Chapter 12, Problem 15P

To determine

(a)

The after-tax present worth.

Answer to Problem 15P

The after-tax present worth is, $\$30,923.99$.

Explanation of Solution

Given:

Capacity of each carton is $100\text{bags}$.

Average quantity of bags in each carton is $105.5$.

The percentage loss to the company by overfilling the cartons is $5.5\%$.

Cost of each bag is $\$0.03$.

Cost of the equipment is $\$18600$.

Life of the equipment is $10\text{years}$.

Salvage value of the machine is $\$3600$.

Maintenance cost of the machine is $\$16000$.

Federal plus state tax is $40\%$.

MARR after tax is $15\%$.

Concept used:

Write the formula to calculate the taxable income.

$\text{TI}=\text{DI}-\text{BTCF}$

Here, the taxable income is $\text{TI}$, the depreciation is $\text{DI}$ and the before tax cash flow is $\text{BTCF}$.

Write the formula to calculate the taxable amount.

$\text{TAX}=\text{TI}\times \text{Tax}\text{rate}$

Here, the taxable amount is $\text{TAX}$.

Write the formula to calculate the after tax cash flow.

$\text{ATCF}=\text{TAX}-\text{BTCF}$

Here, the after tax cash flow is $\text{ATCF}$.

Calculation:

Write the formula to calculate the straight line depreciation.

$\text{Depreciation}=\frac{B-S}{N}$ ....... (I)

Here, the asset value is $B$, salvage value of the machinery is $S$, and life of the machinery is $N$.

Substitute $\$18600$ for $B$, $\$3600$ for $S$, and $10\text{years}$ for $N$ in Equation (I).

$\begin{array}{c}\text{Depreciation}=\frac{\$18600-\$3600}{10}\\ =\$1500\end{array}$

Calculate savings of the company as shown below.

$\text{Savings}=\left(\text{cost}\text{of}\text{one}\text{bag}\right)\times \text{Quantity}\left(\begin{array}{l}\text{Average}\text{bags}\text{packed}\text{by}\text{hand}\\ -\text{average}\text{bags}\text{packed}\text{by}\text{machine}\end{array}\right)$

Substitute $\$0.03$ for cost of one bag, $200,000$ for quantity, $105.5$ for average bags packed by hand, and $100.3$ for average bags packed by machine.

$\begin{array}{c}\text{Savings}=\$0.03\times 200,000\left(105.5-100.3\right)\\ =\$31,200\end{array}$

Calculate the After cash flow of as shown in table below.

Year	Cost	Savings	Net BTCF	DI	TI	Tax	ATCF
$0$	$-\$18600$		$-\$18600$				$-\$18600$
$1$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$2$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$3$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$4$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$5$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$6$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$7$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$8$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$9$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$10$	$-\$16000+3600=\$12400$	$\$31200$	$\$18800$	$\$1500$	$\$15200$	$\$6080$	$\$12720$
						NPV	$\$30923.99$

Conclusion:

Therefore, after tax present worth is, $\$30,923.99$.

To determine

(b)

The after-tax internal rate of return.

Answer to Problem 15P

The after-tax internal rate of return is $52\%$.

Explanation of Solution

Calculate the After cash flow of as shown in table below.

Year	Cost	Savings	Net BTCF	DI	TI	Tax	ATCF
$0$	$-\$18600$		$-\$18600$				$-\$18600$
$1$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$2$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$3$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$4$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$5$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$6$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$7$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$8$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$9$	$-\$16000$	$\$31200$	$\$15200$	$\$1500$	$\$13700$	$\$5480$	$\$9720$
$10$	$-\$16000+3600=\$12400$	$\$31200$	$\$18800$	$\$1500$	$\$15200$	$\$6080$	$\$12720$
						NPV	$\$30923.99$

Calculate the after tax internal rate of return as shown below.

Write the expression to calculate the rate of return after tax as shown below.

$P=A\left(\frac{P}{A},i,n\right)$ ....... (II)

Substitute $\$18600$ for P and $\$9720$ for A and $10\text{years}$ for n in Equation (II).

$\begin{array}{l}\$18600=\$9720\left(\frac{P}{A},i,10\right)\\ 1.91=\left(\frac{P}{A},i,10\right)\\ 1.91=\left(\frac{{\left(1+i\right)}^{10}-1}{i{\left(1+i\right)}^{10}}\right)\end{array}$

Substitute $52\%$ as rate and use trial and error method and solve the right hand side.

$\begin{array}{c}\text{RHS}=\left(\frac{{\left(1+52\%\right)}^{10}-1}{52\%{\left(1+52\%\right)}^{10}}\right)\\ =\frac{64.832}{34.232}\\ =1.9\end{array}$

The value of right hand side is nearly equal to the left hand side.

Conclusion:

Therefore, the after-tax internal rate of return is $52\%$.

To determine

(c)

The after-tax simple payback period.

Answer to Problem 15P

The after-tax simple payback period is, $1.91\text{years}$.

Explanation of Solution

The initial investment is $\$18600$. The net worth flow is $\$9720$ in the first year, so investment cannot be recovered in the first year. Some negative balance will occur which is calculated as follows.

$\begin{array}{c}\text{Negative}\text{balance}=\$18600-\$9720\\ =\$8880\end{array}$

Calculate the payback period as shown below.

$\text{Payback}\text{period}=1+\frac{\text{Negative}\text{balance}}{\text{ATCF}\text{of}\text{firts}\text{year}}$

Substitute $\$8880$ for negative balance and $\$9720$ for ATCF of the first year.

$\begin{array}{c}\text{Payback}\text{period}=1+\left(\frac{\$8880}{\$9720}\right)\\ =1.91\text{years}\end{array}$.

Conclusion:

Therefore, the after-tax simple payback period is, $1.91\text{years}$.