Chapter 12, Problem 17P

To determine

(a)

The after tax rate of return for the investment.

Answer to Problem 17P

The internal rate if return is $7.72\%$.

Explanation of Solution

Given:

The cost of installing a machine is $\$200,000$.

The useful life is 20 years.

Salvage value is $\$50000$.

Per month income is $\$2500$.

Per month expenses is $\$200$.

Interest rate is $38\%$.

Concept used:

Write the expression for straight line depreciation.

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

Here, cost of installing is $B$, useful life in years is $N$ and salvage value is $S$.

Write the expression for taxable income.

$\text{Taxable income }\left(\text{c}\right)=a-b$ ....... (II)

Here, before tax cash flow is $a$ and depreciation is $b$.

Write the expression for income tax.

$\text{Income tax}\left(d\right)=-\left[\left(c\right)\times 0.40\right]$ ....... (III)

Here, taxable income is $c$.

Write the expression for after tax cash flow.

$\text{After tax cash flow }\left(e\right)=a+d$ ....... (IV)

Write the expression for after tax rate of return.

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

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 $n$.

Write the expression for rate of return by linear interpolation.

$IRR=i{\%}_{low}+\left(i{\%}_{high}-i{\%}_{low}\right)\left[\frac{F_{low}-P}{F_{low}-F_{high}}\right]$ ....... (VI)

Here, the internal rate of return is $IRR$, lower interest rate is $i{\%}_{low}$, higher interest rate is $i{\%}_{high}$, factor of lower interest rate is $F_{low}$, factor of higher interest rate is $F_{high}$,

If after-tax $IRR$ is less than expected, then the investment is not considered.

If after-tax $IRR$ is greater than expected, then the investment is considered.

Calculation:

Calculate the depreciation of the machine.

Substitute $\$200,000$ for $B$, $\$50000$ for $S$ and $20$ for $N$ in Equation (I).

$\begin{array}{c}\text{SLN Depreciation}=\frac{\$200,0000-\$50000}{20}\\ =\$7500\end{array}$

The depreciation charged in the machine would be $\$7500$.

Calculate the before tax cash flow.

$\begin{array}{c}\text{Net income}=\text{income}-\text{expenses}\\ \text{=}\$2500-\$200\\ =\$2300\text{/month}\end{array}$

$\begin{array}{c}\text{BTCF}=\$2300\text{/month}\left(\frac{12\text{ month}}{1\text{ year}}\right)\\ =\$27600\text{/year}\end{array}$

Here, the before tax cash flow is BTCF.

Calculate the taxable income.

Substitute $\$27600$ for $a$ and $\$7500$ for $b$ in Equation (II).

$\begin{array}{c}\text{Taxable income }\left(\text{c}\right)=\$27600-\$7500\\ =\$20100\end{array}$

Calculate the income tax.

Substitute $\$20100$ for $c$ in Equation (III).

$\begin{array}{c}\text{Income tax}\left(d\right)=-\left[\left(\$14,000\right)\times 0.38\right]\\ =-\$7638\end{array}$

Calculate the after tax cash flow.

Substitute $-\$7638$ for $d$ and $\$27600$ for $a$ in Equation (IV).

$\begin{array}{c}\text{After tax cash flow }\left(e\right)=\$27600+\left(-\$7638\right)\\ =\$19962\end{array}$

The table for computation of the after tax rate of return for investment is shown below.

	(a)	(b)	(c)	(d)	(e)
Year	Before tax cash flow	Depreciation	Taxable Income $\left(a-b\right)$	Income tax $-\left[\left(c\right)\times 0.40\right]$	After tax cash flow
$0$	$-\$200,000$				$-\$200,000$
$1$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
$2$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
$3$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
$4$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
$5$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
6	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
7	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
8	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
9	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
10	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
11	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
12	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
13	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
14	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
15	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
16	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
17	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
18	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
19	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
20	$\$50000$				$\$50000$

Table (1)

Calculate the after tax rate of return.

Substitute $\$200,000$ for $P$, $\$19962$ for $F$ in Equation (V).

$\begin{array}{l}\$200,000=\left[\begin{array}{l}\$19962\left(P/F,i,1\right)+\$19962\left(P/F,i,2\right)\\ +\mathrm{......}+\$19962\left(P/F,i,20\right)\end{array}\right]\\ 10.02i{\left(1+i\right)}^{20}-\left(1+i\right)20+1=0\end{array}$

Solve the equation for $i$.

$i=7.72\%$

The internal rate if return is $7.72\%$.

Conclusion:

The internal rate if return is $7.72\%$.

To determine

(b)

The after tax rate of return for the investment.

Answer to Problem 17P

The internal rate if return is $7.72\%$.

Explanation of Solution

Given:

The cost of installing a machine is $\$200,000$.

The useful life is 20 years.

Salvage value is $\$50000$.

The market price is $\$150000$

Per month income is $\$2500$.

Per month expenses is $\$200$.

Interest rate is $38\%$.

Calculation:

Calculate the depreciation of the machine.

Substitute $\$200,000$ for $B$, $\$50000$ for $S$ and $20$ for $N$ in Equation (I).

$\begin{array}{c}\text{SLN}\text{Depreciation}=\frac{\$200,0000-\$50000}{20}\\ =\$7500\end{array}$

The depreciation charged in the machine would be $\$7500$.

Calculate the before tax cash flow.

$\begin{array}{c}\text{Net income}=\text{income}-\text{expenses}\\ \text{=}\$2500-\$200\\ =\$2300\text{/month}\end{array}$

$\begin{array}{c}\text{BTCF}=\$2300\text{/month}\left(\frac{12\text{ month}}{1\text{ year}}\right)\\ =\$27600\text{/year}\end{array}$

Calculate the taxable income.

Substitute $\$27600$ for $a$ and $\$7500$ for $b$ in Equation (II).

$\begin{array}{c}\text{Taxable income }\left(\text{c}\right)=\$27600-\$7500\\ =\$20100\end{array}$

Calculate the income tax.

Substitute $\$20100$ for $c$ in Equation (III).

$\begin{array}{c}\text{Income tax}\left(d\right)=-\left[\left(\$14,000\right)\times 0.38\right]\\ =-\$7638\end{array}$

Calculate the after tax cash flow.

Substitute $-\$7638$ for $d$ and $\$27600$ for $a$ in Equation (IV).

$\begin{array}{c}\text{After tax cash flow }\left(e\right)=\$27600+\left(-\$7638\right)\\ =\$19962\end{array}$

The table for computation of the after tax rate of return for investment is shown below.

	(a)	(b)	(c)	(d)	(e)
Year	Before tax cash flow	Depreciation	Taxable Income $\left(a-b\right)$	Income tax $-\left[\left(c\right)\times 0.40\right]$	After tax cash flow
$0$	$-\$200,000$				$-\$200,000$
$1$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
$2$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
$3$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
$4$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
$5$	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
6	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
7	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
8	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
9	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
10	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
11	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
12	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
13	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
14	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
15	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
16	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
17	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
18	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
19	$\$27600$	$\$7500$	$\$20100$	$-\$7638$	$\$19962$
20	$\$150000$		$\$100000$	$-\$38000$	$-\$112000$

Table (1)

Calculate the after tax rate of return.

Substitute $\$200,000$ for $P$, $\$19962$ for $F$ in Equation (V).

$\begin{array}{l}\$200,000=\left[\begin{array}{l}\$19962\left(P/F,i,1\right)+\$19962\left(P/F,i,2\right)\\ +\mathrm{......}+\$19962\left(P/F,i,20\right)\end{array}\right]\\ 10.02i{\left(1+i\right)}^{20}-\left(1+i\right)20+1=0\end{array}$ ....... (VI)

Solve the equation for $i$

$i=7.72\%$

The internal rate if return is $7.72\%$.

Conclusion:

The internal rate if return is $7.72\%$.