Chapter 12, Problem 64P

To determine

(a)

The rate of return before income taxes.

Answer to Problem 64P

The rate of return before income taxes is $15.4\%$.

Explanation of Solution

Given:

Combined tax rate is $40\%$.

Initial cost is $\$2\text{M}$.

Annual receipt is $\$650000$.

Annual disbursements is $\$225000$.

Duration is $9\text{years}$.

Calculation:

Write the expression to calculate the depreciation using straight line method.

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

Here, the depreciation cost is $D$, the initial cost is $B$, the salvage value is $S$ and the total no of period is $N$

Substitute $\$2000000$ for $B$

$0$ for $S$ and $9$ for $N$ in Equation (I).

$\begin{array}{c}D=\frac{\$2000000-S}{9}\\ =\$222222\end{array}$

Make the table to compute the before tax and after tax cash flow.

Year	BTFC $\left(a\right)$	Depreciation $\left(b\right)$	Taxable Income $c=\left(a-b\right)$	Income taxes $d=-\left(c\times 0.40\right)$	ATFC $e=\left(a+d\right)$
$0$	$-\$2000000$			$\$200000$	$-\$1800000$
$1$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$2$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$3$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$4$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$5$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$6$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$7$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$8$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$9$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$

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

Write the expression to calculate the investment credit.

$ITC=IC\times 0.10$ ...... (II).

Here, the investment tax credit is $ITC$ and initial cost is $IC$.

Substitute $\$2000000$ for $IC$ in Equation (II).

$\begin{array}{c}ITC=\$2000000\times 0.10\\ =\$200000\end{array}$

Write the expression to calculate the after tax net present worth.

$NPW=-P+A\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)$ ...... (III)

Here, the net present worth is $NPW$, the initial cost is $P$, the annual cost is $A$, the interest rate is $i$ and the number of period is $n$.

Substitute $\$2000000$ for $P$, $\$425000$ for $A$, $0$ for $NPW$ and $9$ for $n$ in Equation (III).

$0=-\$2000000+\$425000\left(\frac{{\left(1+i\right)}^9-1}{i{\left(1+i\right)}^9}\right)$

Solve the equation for the value of $i$.

$\begin{array}{c}i=\left(0.154\times 100\right)\%\\ =15.4\%\end{array}$.

Conclusion:

Thus, the rate of return before income taxes is $15.4\%$.

To determine

(b)

The rate of return after taxes.

Answer to Problem 64P

The rate of return after taxes is $12.47\%$.

Explanation of Solution

Given:

Combined tax rate is $40\%$.

Initial cost is $\$2\text{M}$.

Annual receipt is $\$650000$.

Annual disbursements is $\$225000$.

Duration is $9\text{years}$.

Calculation:

Make the table to compute the after tax cash flows.

Year	BTFC $\left(a\right)$	Depreciation $\left(b\right)$	Taxable Income $c=\left(a-b\right)$	Income taxes $d=-\left(c\times 0.40\right)$	ATFC $e=\left(a+d\right)$
$0$	$-\$2000000$			$\$200000$	$-\$1800000$
$1$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$2$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$3$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$4$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$5$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$6$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$7$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$8$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$
$9$	$\$425000$	$\$222222$	$\$202778$	$-\$81111$	$\$343888$

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

Write the expression to calculate the investment credit.

$ITC=IC\times 0.10$ ...... (IV).

Here, the investment tax credit is $ITC$ and initial cost is $IC$.

Substitute $\$2000000$ for $IC$ in Equation (IV).

$\begin{array}{c}ITC=\$2000000\times 0.10\\ =\$200000\end{array}$

Write the expression to calculate the after tax net present worth.

$NPW=-P+A\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)$ ...... (V)

Here, the net present worth is $NPW$, the initial cost is $P$, the annual cost is $A$, the interest rate is $i$ and the number of period is $n$.

Substitute $\$1800000$ for $P$, $\$343888$ for $A$, $0$ for $NPW$ and $9$ for $n$ in Equation (V).

$0=-\$1800000+\$343888\left(\frac{{\left(1+i\right)}^9-1}{i{\left(1+i\right)}^9}\right)$

Solve the equation for the value of $i$.

$\begin{array}{c}i=\left(0.1247\times 100\right)\%\\ =12.47\%\end{array}$.

Conclusion:

Thus, the rate of return after taxes is $12.47\%$.

To determine

(c)

The after tax rate of return after $20$ years.

Answer to Problem 64P

The after tax rate of return after $20$ year is $10.11\%$.

Explanation of Solution

Given:

Combined tax rate is $40\%$.

Initial cost is $\$2\text{M}$.

Annual receipt is $\$650000$.

Annual disbursements is $\$225000$.

Duration is $9\text{years}$.

Calculation:

Calculate the depreciation using straight line method.

Substitute $\$2000000$ for $B$

$0$ for $S$ and $20$ for $N$ in Equation (I).

$\begin{array}{c}D=\frac{\$2000000-S}{20}\\ =\$100000\end{array}$

Make the table to compute the after tax cash flows.

Year	BTFC $\left(a\right)$	Depreciation $\left(b\right)$	Taxable Income $c=\left(a-b\right)$	Income taxes $d=-\left(c\times 0.40\right)$	ATFC $e=\left(a+d\right)$
$0$	$-\$2000000$			$\$200000$	$-\$1800000$
$1$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$2$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$3$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$4$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$5$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$6$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$7$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$8$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$9$	$\$425000$	$\$100000$	$\$325000$	$-\$130000$	$\$295000$
$10$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$11$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$12$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$13$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$14$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$15$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$16$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$17$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$18$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$19$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$
$20$	$0$	$\$100000$	$-\$100000$	$\$40000$	$\$40000$

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

Calculate the investment credit.

Substitute $\$2000000$ for $IC$ in Equation (IV).

$\begin{array}{c}ITC=\$2000000\times .10\\ =\$200000\end{array}$

Write the expression to calculate the after tax net present worth.

$NPW=-P+A_1\left(\frac{{\left(1+i\right)}^9-1}{i{\left(1+i\right)}^9}\right)+A_2\left(\frac{{\left(1+i\right)}^{11}-1}{i{\left(1+i\right)}^{11}}\right)\left(\frac{1}{{\left(1+i\right)}^9}\right)$ ...... (VI)

Here, the net present worth is $NPW$, the initial cost is $P$, the annual cost for $\left(1-9\right)\text{year}$ is $A_1$, the annual cost for year $10-20$ is $A_2$, the interest rate is $i$ and the number of period is $n$.

Substitute $\$1800000$ for $P$, $\$295000$ for $A_1$, $\$40000$ for $A_2$, $0$ for $NPW$ in Equation (VI).

$0=-\$1800000+\$295000\left(\frac{{\left(1+i\right)}^9-1}{i{\left(1+i\right)}^9}\right)+\$40000\left(\frac{{\left(1+i\right)}^{11}-1}{i{\left(1+i\right)}^{11}}\right)\left(\frac{1}{{\left(1+i\right)}^9}\right)$

Solve the equation for the value of $i$

$\begin{array}{c}i=\left(0.1011\times 100\right)\%\\ =10.11\%\end{array}$.

Conclusion:

Thus, the after tax rate of return after $20$ year is $10.11\%$.