Chapter 9, Problem 55P

To determine

(a)

The alternative that should be selected.

Answer to Problem 55P

The alternative F should be selected.

Explanation of Solution

Calculations:

The table for cash flow and cumulative cash flow for project E is shown below.

Year	Cash flow	Cumulative cash flow
0	$-\$90$	$-\$90$
1	$\$20$	$-\$70$
2	$\$20$	$-\$50$
3	$\$20$	$-\$30$
4	$\$20$	$-\$10$
5	$\$20$	$\$10$
6	$\$20$	$\$30$

Table (1)

Here, the cumulative cash flow becomes positive in $5^{\text{th}}$ year. At the end of $4^{\text{th}}$ year the cash inflow is only $\$10$ less than the cash outflow. Thus $\$10$ is required to equal the benefits and costs. In the $5^{\text{th}}$ year total cash inflow is $\$20$.

Calculate the payback period for project E.

$\begin{array}{c}\text{Payback period}=\left(4+\frac{\$10}{\$20}\right)\\ =\left(4+0.5\right)\\ =4.5\text{ years}\end{array}$

The table for cash flow and cumulative cash flow for project F is shown below.

Year	Cash flow	Cumulative cash flow
0	$-\$110$	$-\$110$
1	$\$35$	$-\$75$
2	$\$35$	$-\$40$
3	$\$35$	$-\$5$
4	$\$35$	$\$30$
5	$\$0$	$\$30$
6	$\$0$	$\$30$

Table (2)

Here, the cumulative cash flow becomes positive in $4^{\text{th}}$ year. At the end of $3^{\text{rd}}$ year the cash inflow is only $\$5$ less than the cash outflow. Thus $\$5$ is required to equal the benefits and costs. In $4^{\text{th}}$ year total cash inflow is $\$35$.

Calculate the payback period for project F.

$\begin{array}{c}\text{Payback period}=\left(3+\frac{\$5}{\$35}\right)\\ =\left(3+0.14\right)\\ =3.14\text{ years}\end{array}$

The table for cash flow and cumulative cash flow for project G is shown below.

Year	Cash flow	Cumulative cash flow
0	$-\$100$	$-\$100$
1	$\$0$	$-\$100$
2	$\$10$	$-\$90$
3	$\$20$	$-\$70$
4	$\$30$	$-\$40$
5	$\$40$	$\$0$
6	$\$50$	$\$50$

Table (3)

Here, the cumulative cash flow becomes NIL in $5^{\text{th}}$ year. The profit equals cost in $5^{\text{th}}$ year. The payback period will be $5^{\text{th}}$ year.

The table for cash flow and cumulative cash flow for project H is shown below.

Year	Cash flow	Cumulative cash flow
0	$-\$120$	$-\$120$
1	$\$0$	$-\$120$
2	$\$0$	$-\$120$
3	$\$0$	$-\$120$
4	$\$0$	$-\$120$
5	$\$0$	$-\$120$
6	$\$180$	$\$60$

Table (4)

Here, the cumulative cash flow becomes positive in $6^{\text{th}}$ year. At the end of $5^{\text{th}}$ year the cash inflow is only $\$120$ less than the cash outflow. Thus $\$120$ is required to equal the benefits and costs. In $6^{\text{th}}$ year total cash inflow is $\$180$.

Calculate the payback period for project H.

$\begin{array}{c}\text{Payback period}=\left(5+\frac{\$120}{\$180}\right)\\ =\left(5+0.6\right)\\ =5.6\text{ years}\end{array}$

The payback period for project F is minimum which is equal to $3.1$ years.

Thus, the alternative F should be selected.

Conclusion:

The alternative F should be selected.

To determine

(b)

The alternative that should be selected.

Answer to Problem 55P

The alternative G should be selected.

Explanation of Solution

Given:

The interest rate is $5\%$.

Concept used:

Write the expression for future worth factor.

$\text{factor}={\left(1+i\right)}^n$ ...... (I)

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

Calculations:

The table for future worth analysis of alternatives is shown below.

Year	Future worth factor ${\left(1.05\right)}^n$ 1	Project (E)		Project (F)		Project (G)		Project (H)
		Cash flow2	Future worth3 $\left(1\times 2\right)$	Cash flow4	Future worth5 $\left(1\times 4\right)$	Cash flow6	Future worth7 $\left(1\times 6\right)$	Cash flow8	Future worth9 $\left(1\times 8\right)$
0	$1.3401$	$-\$90$	$-\$120.61$	$-\$110$	$-\$147.411$	$-\$100$	$-\$134$	$-\$120$	$-\$160.8$
1	$1.2762$	$\$20$	$\$25.52$	$\$35$	$\$46.67$	$\$0$	$\$0$	$\$0$	$\$0$
2	$1.2155$	$\$20$	$\$24.31$	$\$35$	$\$42.5425$	$\$10$	$\$12.155$	$\$0$	$\$0$
3	$1.57625$	$\$20$	$\$31.525$	$\$35$	$\$55.17$	$\$20$	$\$31.52$	$\$0$	$\$0$
4	$1.1025$	$\$20$	$\$22.05$	$\$35$	$\$38.587$	$\$30$	$\$30.75$	$\$0$	$\$0$
5	$1.05$	$\$20$	$\$21$	$\$0$	$\$0$	$\$40$	$\$42$	$\$0$	$\$0$
6	$1.00$	$\$20$	$\$20$	$\$0$	$\$0$	$\$50$	$\$50$	$\$180$	$\$180$
	Total		$\$23.8$		$-\$6.45$		$\$32.425$		$\$19.2$

Table (5)

The alternative G has the maximum future worth.

Thus, thee alternative G should be selected.

Conclusion:

The alternative G should be selected.

To determine

(c)

The alternative that should be selected.

Answer to Problem 55P

The alternative G should be selected.

Explanation of Solution

Given:

The interest rate is $20\%$.

Calculations:

The table for future worth analysis of alternatives is shown below.

Year	Future worth factor ${\left(1.05\right)}^n$ 1	Project (E)	Project (F)	Project (G)	Project (H)
Cash flow 2	Future worth 3 $\left(1\times 2\right)$	Cash flow 4	Future worth 5 $\left(1\times 4\right)$	Cash flow 6	Future worth 7 $\left(1\times 6\right)$	Cash flow 8	Future worth 9 $\left(1\times 8\right)$
0	$2.986$	$-\$90$	$-\$268.74$	$-\$110$	$-\$328.46$	$-\$100$	$-\$298.6$	$-\$120$	$-\$358.32$
1	$2.488$	$\$20$	$\$49.76$	$\$35$	$\$87.08$	$\$0$	$\$0$	$\$0$	$\$0$
2	$2.073$	$\$20$	$\$41.47$	$\$35$	$\$72.57$	$\$10$	$\$20.73$	$\$0$	$\$0$
3	$1.728$	$\$20$	$\$34.56$	$\$35$	$\$60.48$	$\$20$	$\$34.56$	$\$0$	$\$0$
4	$1.44$	$\$20$	$\$28.8$	$\$35$	$\$50.04$	$\$30$	$\$43.2$	$\$0$	$\$0$
5	$1.2$	$\$20$	$\$25.2$	$\$0$	$\$0$	$\$40$	$\$48$	$\$0$	$\$0$
6	$1.00$	$\$20$	$\$20$	$\$0$	$\$0$	$\$50$	$\$50$	$\$180$	$\$180$
	Total		$-\$68.95$		$-\$58.29$		$-\$102.1$		$-\$178.32$

Table (6)

The alternative G has the maximum future worth.

Thus, the alternative G should be selected.

Conclusion:

The alternative G should be selected.

To determine

(d)

The benefit cost ratio for alternate G.

Answer to Problem 55P

The benefit cost ratio for alternate G is $0.96$.

Explanation of Solution

Given:

The interest rate is $10\%$.

Concept used:

Write the expression for present worth of benefits.

$P{W}_B=G\left[\frac{{\left(1+i\right)}^n-in-1}{i^2{\left(1+i\right)}^n}\right]$ ...... (I)

Here rate of interest is $i$, time period is $n$ and uniform gradient is $G$.

Write the expression for benefit cost ratio.

$B-C\text{ ratio}=\frac{P{W}_B}{P{W}_C}$ ...... (II)

Here, the present worth of costs is $P{W}_C$.

Calculations:

Calculate the present worth of benefits.

Substitute $\$10$ for $G$, $10\%$ for $i$ and $6$ for $n$ in Equation (I).

$\begin{array}{c}P{W}_B=\$10\left[\frac{{\left(1+0.1\right)}^6-0.1\times 6-1}{{0.1}^2{\left(1+0.1\right)}^6}\right]\\ =\$10\left[\frac{1.77-1.6}{0.01\times 1.77}\right]\\ =\$10\left(9.6\right)\\ =\$96\end{array}$

Calculate the benefit cost ratio.

Substitute $\$96$ for $P{W}_B$, and $\$100$ for $P{W}_C$ in Equation (I).

$\begin{array}{c}B-C\text{ ratio}=\frac{\$96}{\$100}\\ =0.96\end{array}$

Thus, the benefit cost ratio for alternate G is $0.96$.

Conclusion:

The benefit cost ratio for alternate G is $0.96$.