Chapter 12, Problem 37P

To determine

(a)

The after-tax cash flow for 10-year project life.

Answer to Problem 37P

The after tax cash flow for $10\text{years}$ is $\$19,415$.

Explanation of Solution

Given data:

Investing amount is $\$18,000$.

Estimate gross income is $\$4500$.

Increment per year is $\$500$.

Expenses are $\$300$.

Increment per year is $\$250$.

Time period is $10\text{years}$.

Combined income tax rate is $45\%$.

Concept used:

According to MACRS the rate of depreciation is shown below.

Year	MACRS Depreciation rate
$1$	$14.29\%$
$2$	$24.49\%$
$3$	$17.49\%$
$4$	$12.49\%$
$5$	$8.93\%$
6	$8.92\%$
7	$8.93\%$
8	$4.46\%$

Write the expression to calculate the taxable income.

$TI=BTCF-\left(\text{Depreciation}\right)$ .... (I)

Here before tax cash flow is $BTCF$.

Write the expression to calculate the tax to be paid.

$\text{Tax}=45\%\text{of}TI$ .... (II)

Write the expression to calculate the after tax cash flow.

$ATCF=BTCF-\left(\text{Tax}\right)$ .... (III).

Calculation:

Calculate the depreciation.

Year	Cost basis $\left(\text{a}\right)$	MACRS Depreciation rate $\left(\text{b}\right)$	Depreciation $\left(\text{c}\right)$ $\left(\text{a}\times \text{b}\right)$
$1$	$\$18,000$	$14.29\%$	$\$2,572.2$
$2$	$\$18,000$	$24.49\%$	$\$4,408.2$
$3$	$\$18,000$	$17.49\%$	$\$3,148.2$
$4$	$\$18,000$	$12.49\%$	$\$2,248.2$
$5$	$\$18,000$	$8.93\%$	$\$1,607.4$
6	$\$18,000$	$8.92\%$	$\$1,605.6$
7	$\$18,000$	$8.93\%$	$\$1,607.4$
8	$\$18,000$	$4.46\%$	$\$802.8$

Write the expression to calculate before tax cash flow.

$BTCF=\left(\text{Gross}\text{income}\right)-\left(\text{Expenses}\right)$ .... (I)

Substitute $\$4500$ for Gross income, $\$300$ for Expenses in Equation (I).

Calculate the before tax cash flow for $10\text{years}$.

Year	Gross income	Expenses	Before tax cash flow
0			$-\$18,000$
1	$\$4500$	$\$300$	$\$4,200$
2	$\$5,000$	$\$550$	$\$4,450$
3	$\$5,500$	$\$800$	$\$4,700$
4	$\$6,000$	$\$1,050$	$\$4,950$
5	$\$6,500$	$\$1,300$	$\$5,200$
6	$\$7,000$	$\$1,550$	$\$5,500$
7	$\$7,500$	$\$1800$	$\$5,700$
8	$\$8,000$	$\$2,050$	$\$5,950$
9	$\$8,500$	$\$2,300$	$\$6,200$
10	$\$9,000$	$\$2,550$	$\$6,450$

Calculate the taxable income for the first year.

Substitute $\$4,200$ for BTCF and $\$2,572.2$ for depreciation in Equation (I).

$\begin{array}{c}TI=\$4,200-\$2,572.2\\ =\$1,627.8\end{array}$

Calculate the tax for the first year.

Substitute $\$1,627.8$ for $TI$ in Equation (II).

$\begin{array}{c}\text{Tax}=45\%\text{of}\$1,627.8\\ =\frac{45}{100}\times \$1,627.8\\ =\$732.51\end{array}$

Calculate the after tax flow cash for the first year.

Substitute, $\$4,200$ for BTCF and $\$732.51$ for tax in Equation (III).

$\begin{array}{c}\text{ATCF}=\$4,200-\$732.51\\ =\$3,467.49\end{array}$

Similarly calculate the value for 10 years and enter them in the table below.

Year	Before tax cash flow	Depreciation	Taxable Income	Tax	After Tax Cash Flow
$0$	$-\$18,000$				$-\$18,000$
$1$	$\$4,200$	$\$2,572.2$	$\$1,627.8$	$\$732.51$	$\$3,467.49$
$2$	$\$4,450$	$\$4,408.2$	$\$41.8$	$\$18.81$	$\$4,431.19$
$3$	$\$4,700$	$\$3,148.2$	$\$1,551.8$	$\$698.31$	$\$4,001.69$
$4$	$\$4,950$	$\$2,248.2$	$\$2,701.8$	$\$1,215.81$	$\$3,734.19$
$5$	$\$5,200$	$\$1,607.4$	$\$3592.6$	$\$1,616.67$	$\$3,583.33$
6	$\$5,500$	$\$1,605.6$	$\$3894.4$	$\$1,752.48$	$\$3,747.52$
7	$\$5,700$	$\$1,607.4$	$\$4,092.6$	$\$1,841.67$	$\$3,858.33$
8	$\$5,950$	$\$802.8$	$\$5,147.2$		$\$2,316.24$
9	$\$6,200$	0	$\$6,200$	$\$2,790$	$\$3,410$
10	$\$6,450$	0	$\$6,450$	$\$2,902.5$	$\$3,547.5$

Calculate the total ATCF.

$\text{Total}ATCF=\$19,415$.

Conclusion:

Thus, the after tax cash flow for $10\text{years}$ is $\$19,415$.

To determine

(b)

The after rate tax return.

Answer to Problem 37P

The after rate tax return is $8.5\%$.

Explanation of Solution

Calculate the after tax rate of return.

Write the expression for after tax rate of return.

$P=\left[\begin{array}{l}{F}_1\left(P/F,i,n\right)+F_2\left(P/F,i,n\right)\\ +F_3\left(P/F,i,n\right)+F_4\left(P/F,i,n\right)\\ +F_5\left(P/F,i,n\right)+F_6\left(P/F,i,n\right)\\ +F_7\left(P/F,i,n\right)+F_8\left(P/F,i,n\right)\\ +F_9\left(P/F,i,n\right)+F_{10}\left(P/F,i,n\right)\end{array}\right]$ .... (IV)

Here, the present value of the annuity is $P$, future value of the series is $F$, interest rate is $i$ and number of periods is $n$.

Substitute $\$18,000$ for $P$, $\$3,467.49$ for $F_1$, $\$4,431.19$ for $F_2$, $\$4,001.69$ for $F_3$, $\$3,734.19$ for $F_4$, $\$3,583.33$ for $F_5$, $\$3,747.52$ for $F_6$, , $\$3,858.33$ for $F_7$, $\$3633.76$ for $F_8$, $\$3,410$ for $F_9$, $\$3,547.5$ for $F_{10}$ in Equation (IV).

$\begin{array}{l}\$18,000=\left[\begin{array}{l}\$3,467.49\left(P/F,i,1\right)+\$4,431.19\left(P/F,i,2\right)\\ +\$4,001.69\left(P/F,i,3\right)+\$3,7341.19\left(P/F,i,4\right)\\ +\$3,583.33\left(P/F,i,5\right)+\$3,747.52\left(P/F,i,6\right)\\ +\$3,858.33\left(P/F,i,7\right)+\$3,633.76\left(P/F,i,8\right)\\ +\$3,410\left(P/F,i,9\right)+\$3,547.5\left(P/F,i,10\right)\end{array}\right]\\ \end{array}$ .... (V)

Solve Equation (V) and calculate the value of $i$

$i=8.5\%$.

Conclusion:

Thus, the after rate tax return is $8.5\%$.

To determine

(c)

The rate of return for at the end of fifth year.

Answer to Problem 37P

The rate of return in this case is $8.97\%$.

Explanation of Solution

Given data:

Selling price is $\$10,000$.

Calculation:

Calculate the depreciation.

Year	Cost basis $\left(\text{a}\right)$	MACRS Depreciation rate $\left(\text{b}\right)$	Depreciation $\left(\text{c}\right)$ $\left(\text{a}\times \text{b}\right)$
$1$	$\$10,000$	$14.29\%$	$\$1,429$
$2$	$\$10,000$	$24.49\%$	$\$2,449$
$3$	$\$10,000$	$17.49\%$	$\$1,749$
$4$	$\$10,000$	$12.49\%$	$\$1,249$
$5$	$\$10,000$	$8.93\%$	$\$893$

Calculate before tax cash flow

Year	Gross incomea	Expensesb	Before tax cash flowc $\left(a-b\right)$
0			$-\$10,000$
1	$\$4500$	$\$300$	$\$4,200$
2	$\$5,000$	$\$550$	$\$4,450$
3	$\$5,500$	$\$800$	$\$4,700$
4	$\$6,000$	$\$1,050$	$\$4,950$
5	$\$6,500$	$\$1,300$	$\$5,200$

Calculate after tax cash flow.

Year	Before tax cash flow	Depreciation	Taxable Income	Tax	After Tax cash Flow
$0$					$-\$10,000$
$1$	$\$4,200$	$\$2,572.2$	$\$1,627.8$	$\$732.51$	$\$3,467.49$
$2$	$\$4,450$	$\$4,408.2$	$\$41.8$	$\$18.81$	$\$4,431.19$
$3$	$\$4,700$	$\$3,148.2$	$\$1,551.8$	$\$698.31$	$\$4,001.69$
$4$	$\$4,950$	$\$2,248.2$	$\$2,701.8$	$\$1,215.81$	$\$3,734.19$
$5$	$\$5,200$	$\$1,607.4$	$\$3592.6$	$\$1,616.67$	$\$3,583.33$

Write the expression for after tax rate of return.

$P=\left[\begin{array}{l}{F}_1\left(P/F,i,n\right)+F_2\left(P/F,i,n\right)\\ +F_3\left(P/F,i,n\right)+F_4\left(P/F,i,n\right)\\ +F_5\left(P/F,i,n\right)\end{array}\right]$ .... (I)

Here, the present value of the annuity is $P$, future value of the series is $F$, interest rate is $i$ and number of periods is $n$.

Substitute $\$10,000$ for $P$, $\$3,467.49$ for $F_1$, $\$4,431.19$ for $F_2$, $\$4,001.69$ for $F_3$, $\$3,734.19$ for $F_4$, $\$3,583.33$ for $F_5$ in Equation (I).

$\begin{array}{l}\$10,000=\left[\begin{array}{l}\$3,467.49\left(P/F,i,1\right)+\$4,431.19\left(P/F,i,2\right)\\ +\$4,001.69\left(P/F,i,3\right)+\$3,7341.19\left(P/F,i,4\right)\\ +\$3,583.33\left(P/F,i,5\right)\end{array}\right]\\ \end{array}$ .... (II)

Solve Equation (II) and calculate the value of $i$

$i=8.97\%$.

Conclusion:

Thus, the rate of return in this case is $8.97\%$.