Chapter 14, Problem 85P

To determine

(a)

The after tax rate of return.

Answer to Problem 85P

The after tax rate of return will be $5.25\%$.

Explanation of Solution

Given:

Cost of land is $\$17,500$.

Cost of the house is $\$67,500$.

Net income is $\$8000$.

Calculation:

Calculate the straight line depreciation.

$SL\left(d_1\right)=\frac{B-S}{N}$ ...... (I)

Here, book value is B, salvage value is S and number of years is N.

Substitute $\$67,500$ for B, $\$0$ for S and $45$ years for N.

$\begin{array}{c}SL\left(d_1\right)=\frac{\$67,500-\$0}{45}\\ =\$1500\end{array}$

Calculate the depreciation and after tax cash flows.

Year	Before tax cash flow $\left(\text{a}\right)$	Straight line Depreciation $\left(\mathrm{b}\right)$	Taxable Income $\left(\text{c}\right)$ $\left(\text{a}-\mathrm{b}\right)$	Income taxes $34\%$ $\begin{array}{c}\left(\text{d}\right)\\ \left(\text{c}\times 34\%\right)\end{array}$	After tax cash flowActual dollars $\left(\text{c}-\text{d}\right)$
$0$	$\$85,000$				$-\$85,000$
$1$	$\$8,000$	$\$1500$	$\$6500$	$\$2210$	$\$5790$
$2$	$\$8,000$	$\$1500$	$\$6500$	$\$2210$	$\$5790$
$3$	$\$8,000$	$\$1500$	$\$6500$	$\$2210$	$\$5790$
$4$	$\$8,000$	$\$1500$	$\$6500$	$\$2210$	$\$5790$
$5$	$\$8,000$	$\$1500$	$\$6500$	$\$2210$	$\$83290$
Total		$\$7500$

Calculate the book value at the end of $5$ years.

$\text{Book}\text{value}=\text{Cost}\text{of}\text{property}-n\left(SL\left(d_1\right)\right)$ ...... (II)

Substitute $\$85,000$ for cost of property, and $5$ years for n in Equation (II).

$\begin{array}{c}\text{Book}\text{value}=\$85,000-5\left(1500\right)\\ =\$77,500\end{array}$

Calculate the after tax cash flow at the end of $5$ years.

$ATC{F}_5=\text{Book}\text{value}+ATC{F}_4$ ...... (III)

Substitute $\$77,500$ for book value at the end of $5$ years and $\$5790$ for ${\text{ATCF}}_4$.

$\begin{array}{c}ATC{F}_5=\$77,500+\$5790\\ =\$83,290\end{array}$

Determine the rate of return after tax.

$\begin{array}{c}\text{Cost}\text{of}\text{property}=\left[\begin{array}{l}\left(ATCF\right)\left(\frac{P}{A},i,n\right)\\ +\text{Book}\text{value}\text{at}\text{end}\text{of}\text{5}\text{years}\left(\frac{P}{F},i,n\right)\end{array}\right]\\ =\left[\begin{array}{c}\left(ATCF\right)\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)\\ +\text{Book}\text{value}\text{at}\text{end}\text{of}\text{5}\text{years}\left(\frac{1}{{\left(1+i\right)}^n}\right)\end{array}\right]\end{array}$ ...... (IV)

Use trial and error method.

Try $i=5\%$.

Substitute $\$85,000$ for cost of property, $\$5790$ for $ATCF$, $5\%$ for $i$ and $\$77500$ for Book value in Equation (IV).

$\begin{array}{c}\$85,000=\left[\left(5790\right)\left(\frac{{\left(1+0.05\right)}^5-1}{0.05{\left(1+0.05\right)}^5}\right)+\$77500\left(\frac{1}{{\left(1+0.05\right)}^5}\right)\right]\\ \$85,000=\left(\$5790\left(4.328\right)+60727.15\right)\\ \$85,000\ne \$85788\end{array}$

Try $i=5.25\%$.

Substitute $\$85,000$ for cost of property, $\$5790$ for $ATCF$, $5\%$ for $i$ and $\$77500$ for Book value in Equation (IV).

$\begin{array}{c}\$85,000=\left[\left(5790\right)\left(\frac{{\left(1+0.0525\right)}^5-1}{0.05{\left(1+0.05\right)}^5}\right)+\$77500\left(\frac{1}{{\left(1+0.0525\right)}^5}\right)\right]\\ \$85,000=\left(\$5790\left(4.3744\right)+60,005.51\right)\\ \$85,000\approx \$85332\end{array}$

Thus, the rate of return after tax is $5.25\%$.

Conclusion:

The after tax rate of return will be $5.25\%$.

To determine

(b)

The after tax rate of return considering inflation.

Answer to Problem 85P

The after tax rate of return in actual dollars is $14.9\%$.

The after tax rate of return in year- $0$ dollars is $7.4\%$.

Explanation of Solution

Given:

Inflation rate is $7\%$.

Calculation:

Calculate the inflation applied cash flows.

Determine the before tax cash flow for year $1$.

$BTC{F}_1=BTCF+\left(BTCF\times f\right)$ ...... (V)

Here, f is the inflation rate.

Substitute $\$8000$ for BTCF and $7\%$ for f in Equation (V).

$\begin{array}{c}BTC{F}_1=\$8000+\left(\$8000\times 0.07\right)\\ =\$8560\end{array}$

Calculate the depreciation when inflation is applied.

Year	Before tax cash flow $\left(\text{a}\right)$ $\begin{array}{c}\text{increasing}\\ 7\%\text{per}\text{year}\end{array}$	Straight line Depreciation $\left(\mathrm{b}\right)$	Taxable Income $\left(\text{c}\right)$ $\left(\text{a}-\mathrm{b}\right)$	Income taxes $34\%$ $\begin{array}{c}\left(\text{d}\right)\\ \left(\text{c}\times 34\%\right)\end{array}$	After tax cash flowActual dollars $\left(\text{c}-\text{d}\right)$
$0$	$\$85,000$				$-\$85,000$
$1$	$\$8,560$	$\$1500$	$\$7060$	$\$2400$	$\$6160$
$2$	$\$9159$	$\$1500$	$\$7659$	$\$2604$	$\$6555$
$3$	$\$9800$	$\$1500$	$\$8300$	$\$2822$	$\$6978$
$4$	$\$10486$	$\$1500$	$\$8986$	$\$3055$	$\$7731$
$5$	$\$11220$	$\$1500$	$\$9720$	$\$3305$	$\$131913$
Total		$\$7500$

Calculate the selling price.

$\begin{array}{c}\text{Selling}\text{price}=\left[\left(\text{Cost}\text{of}\text{property}\right)\left(\frac{F}{P},i,n\right)\right]\\ =\text{Cost}\text{of}\text{property}\left[{\left(1+i\right)}^n\right]\end{array}$ ...... (VI)

Substitute $\$85000$ for cost of property and $5$ years for n in Equation (VI).

$\begin{array}{c}\text{Selling}\text{price}=\$85000\left[{\left(1+0.10\right)}^5\right]\\ =\$136850\end{array}$

Calculate the capital gain.

$\begin{array}{c}\text{Capital}\text{gain}=\left(\text{selling}\text{price}\right)-\left(\text{cost}\text{of}\text{property}\right)\\ =\$136850-85000\\ =\$51850\end{array}$

Calculate the tax on capital gain.

$\begin{array}{c}\text{Tax}\text{on}\text{capital}\text{gain}=\left(\text{Tax}\text{rate}\right)\times \left(\text{Capital}\text{gain}\right)\\ =0.2\times 51850\\ =\$10370\end{array}$

Use trial and error method.

Try $i=10\%$.

Substitute $\$85,000$ for cost of property, $\$6160$ for $ATCF$, $10\%$ for $i$ and $\$77500$ for Book value in Equation (IV).

$\begin{array}{c}\$85,000=\left[\left(6160\right)\left(\frac{{\left(1+0.10\right)}^5-1}{0.10{\left(1+0.10\right)}^5}\right)+\$77500\left(\frac{1}{{\left(1+0.10\right)}^5}\right)\right]\\ \$85,000=\left(\$6160\left(10\right)+\$77500\left(0.6209\right)\right)\\ \$85,000\ne \$109719\end{array}$

Repeat the same procedure to find the after tax rate of return.

The after tax rate of return is coming out to be $14.9\%$.

Calculate the after tax cash flow for year $0$ dollars.

Year	After tax cash flowActual dollars $\left(\text{a}\right)$	Multiply by ${\left(1+0.07\right)}^{-n}$ $\left(\mathrm{b}\right)$	ATCF year $0$ dollars $\left(\text{c}\right)$ $\left(\text{c}\right)$
$0$	$-\$85,000$	$1$
$1$	$\$6160$	${1.07}^{-1}$	$\$5757$
$2$	$\$6555$	${1.07}^{-2}$	$\$5725$
$3$	$\$6978$	${1.07}^{-3}$	$\$5696$
$4$	$\$7731$	${1.07}^{-4}$	$\$5669$
$5$	$\$131913$	${1.07}^{-5}$	$\$94052$

Use trial and error method.

Try $i=7\%$.

Substitute $\$85,000$ for cost of property, $\$5757$ for $ATCF$, $7\%$ for $i$ and $\$77500$ for Book value in Equation (IV).

$\begin{array}{c}\$85,000=\left[\left(5757\right)\left(\frac{{\left(1+0.07\right)}^5-1}{0.07{\left(1+0.07\right)}^5}\right)+\$77500\left(\frac{1}{{\left(1+0.07\right)}^5}\right)\right]\\ \$85,000=\left(\$5757\left(4.102\right)+\$77500\left(0.7129\right)\right)\\ \$85,000\ne \$78864\end{array}$

Repeat the same procedure to find the after tax rate of return.

The after tax rate of return for year $0$ dollars is coming out to be $7.4\%$.

Conclusion:

The after tax rate of return in actual dollars is $14.9\%$.

The after tax rate of return in year- $0$ dollars is $7.4\%$.