Chapter 14, Problem 83P

To determine

(a)

The actual dollar after tax rate of return.

Answer to Problem 83P

The actual dollar after tax rate of return is $16.17\%$.

Explanation of Solution

Given:

Net annual income to the owner is $10.5\%$ of the market value.

Purchase amount of the property is $\$150000$.

Cost of the land is $\$46500$.

The selling price $59\text{ months}$ later is equal to the original purchase price.

The income tax rate is $35\%$.

Calculation:

Calculate the before tax cash flow for year 1.

$BTC{F}_1=f\times 150000$ ...... (I)

Here, the inflation rate is $f$.

Substitute $8\%$ for $f$ in Equation (I).

$\begin{array}{c}BTC{F}_1=0.08\times \$150000\\ =\$12000\end{array}$

Calculate the before tax cash flow for year 2.

$BTC{F}_2=f\times 150000+\left(f\times BTC{F}_1\right)+\left(f_1\times BTC{F}_1\right)$ ...... (II)

Here, the inflation rate is $f$ and the market interest rate is $f_1$.

Substitute $10\%$ for $f$, $2\%$ for $f_1$ and $\$12000$ for $BTCF$ in Equation (II).

$\begin{array}{c}BTC{F}_2=\$12000+\left(0.10\times \$12000\right)+\left(0.02\times \$12000\right)\\ =\$12000+\$1200+\$240\\ =\$13440\end{array}$

Calculate the before tax cash flow for year $3$.

$BTC{F}_3=BTC{F}_2+\left(BTC{F}_2\times 012\right)$ ...... (III)

Substitute $\$13440$ for $BTC{F}_2$ in Equation (III).

$\begin{array}{c}BTC{F}_3=\$13440+\left(\$13440\times 0.12\right)\\ =\$15053\end{array}$

Calculate the before tax cash flow for year $4$.

$BTC{F}_4=BTC{F}_3+\left(BTC{F}_3\times 0.12\right)$ ...... (IV)

Substitute $\$15053$ for $BTC{F}_3$ in Equation (IV).

$\begin{array}{c}BTC{F}_4=\$15053+\left(\$15053\times 0.12\right)\\ =\$16859\end{array}$

Calculate the before tax cash flow for year $5$.

$BTC{F}_5=BTC{F}_4+\left(BTC{F}_4\times 0.12\right)$ ...... (V)

Substitute $\$16859$ for $BTC{F}_4$ in Equation (V).

$\begin{array}{c}BTC{F}_5=\$16859+\left(\$16859\times 0.12\right)\\ =\$18882\end{array}$

Calculate the cost of home.

$\text{Home}\text{Cost}=\text{PC}-\text{LC}$ ..... (VI)

Here, the property cost is $\text{PC}$ and land cost is $\text{LC}$.

Substitute $\$150000$ for $\text{PC}$ and $\$46500$ for $\text{LC}$ in Equation (VI).

$\begin{array}{c}\text{Home}\text{Cost}=\text{PC}-\text{LC}\\ \text{=\$150000}-\$46500\\ =\$103500\end{array}$

Write the expression to calculate the market value plus $12\%$ per year.

$M{V}_{+12\%}=\left(\left(M{V}_{n-1}\right)+\left(0.12\times M{V}_{n-1}\right)\right)$ ...... (VII)

Here, the market value of the precious year is $M{V}_{n-1}$ and the market value of the present year with $12\%$ increase is $M{V}_{+12\%}$.

Calculate the market value plus $12\%$ per year for year $1$.

Substitute $\$150000$ for $M{V}_{n-1}$ in Equation (VII).

$\begin{array}{c}M{V}_{+12\%}=\left(\left(\$150000\right)+\left(0.12\times \$150000\right)\right)\\ =\$168000\end{array}$

Calculate the market value plus $12\%$ per year for year $2$.

Substitute $\$168000$ for $M{V}_{n-1}$ in Equation (VII).

$\begin{array}{c}M{V}_{+12\%}=\left(\left(\$168000\right)+\left(0.12\times \$168000\right)\right)\\ =\$188160\end{array}$

Calculate the market value plus $12\%$ per year for year $3$.

Substitute $\$188160$ for $M{V}_{n-1}$ in Equation (VII).

$\begin{array}{c}M{V}_{+12\%}=\left(\left(\$188160\right)+\left(0.12\times \$188160\right)\right)\\ =\$210775.2\end{array}$

Calculate the market value plus $12\%$ per year for year $3$.

Substitute $\$188160$ for $M{V}_{n-1}$ in Equation (VII).

$\begin{array}{c}M{V}_{+12\%}=\left(\left(\$188160\right)+\left(0.12\times \$188160\right)\right)\\ =\$210739\end{array}$

Calculate the market value plus $12\%$ per year for year $4$.

Substitute $\$210739$ for $M{V}_{n-1}$ in Equation (VII).

$\begin{array}{c}M{V}_{+12\%}=\left(\left(\$210739\right)+\left(0.12\times \$210739\right)\right)\\ =\$236028\end{array}$

Calculate the market value plus $12\%$ per year for year $5$.

Substitute $\$236028$ for $M{V}_{n-1}$ in Equation (VII).

$\begin{array}{c}M{V}_{+12\%}=\left(\left(\$236028\right)+\left(0.12\times \$236028\right)\right)\\ =\$264351\end{array}$

Make the table to calculate the MARCS $5$ year depreciation using values for residential property.

Year	Initial Cost $\left(a\right)$	MARCS rate $\left(b\right)$	Depreciation $\begin{array}{c}\left(c\right)\\ \left(a\times b\right)\end{array}$
$0$
$1$	$\$103500$	$3.485\%$	$\$3607$
$2$	$\$103500$	$3.636\%$	$\$3763$
$3$	$\$103500$	$3.636\%$	$\$3763$
$4$	$\$103500$	$3.636\%$	$\$3763$
$5$	$\$103500$	$3.485\%$	$\$3607$

Make the table to calculate the after tax cash flow.

Year	BTCF $\left(a\right)$	Market Value of property $+12\%$	Depreciation $\left(b\right)$	TaxableIncome $\left(c\right)$	Income taxes $\begin{array}{c}\left(d\right)\\ -\left(c\times 0.35\right)\end{array}$	ATCF $\begin{array}{c}\left(e\right)\\ \left(a+d\right)\end{array}$
$0$	$-\$150000$	$\$150000$				$-\$150000$
$1$	$\$12000$	$\$168000$	$\$3607$	$\$8393$	$-\$2938$	$\$9062$
$2$	$\$13440$	$\$188160$	$\$3763$	$\$9677$	$-\$3387$	$\$10053$
$3$	$\$15053$	$\$210739$	$\$3763$	$\$11290$	$-\$3951$	$\$11101$
$4$	$\$16859$	$\$236028$	$\$3763$	$\$13096$	$-\$4584$	$\$12275$
$5$	$\$18882$	$\$264351$	$\$3607$			$\$245469$

The $ATC{F}_5$ is $\$254469$ as mentioned and it will be equal to the original price of the property.

Thus the selling price of the property is $\$261991$.

Write the expression to calculate the capital gain.

$\text{Capital}\text{gain}=\left(\text{Selling}\text{Price}\right)-\left(\text{Cost}\text{of}\text{the}\text{property}\right)$ ..... (VIII)

Substitute $\$261991$ for $\text{Selling}\text{Price}$ and $\$150000$ for $\text{Cost}\text{of}\text{the}\text{property}$ in Equation (VIII).

$\begin{array}{c}\text{Capital}\text{gain}=\text{\$261991}-\$150000\\ =\$111991\end{array}$

Calculate the tax on capital gain.

$\begin{array}{c}\text{Tax}\text{on}\text{capital}\text{gain}=0.02\times \$111991\\ =\$22398\end{array}$

Write the expression to calculate the recaptured depreciation.

$RD={\displaystyle \sum MACR\text{}depreciation}$

Substitute $\$18503$ for ${\displaystyle \sum MACR\text{}depreciation}$ in Equation (IV).

$RD=\$18503$

Calculate the tax on recaptured depreciation.

$\begin{array}{c}\text{Tax}\text{on}RD=0.35\times \$18503\\ =\$6476\end{array}$

Calculate the total tax on disposal.

$\text{Total}\text{tax}\text{on}\text{disposal}=\left(\text{Tax}\text{on}\text{capital}\text{gain}\right)+\left(\text{Tax}\text{on}RD\right)$ ....... (IX)

Substitute $\$22398$ for $\text{Tax}\text{on}\text{capital}\text{gain}$ and $\$6476$ for $\text{Tax}\text{on}RD$ in Equation (IX).

$\begin{array}{c}\text{Total}\text{tax}\text{on}\text{disposal}=\$22398+\$6476\\ =\$28874\end{array}$

Calculate the rate of return.

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

Here, the property cost is $P$ and the land cost is $A$.

Substitute $\$150000$ for $P$, $\$46500$ for $A$ and $4.92$ for $n$ in Equation (X).

$\$150000=\$46500\frac{{\left(1+i\right)}^{4.92}-1}{{\left(1+i\right)}^{4.92}i}$

Solve the equation for the value of $i$.

$\begin{array}{l}i=0.1617\times 100\%\\ i=16.17\%\end{array}$

Here, the value of $n$ has been taken $4.92\text{years}$ because the rate of return has to be calculated for $59\text{months}$ which is equal to $4.92\text{years}$.

Conclusion:

The actual dollar after tax rate of return is $16.17\%$.

To determine

(b)

The after tax return for the owner.

Answer to Problem 83P

The after tax return for the owner is $21.9\%$.

Explanation of Solution

Year	ATCF $\left(a\right)$	ATCF Year $0$ $\begin{array}{c}\left(b\right)\\ \left(a\times {1.07}^{-n}\right)\end{array}$
$0$	$-\$150000$	$-\$150000$
$1$	$\$9062$	$\$8469$
$2$	$\$10053$	$\$8781$
$3$	$\$11101$	$\$9061$
$4$	$\$12275$	$\$9365$
$5$	$\$245469$	$\$344283$

Calculate the after tax rate of return in year $0$.

Write the expression to calculate the after tax rate of return.

$P=\frac{A_1}{{\left(1+i\right)}^1}+\frac{A_2}{{\left(1+i\right)}^2}+\frac{A_3}{{\left(1+i\right)}^3}+\frac{A_4}{{\left(1+i\right)}^4}+\frac{A_5}{{\left(1+i\right)}^5}$ ...... (XI)

Here, the present worth is P, the after tax cash flow is A, the rate of return is i.

Substitute $-\$150000$ for $P$, $\$8469$ for $A_1$

$\$8781$ for $A_2$, $\$9061$ for $A_3$, $\$9365$ for $A_4$ and $\$344283$ for $A_5$ in Equation (XI).

$\$150000=\frac{\$8469}{{\left(1+i\right)}^1}+\frac{\$8781}{{\left(1+i\right)}^2}+\frac{\$9061}{{\left(1+i\right)}^3}+\frac{\$9365}{{\left(1+i\right)}^4}+\frac{\$344283}{{\left(1+i\right)}^5}$

Salve for the value of $i$

$\begin{array}{c}i=0.219\\ =0.219\times 100\%\\ =21.9\%\end{array}$.

Conclusion:

The after tax return of the owner is $21.9\%$.