Chapter 15, Problem 22P

To determine

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

Answer to Problem 22P

Projects $E,C,H,B,A\text{and}G$ should be funded.

The opportunity cost of the capital is $13.4\%$.

Explanation of Solution

Given:

Project	First cost	Annual benefits	Life (years)
A	$\$150,000$	$\$44,290$	$5$
B	$\$200,000$	$\$61,730$	$5$
C	$\$300,000$	$\$98,780$	$5$
D	$\$250,000$	$\$62610$	$6$
E	$\$400,000$	$\$119,330$	$6$
F	$\$500,000$	$\$115,500$	$6$
G	$\$350,000$	$\$67940$	$10$
H	$\$600,000$	$\$126920$	$10$
I	$\$750,000$	$\$140580$	$10$

The budget of the company is $\$2\text{million}$.

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 A.

Substitute $0$ for PW, $\$150,000$ for First cost, $\$44,290$ for Benefit and $5$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$150,000+\$44,290\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ 3.387=\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ i=15.12\%\end{array}$

For project B.

Substitute $0$ for PW, $\$200,000$ for First cost, $\$61,730$ for Benefit and $5$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$200,000+\$61,730\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ 3.24=\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ i=16.10\%\end{array}$

For project C.

Substitute $0$ for PW, $\$300,000$ for First cost, $\$98,780$ for Benefit and $5$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$300,000+\$98,780\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ 3.04=\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)\\ i=19.04\%\end{array}$

For project D.

Substitute $0$ for PW, $\$250,000$ for First cost, $\$62610$ for Benefit and $6$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$250,000+\$62,610\left(\frac{{\left(1+i\right)}^6-1}{i{\left(1+i\right)}^6}\right)\\ 3.99=\left(\frac{{\left(1+i\right)}^6-1}{i{\left(1+i\right)}^6}\right)\\ i=13.4\%\end{array}$

For project E.

Substitute $0$ for PW, $\$400,000$ for First cost, $\$119,330$ for Benefit and $6$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$400,000+\$119,330\left(\frac{{\left(1+i\right)}^6-1}{i{\left(1+i\right)}^6}\right)\\ 3.352=\left(\frac{{\left(1+i\right)}^6-1}{i{\left(1+i\right)}^6}\right)\\ i=20\%\end{array}$

For project F.

Substitute $0$ for PW, $\$500,000$ for First cost, $\$115,500$ for Benefit and $6$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$500,000+\$115,500\left(\frac{{\left(1+i\right)}^6-1}{i{\left(1+i\right)}^6}\right)\\ 4.33=\left(\frac{{\left(1+i\right)}^6-1}{i{\left(1+i\right)}^6}\right)\\ i=10.16\%\end{array}$

For project G.

Substitute $0$ for PW, $\$350,000$ for First cost, $\$67940$ for Benefit and $10$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$350,000+\$67940\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ 5.15=\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ i=14.27\%\end{array}$

For project H.

Substitute $0$ for PW, $\$600,000$ for First cost, $\$126920$ for Benefit and $10$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$600,000+\$126920\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ 4.73=\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ i=17.14\%\end{array}$

For project I.

Substitute $0$ for PW, $\$750,000$ for First cost, $\$140580$ for Benefit and $10$ for n in Equation (I).

$\begin{array}{l}\text{0}=-\$750,000+\$140580\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ 5.335=\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\\ i=13.37\%\end{array}$

Tabulate the results.

Project	First cost	IRR
A	$\$150,000$	$15.12\%$
B	$\$200,000$	$16.10\%$
C	$\$300,000$	$19.04\%$
D	$\$250,000$	$13.4\%$
E	$\$400,000$	$20\%$
F	$\$500,000$	$10.16\%$
G	$\$350,000$	$14.27\%$
H	$\$600,000$	$17.14\%$
I	$\$750,000$	$13.37\%$

Projects $E,C,H,B,A\text{and}G$ should be chosen as they have the highest IRR and a combined initial cost of $\$2\text{million}$.

Thus, the opportunity cost of the capital is $13.4\%$.

Conclusion:

Projects $E,C,H,B,A\text{and}G$ should be funded.

The opportunity cost of the capital is $13.4\%$.