}$.

Thus, the opportunity cost of the capital is $15.24\%$.

Conclusion:

Projects $1,2,3,4,6\text{and}7$ should be funded.

Thus, the opportunity cost of the capital is $15.24\%$.