Chapter 9, Problem 50P

To determine

(a)

The best alternative based on payback period.

Answer to Problem 50P

The Alternative- C should be selected.

Explanation of Solution

Write the Equation for Payback period.

$\text{Payback}\text{period}=\frac{\text{Cost}}{\text{Annual}\text{Benefit}}$ ...... (I)

Calculate the Payback period for Alternative-A.

Substitute $\$75$ for cost, and $\$18.8$ for annual benefit in Equation (I).

$\begin{array}{c}\text{Payback}\text{period}=\frac{\$75}{\text{\$18}\text{.8}}\\ =3.98\text{Years}\end{array}$

Calculate the Payback period for Alternative-B.

Substitute $\$50$ for cost, and $\$13.9$ for annual benefit in Equation (I).

$\begin{array}{c}\text{Payback}\text{period}=\frac{\$50}{\text{\$13}\text{.9}}\\ =3.6\text{Years}\end{array}$

Calculate the Payback period for Alternative-C.

Substitute $\$15$ for cost, and $\$4.5$ for annual benefit in Equation (I).

$\begin{array}{c}\text{Payback}\text{period}=\frac{\$15}{\text{\$4}\text{.5}}\\ =3.3\text{Years}\end{array}$

Calculate the Payback period for Alternative-D.

Substitute $\$90$ for cost, and $\$23.8$ for annual benefit in Equation (I).

$\begin{array}{c}\text{Payback}\text{period}=\frac{\$90}{\text{\$23}\text{.8}}\\ =3.8\text{Years}\end{array}$

The Alternative that has minimum value of Payback period should be selected.

Conclusion:

Therefore, the Alternative- C should be selected.

To determine

(b)

The best alternative based on Future worth analysis.

Answer to Problem 50P

The Alternative- B should be selected.

Explanation of Solution

Calculate the net future worth for alternative-A.

$\text{NFW}=\$18.8\left(\frac{F}{A},10\%,5\right)-\$75\left(\frac{F}{P},10\%,5\right)$ ...... (II)

Here, the net future worth is NFW, the future value is F, the Annual benefit is A and the initial cost is P.

Calculate the factor $\left(\frac{F}{A},10\%,5\right)$.

$\begin{array}{c}\left(\frac{F}{A},10\%,5\right)=\left[\frac{{\left(1+0.10\right)}^5-1}{0.10}\right]\\ =6.105\end{array}$

Calculate the factor $\left(\frac{F}{P},10\%,5\right)$.

$\begin{array}{c}\left(\frac{F}{P},10\%,5\right)={\left(1+0.10\right)}^5\\ =1.610\end{array}$

Substitute $\$6.105$ for $\left(\frac{F}{A},10\%,5\right)$, and $1.610$ for $\left(\frac{F}{P},10\%,5\right)$ in Equation (II).

$\begin{array}{c}\text{NFW}=\left(\$18.8\times 6.105\right)-\left(\$75\times 1.610\right)\\ =\$114.774-\$120.75\\ =-\$6\end{array}$

Calculate the net future worth for alternative-B.

$\text{NFW}=\$13.9\left(\frac{F}{A},10\%,5\right)-\$50\left(\frac{F}{P},10\%,5\right)$ ...... (III)

Substitute $\$6.105$ for $\left(\frac{F}{A},10\%,5\right)$, and $1.610$ for $\left(\frac{F}{P},10\%,5\right)$ in Equation (III).

$\begin{array}{c}\text{NFW}=\$13.9\times 6.105-\$50\times 1.610\\ =\$84.86-\$80.5\\ =\$4.36\end{array}$

Calculate the net future worth for alternative-C.

$\text{NFW}=\$4.5\left(\frac{F}{A},10\%,5\right)-\$15\left(\frac{F}{P},10\%,5\right)$ ...... (IV)

Substitute $\$6.105$ for $\left(\frac{F}{A},10\%,5\right)$, and $1.610$ for $\left(\frac{F}{P},10\%,5\right)$ in Equation (IV).

$\begin{array}{c}\text{NFW}=\$4.5\times 6.105-\$15\times 1.610\\ =\$27.47-\$24.15\\ =\$3.32\end{array}$

Calculate the net future worth for alternative-D.

$\text{NFW}=\$23.8\left(\frac{F}{A},10\%,5\right)-\$90\left(\frac{F}{P},10\%,5\right)$ ...... (V)

Substitute $\$6.105$ for $\left(\frac{F}{A},10\%,5\right)$, and $1.610$ for $\left(\frac{F}{P},10\%,5\right)$ in Equation (V).

$\begin{array}{c}\text{NFW}=\$23.8\times 6.105-\$90\times 1.610\\ =\$145.30-\$145\\ =\$0.30\end{array}$

The Alternative that has maximum value of Future worth should be selected.

Conclusion:

Therefore, the Alternative- B should be selected.

To determine

(c)

The best alternative based on B/C ratio analysis.

Answer to Problem 50P

All Alternatives should be selected except A.

Explanation of Solution

Calculate the factor $\left(\frac{P}{A},10\%,5\right)$.

$\begin{array}{c}\left(\frac{P}{A},10\%,5\right)=\left[\frac{{\left(1+0.10\right)}^5-1}{0.10{\left(1+0.10\right)}^5}\right]\\ =3.790\end{array}$

Write the equation for B/C ratio.

$\text{B}/\text{C}\text{ratio}=\frac{\text{EUAB}}{\text{EUAC}}$ ...... (VI)

Calculate EUAB for Alternative-A.

$\text{EUAB}=\$18.8\left(\frac{P}{A},10\%,5\right)$ ...... (VII).

Substitute $3.790$ for $\left(\frac{P}{A},10\%,5\right)$ in Equation (VII).

$\begin{array}{c}\text{EUAB}=\$18.8\times 3.790\\ =\$71.252\end{array}$

Calculate the B/C ratio for Alternative-A.

Substitute $\$71.252$ for EUAB, and $\$75$ for EUAC in Equation (VI).

$\begin{array}{c}\text{B}/\text{C}\text{ratio}=\frac{\$71.252}{\$75}\\ =0.95\end{array}$

Calculate EUAB for Alternative-B.

$\text{EUAB}=\$13.9\left(\frac{P}{A},10\%,5\right)$ ...... (VIII).

Substitute $3.790$ for $\left(\frac{P}{A},10\%,5\right)$ in Equation (VIII).

$\begin{array}{c}\text{EUAB}=\$13.9\times 3.790\\ =\$52.7\end{array}$

Calculate the B/C ratio for Alternative-B.

Substitute $\$52.7$ for EUAB, and $\$50$ for EUAC in Equation (VI).

$\begin{array}{c}\text{B}/\text{C}\text{ratio}=\frac{\$52.7}{\$50}\\ =1.05\end{array}$

Calculate EUAB for Alternative-C.

$\text{EUAB}=\$4.5\left(\frac{P}{A},10\%,5\right)$ ...... (IX).

Substitute $3.790$ for $\left(\frac{P}{A},10\%,5\right)$ in Equation (IX).

$\begin{array}{c}\text{EUAB}=\$4.5\times 3.790\\ =\$17.055\end{array}$

Calculate the B/C ratio for Alternative-C.

Substitute $\$17.05$ for EUAB, and $\$15$ for EUAC in Equation (VI).

$\begin{array}{c}\text{B}/\text{C}\text{ratio}=\frac{\$17.05}{\$15}\\ =1.14\end{array}$

Calculate EUAB for Alternative-D.

$\text{EUAB}=\$23.8\left(\frac{P}{A},10\%,5\right)$ ...... (X).

Substitute $3.790$ for $\left(\frac{P}{A},10\%,5\right)$ in Equation (X).

$\begin{array}{c}\text{EUAB}=\$23.8\times 3.790\\ =\$90.202\end{array}$

Calculate the B/C ratio for Alternative-C.

Substitute $\$90.202$ for EUAB, and $\$90$ for EUAC in Equation (VI).

$\begin{array}{c}\text{B}/\text{C}\text{ratio}=\frac{\$90.202}{\$90}\\ =1\end{array}$.

Conclusion:

Therefore, all Alternatives should be selected except A.