Chapter 12, Problem 20P

To determine

The after tax rate of return.

Answer to Problem 20P

The after tax rate of return is $11.62\%$.

Explanation of Solution

Concept used:

Write the expression for depreciation.

$D=\frac{2}{n}\times B{V}_{t-1}$ ....... (I)

Here, book value is $BV$ and useful life is $n$.

Write the expression for taxable income.

${\left(\text{Taxable income}\right)}_t=BTC{F}_t-D_t$ ....... (II)

Here before tax cash flow is $BTCF$.

Write the expression for income tax.

${\left(\text{Tax}\right)}_t={\left(\text{Taxable income}\right)}_t\times \text{tax rate}$ ....... (III)

Write the expression for after tax cash flow.

${\left(ATCF\right)}_t=BTC{F}_t-Ta{x}_t$ ....... (IV)

Write the expression for after tax rate of return.

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

Calculation:

Calculate depreciation.

Substitute $\$100000$ for $B{V}_{t-1}$, $4$ for $n$ and $1$ for $t$ in Equation (I).

$\begin{array}{c}{D}_1=\frac{2}{4}\times \$100000\\ =\$50000\end{array}$

Substitute $\$50000$ for $B{V}_{t-1}$, $4$ for $n$ and $2$ for $t$ in Equation (I).

$\begin{array}{c}{D}_2=\frac{2}{4}\times \$50000\\ =\$25000\end{array}$

Substitute $\$25000$ for $B{V}_{t-1}$, $4$ for $n$ and $3$ for $t$ in Equation (I).

$\begin{array}{c}{D}_3=\frac{2}{4}\times \$25000\\ =\$12500\end{array}$

Substitute $\$12500$ for $B{V}_{t-1}$, $4$ for $n$ and $4$ for $t$ in Equation (I).

$\begin{array}{c}{D}_4=\frac{2}{4}\times \$12500\\ =\$6250\end{array}$

Calculate taxable income.

Substitute $\$30000$ for $BTCF$, $\$50000$ for $D_1$ and $1$ for $t$ in Equation (II).

$\begin{array}{c}{\left(\text{Taxable income}\right)}_1=\$30000-\$50000\\ =-\$20000\end{array}$

Substitute $\$30000$ for $BTCF$, $\$25000$ for $D_2$ and $2$ for $t$ in Equation (II).

$\begin{array}{c}{\left(\text{Taxable income}\right)}_2=\$30000-\$25000\\ =\$5000\end{array}$

Substitute $\$35000$ for $BTCF$, $\$12500$ for $D_3$ and $3$ for $t$ in Equation (II).

$\begin{array}{c}{\left(\text{Taxable income}\right)}_3=\$35000-\$12500\\ =\$22500\end{array}$

Substitute $\$40000$ for $BTCF$, $\$62500$ for $D_4$ and $4$ for $t$ in Equation (II).

$\begin{array}{c}{\left(\text{Taxable income}\right)}_4=\$40000-\$62500\\ =\$33750\end{array}$

Substitute $\$10000$ for $BTCF$, $0$ for $D_5$ and $5$ for $t$ in Equation (II).

$\begin{array}{c}{\left(\text{Taxable income}\right)}_5=\$10000-0\\ =\$10000\end{array}$

Substitute $\$10000$ for $BTCF$, $0$ for $D_6$ and $6$ for $t$ in Equation (II).

$\begin{array}{c}{\left(\text{Taxable income}\right)}_6=\$10000-0\\ =\$10000\end{array}$

Calculate income tax.

Substitute $-\$20000$ for taxable income, $0.46$ for tax rate and $1$ for $t$ in Equation (III).

$\begin{array}{c}{\left(\text{Tax}\right)}_1=-\$20000\times 0.46\\ =-\$9200\end{array}$

Substitute $\$5000$ for taxable income, $0.46$ for tax rate and $2$ for $t$ in Equation (III).

$\begin{array}{c}{\left(\text{Tax}\right)}_2=\$5000\times 0.46\\ =\$2300\end{array}$

Substitute $\$22500$ for taxable income, $0.46$ for tax rate and $3$ for $t$ in Equation (III).

$\begin{array}{c}{\left(\text{Tax}\right)}_3=\$22500\times 0.46\\ =\$10350\end{array}$

Substitute $\$33750$ for taxable income, $0.46$ for tax rate and $4$ for $t$ in Equation (III).

$\begin{array}{c}{\left(\text{Tax}\right)}_4=\$33750\times 0.46\\ =\$15525\end{array}$

Substitute $\$10000$ for taxable income, $0.46$ for tax rate and $5$ for $t$ in Equation (III).

$\begin{array}{c}{\left(\text{Tax}\right)}_5=\$10000\times 0.46\\ =\$4600\end{array}$

Substitute $\$10000$ for taxable income, $0.46$ for tax rate and $6$ for $t$ in Equation (III).

$\begin{array}{c}{\left(\text{Tax}\right)}_6=\$10000\times 0.46\\ =\$4600\end{array}$

Calculate after tax cash flow.

Substitute $\$30000$ for $BTCF$, $-\$9200$ and $1$ for $t$ for tax in Equation (IV).

$\begin{array}{c}{\left(ATCF\right)}_1=\$30000-\left(-\$9200\right)\\ =\$39200\end{array}$

Substitute $\$30000$ for $BTCF$, $\$2300$ and $2$ for $t$ for tax in Equation (IV).

$\begin{array}{c}{\left(ATCF\right)}_2=\$30000-\$2300\\ =\$27700\end{array}$

Substitute $\$35000$ for $BTCF$, $\$10350$ and $3$ for $t$ for tax in Equation (IV).

$\begin{array}{c}{\left(ATCF\right)}_3=\$35000-\$10350\\ =\$24650\end{array}$

Substitute $\$40000$ for $BTCF$, $\$15525$ and $4$ for $t$ for tax in Equation (IV).

$\begin{array}{c}{\left(ATCF\right)}_4=\$40000-\$15525\\ =\$24475\end{array}$

Substitute $\$10000$ for $BTCF$, $\$4600$ and $5$ for $t$ for tax in Equation (IV).

$\begin{array}{c}{\left(ATCF\right)}_5=\$10000-\$4600\\ =5400\end{array}$

Substitute $\$10000$ for $BTCF$, $\$4600$ and $6$ for $t$ tax in Equation (IV).

$\begin{array}{c}{\left(ATCF\right)}_6=\$10000-\$4600+\$6250\\ =\$11650\end{array}$

The table for the calculated values is shown below.

Year	$BTCF$	$D$	$\text{Taxable income}$	$\text{Tax}$	$ATCF$
0	$-\$100000$				$-\$100000$
1	$\$30000$	$\$50000$	$-\$20000$	$-\$9200$	$\$39200$
2	$\$30000$	$\$25000$	$\$5000$	$\$2300$	$\$27700$
3	$\$35000$	$\$12500$	$\$22500$	$\$10350$	$\$24650$
4	$\$40000$	$\$6250$	$\$33750$	$\$15525$	$\$24475$
5	$\$10000$	$0$	$\$10000$	$\$4600$	$\$5400$
6	$\$10000$	$0$	$\$10000$	$\$4600$	$\$11650$

Table (1)

Calculate the after tax rate of return.

Substitute $\$100000$ for $P$, $\$39200$ for $F_1$, $\$27700$ for $F_2$, $\$24650$ for $F_3$, $\$24475$ for $F_4$, $\$5400$ for $F_5$, $\$11650$ for $F_6$ in Equation (V).

$\begin{array}{l}\$100000=\left[\begin{array}{l}\$39200\left(P/F,i,1\right)+\$27700\left(P/F,i,2\right)\\ +\$24650\left(P/F,i,3\right)+\$24475\left(P/F,i,4\right)\\ +\$5400\left(P/F,i,5\right)+\$11650\left(P/F,i,6\right)\end{array}\right]\\ \end{array}$ ....... (VII)

Consider the table and apply trial and error method equalizing the equation (VII).

Determine an interest rate which equalizes left hand side and right hand side of Equation (VII).

Substitute the tabular values into the right hand side of the equation (VII).

The table for values close to the left hand side of the equation (VII) is shown below.

Interest rate	Value
$10\%$	$\$103693.30$
$12\%$	$\$99149.70$

Table (2)

Calculate the rate of return by linear interpolation.

Substitute $\$100000$ for $P$, $10\%$ for $i{\%}_{low}$, $12\%$ for $i{\%}_{high}$, $\$103693.30$ for $F_{low}$ and $\$99149.70$ for $F_{high}$ in Equation (VI).

$\begin{array}{c}IRR=10\%+\left(12\%-10\%\right)\left[\frac{\$103693.30-\$100000}{\$103693.30-\$99149.70}\right]\\ =10\%+2\%\left[\frac{\$3693.30}{\$4543.60}\right]\\ =0.1+0.02\times 0.8218\\ =0.1162\end{array}$.

Conclusion:

The internal rate if return is $11.62\%$.