Chapter 9, Problem 51P

To determine

(a)

The best alternative based on payback period.

Answer to Problem 51P

The Alternative- B should be selected.

Explanation of Solution

Calculation:

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 $\$50$ for Cost, and $\$30$ for Annual Benefit in Equation (I).

$\begin{array}{c}\text{Payback}\text{period}=\frac{\$50}{\text{\$30}}\\ =1.67\text{Years}\end{array}$

Calculate the Payback period for Alternative-B.

Substitute $\$150$ for Cost, and $\$45$ for Annual Benefit in Equation (I).

$\begin{array}{c}\text{Payback}\text{period}=\frac{\$150}{\text{\$45}}\\ =3.33\text{Years}\end{array}$

Calculate the Payback period for Alternative-C.

Substitute $\$110$ for Cost, and $\$45$ for Annual Benefit in Equation (I).

$\begin{array}{c}\text{Payback}\text{period}=\frac{\$110}{\text{\$45}}\\ =2.44\text{Years}\end{array}$

The alternative that has minimum value of payback period should be selected.

Conclusion:

Therefore, the Alternative- B should be selected.

To determine

(b)

The best alternative based on future worth analysis.

Answer to Problem 51P

The Alternative- B should be selected.

Explanation of Solution

Calculation:

Calculate the net future worth for alternative-A.

$\text{NFW}=\$30\left(\frac{F}{A},10\%,3\right)-\$50\left(\frac{F}{P},10\%,3\right)$ ...... (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 $\left(\frac{F}{A},10\%,3\right)$.

$\begin{array}{c}\left(\frac{F}{A},10\%,3\right)=\left[\frac{{\left(1+0.10\right)}^3-1}{0.10}\right]\\ =3.31\end{array}$

Calculate the factor $\left(\frac{F}{P},10\%,3\right)$.

$\begin{array}{c}\left(\frac{F}{P},10\%,3\right)={\left(1+0.10\right)}^3\\ =1.331\end{array}$

Substitute $\$3.31$ for $\left(\frac{F}{A},10\%,3\right)$, and $1.331$ for $\left(\frac{F}{P},10\%,3\right)$ in Equation (II).

$\begin{array}{c}\text{NFW}=\$30\times 3.31-\$50\times 1.331\\ =\$99.3-\$66.55\\ =\$32.75\end{array}$

Calculate the net future worth (NFW) for alternative-B.

$\text{NFW}=\$45\left(\frac{F}{A},10\%,9\right)-\$150\left(\frac{F}{P},10\%,9\right)$ ...... (III)

Calculate the factor $\left(\frac{F}{A},10\%,9\right)$.

$\begin{array}{c}\left(\frac{F}{A},10\%,9\right)=\left[\frac{{\left(1+0.10\right)}^9-1}{0.10}\right]\\ =\$13.58\end{array}$

Calculate the factor $\left(\frac{F}{P},10\%,9\right)$.

$\begin{array}{c}\left(\frac{F}{P},10\%,9\right)={\left(1+0.10\right)}^9\\ =2.36\end{array}$

Substitute $\$13.58$ for $\left(\frac{F}{A},10\%,9\right)$, and $2.36$ for $\left(\frac{F}{P},10\%,9\right)$ in Equation (III).

$\begin{array}{c}\text{NFW}=\$45\times 13.58-\$150\times 2.36\\ =\$611.1-\$354\\ =\$257.1\end{array}$

Calculate the net future worth (NFW) for alternative-C.

$\text{NFW}=\$45\left(\frac{F}{A},10\%,6\right)-\$110\left(\frac{F}{P},10\%,6\right)$ ...... (IV)

Calculate the factor $\left(\frac{F}{A},10\%,6\right)$.

$\begin{array}{c}\left(\frac{F}{A},10\%,6\right)=\left[\frac{{\left(1+0.10\right)}^6-1}{0.10}\right]\\ =\$7.715\end{array}$

Calculate the factor $\left(\frac{F}{P},10\%,6\right)$.

$\begin{array}{c}\left(\frac{F}{P},10\%,6\right)={\left(1+0.10\right)}^6\\ =1.772\end{array}$

Substitute $\$7.715$ for $\left(\frac{F}{A},10\%,6\right)$, and $1.772$ for $\left(\frac{F}{P},10\%,6\right)$ in Equation (IV).

$\begin{array}{c}\text{NFW}=\$45\times 7.715-\$110\times 1.772\\ =\$347.175-\$194.92\\ =\$152.255\end{array}$

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 51P

The Alternative- C should be selected.

Explanation of Solution

Calculation:

Calculate the factor $\left(\frac{P}{A},10\%,3\right)$.

$\begin{array}{c}\left(\frac{P}{A},10\%,3\right)=\left[\frac{{\left(1+0.10\right)}^3-1}{0.10{\left(1+0.10\right)}^3}\right]\\ =2.49\end{array}$

Write the equation for B/C ratio.

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

Calculate EUAB for Alternative-A.

$\text{EUAB}=\$30\left(\frac{P}{A},10\%,5\right)$ ...... (VI).

Substitute $2.49$ for $\left(\frac{P}{A},10\%,3\right)$ in Equation (VI).

$\begin{array}{c}\text{EUAB}=\$30\times 2.49\\ =\$74.7\end{array}$

Calculate the B/C ratio for Alternative-A.

Substitute $\$74.7$ for EUAB, and $\$50$ for EUAC in Equation (V).

$\begin{array}{c}\text{B}/\text{C}\text{ratio}=\frac{\$74.7}{\$50}\\ =1.494\end{array}$

Calculate the factor $\left(\frac{P}{A},10\%,9\right)$.

$\begin{array}{c}\left(\frac{P}{A},10\%,9\right)=\left[\frac{{\left(1+0.10\right)}^9-1}{0.10{\left(1+0.10\right)}^9}\right]\\ =5.759\end{array}$

Write the Equation for B/C ratio.

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

Calculate EUAB for Alternative-B.

$\text{EUAB}=\$30\left(\frac{P}{A},10\%,9\right)$ ...... (VIII).

Substitute $5.759$ for $\left(\frac{P}{A},10\%,9\right)$ in Equation (VIII)

$\begin{array}{c}\text{EUAB}=\$45\times 5.759\\ =\$259.155\end{array}$

Calculate the B/C ratio for Alternative-B.

Substitute $\$259.155$ for EUAB, and $\$150$ for EUAC in Equation (VII).

$\begin{array}{c}\text{B}/\text{C}\text{ratio}=\frac{\$259.155}{\$150}\\ =1.73\end{array}$

Calculate the factor $\left(\frac{P}{A},10\%,6\right)$.

$\begin{array}{c}\left(\frac{P}{A},10\%,6\right)=\left[\frac{{\left(1+0.10\right)}^6-1}{0.10{\left(1+0.10\right)}^6}\right]\\ =4.355\end{array}$

Write the Equation for B/C ratio.

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

Calculate EUAB for Alternative-B.

$\text{EUAB}=\$30\left(\frac{P}{A},10\%,6\right)$ ...... (X).

Substitute $4.355$ for $\left(\frac{P}{A},10\%,6\right)$ in Equation (X).

$\begin{array}{c}\text{EUAB}=\$45\times 4.355\\ =\$195.975\end{array}$

Calculate the B/C ratio for Alternative-B.

Substitute $\$195.975$ for EUAB, and $\$150$ for EUAC in Equation (VII).

$\begin{array}{c}\text{B}/\text{C}\text{ratio}=\frac{\$195.975}{\$110}\\ =1.78\end{array}$.

Conclusion:

Therefore, the Alternative- C should be selected.

To determine

(d)

The reason behind the difference in answer in parts (a), (b), and (c).

Answer to Problem 51P

Explanation of Solution

In this case, there are three alternatives and rate of return accounts from $10\%$. Since, each alternative generates uniform annual benefits in excess of the cost, during the useful life and the payback period analysis ignores time value of money and only concentrates on time required to recover the investment. Therefore, payback period pertaining to each alternative is less than the alternative's useful life.

Since, least time is taken by alternative A for recovering the cost, however according to part (b), Alternative B is favorable for long-term economic efficiency and Alternative C is favorable according to part (c). Hence, it is ascertained that the answers in part (c), (a) and (b) differs from each other.