Chapter 12, Problem 72P

To determine

The three alternatives for MACRS 3-year property.

Answer to Problem 72P

Choose Alternative C..

Explanation of Solution

Given:

The after-tax MARR is $25\%$.

The project life is $5\text{years}$.

The combined incremental tax rate is $45\%$.

Calculation:

Alternate A:

Write the formula to calculate the depreciation charge for the property at any year.

$d_t=B\times r_t$ .... (I).

Here, depreciation charge in any year $t$ is $d_t$, cost of the property is $B$ and appropriate MACRS percentage rate is $r_t$.

Write the formula to calculate the $\text{Net}\text{book}\text{value}$.

$\text{Net}\text{book}\text{value}=\left(\text{Base}\text{book}\text{value}\right)-\left(\text{Depreciation}\text{charge}\text{for}\text{year}{d}_t\right)$ .... (II).

Determine the values of $r_t$ .throughout the recovery period.

A table would be most suitable to calculate the values of $r_t$.

Recovery year, $t$ (years)	MACRS percentage rate for the recovery year $t$, $r_t$ (in %)
$1$	$33.33\%$
$2$	$44.45\%$
$3$	$14.81\%$

Calculate the depreciation charge for $1^{\text{st}}$ year.

Substitute $\$14000$ for $B$ and $33.33\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{1^{st}}}=\$14000\times \left(\frac{33.33}{100}\right)\\ =\$4666.2\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $1^{\text{st}}$ year.

Substitute $\$14000$ for $\text{Base}\text{book}\text{value}$ and $\$4666.2$ for $\text{Depreciation}\text{charge}\text{for}\text{year}{d}_t$ in Equation (II).

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{1^{\text{st}}}=\$14000-\$4666.2\\ =\$9333.8\end{array}$

Calculate the depreciation charge for $2^{\text{nd}}$ year.

Substitute $\$9333.8$ for $B$ and $44.45\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{2^{\text{nd}}}}=\$9333.8\times \left(\frac{44.45}{100}\right)\\ =\$4148.8\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $2^{\text{nd}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{2^{\text{nd}}}=\$9333.8-\$4148.8\\ =\$5185\end{array}$

Calculate the depreciation charge for $3^{\text{rd}}$ year.

Substitute $\$5185$ for $B$ and $14.81\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{3^{\text{rd}}}}=\$5185\times \left(\frac{14.81}{100}\right)\\ =\$767.9\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $3^{\text{rd}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{3^{\text{rd}}}=\$5158-\$767.9\\ =\$4417.1\end{array}$

Write the values of annual depreciation charge and Net book value in tabular form.

Year, $t$ (years)	Base book value (a)	Depreciation charge for year $d_t$ (b)	Net book value(a-b)
1	$\$14000$	$\$4666.2$	$\$9333.8$
2	$\$9333.8$	$\$4148.8$	$\$5185$
3	$\$5185$	$\$767.9$	$\$4414.1$

Write the formula to calculate the taxable incomes.

$\text{Taxable}\text{Incomes}=\left(\text{Before-tax}\text{cash}\text{flow}\right)-\left(\text{Depreciation}\right)$ .... (III).

Calculate the taxable incomes for $1^{\text{st}}$ year.

Substitute $\$2500$ for $\text{Before-tax}\text{cash}\text{flow}$ and $\$4666.2$ for $\text{Depreciation}$ in Equation (III).

$\begin{array}{c}\text{Taxable}{\text{Incomes}}_{1^{\text{st}}}=\$2500-\$4666.2\\ =-\$2166.2\end{array}$

Calculate the $\text{Income}\text{taxes}$ for $1^{\text{st}}$ year.

$\begin{array}{c}\text{Income}{\text{taxes}}_{1^{\text{st}}}=45\%\text{of}\text{Taxable}{\text{incomes}}_{1^{\text{st}}}\\ =\left(\frac{45}{100}\right)\times \left(-\$2166.2\right)\\ =-\$974.79\end{array}$

Write the formula to calculate the after-tax cash flow.

$\text{After-tax}\text{cash}\text{flow}=\left(\text{Before-tax}\text{cash}\text{flow}\right)-\left(\text{Income}\text{taxes}\right)$ .... (IV).

Calculate the after-tax cash flow.

Substitute $\$2500$ for $\text{Before-tax}\text{cash}\text{flow}$ and $-\$974.79$ for $\text{Income}\text{taxes}$ in Equation (IV).

$\begin{array}{c}\text{After-tax}\text{cash}{\text{flow}}_{1^{\text{st}}}=\$2500-\left(-\$974.79\right)\\ =\$3474.79\end{array}$

Calculate the After-tax cash flow for the remaining years in tabular form.

Period	Before-tax cash flow(p)	MACRS Depreciation(q)	Taxable Incomes $\left(r\right)=\left(p-q\right)$	Income taxes ($45\%$ rate) $\left(s\right)=0.45\times \left(r\right)$	After-tax cash flow $\left(t\right)=\left(p\right)-\left(s\right)$
$0$	$-\$14000$				$-\$14000$
$1$	$\$2500$	$\$4666.2$	$-\$2166.2$	$-\$974.79$	$\$3474.79$
$2$	$\$2500$	$\$4148.8$	$-\$1648.8$	$-\$741.96$	$\$3241.96$
$3$	$\$2500$	$\$767.9$	$\$1732.1$	$\$779.44$	$\$1720.56$
$4$	$\$2500$		$\$2500$	$\$1125$	$\$1375$
$5$	$\$2500$		$\$2500$	$\$1125$	$\$1375$

Write the equation for present worth factor of annuity $\left(PW\right)$.

$\begin{array}{c}PW=D+A\left(\frac{P}{A},i,n\right)+F\left(\frac{P}{F},i,n\right)\\ =D+A\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)+F\left(\frac{1}{{\left(1+i\right)}^n}\right)\end{array}$ .... (V).

Here, initial payment is $D$, present value of the sum of the money is $P$, interest rate is $i$, number of years is $n$, After-tax cash flow per year is $A$ and net salvage amount after three years is $F$.

Calculate present worth factor of annuity.

Substitute $-\$14000$ for $D$, $\$3474.79$ for $A$, $25\%$ for $i$, $5\text{years}$ for $n$ and $\$5000$ for $F$ in Equation (V).

$\begin{array}{c}PW=\left[-14000+\left(\$3474.79\right)\left(\frac{{\left(1+\frac{25}{100}\right)}^5-1}{\left(\frac{25}{100}\right){\left(1+\frac{25}{100}\right)}^5}\right)+\left(\$5000\right)\left(\frac{1}{{\left(1+\frac{25}{100}\right)}^5}\right)\right]\\ =\left[-\$14000+\left(\$3474.79\right)\left(2.6893\right)+\left(\$5000\right)\left(0.3277\right)\right]\\ =\left(-\$14000+\$9344+\$1638.5\right)\\ =-\$3016.75\end{array}$

Thus, the present worth value for Alternate A is $-\$3016.75$.

Alternate B:

Determine the values of $r_t$ .throughout the recovery period.

A table would be most suitable to calculate the values of $r_t$.

Recovery year, $t$ (years)	MACRS percentage rate for the recovery year $t$, $r_t$ (in %)
$1$	$33.33\%$
$2$	$44.45\%$
$3$	$14.81\%$

Calculate the depreciation charge for $1^{\text{st}}$ year.

Substitute $\$18000$ for $B$ and $33.33\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{1^{st}}}=\$18000\times \left(\frac{33.33}{100}\right)\\ =\$5999.4\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $1^{\text{st}}$ year.

Substitute $\$18000$ for $\text{Base}\text{book}\text{value}$ and $\$5999.4$ for $\text{Depreciation}\text{charge}\text{for}\text{year}{d}_t$ in Equation (II).

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{1^{\text{st}}}=\$18000-\$5999.4\\ =\$12000.6\end{array}$

Calculate the depreciation charge for $2^{\text{nd}}$ year.

Substitute $\$12000.6$ for $B$ and $44.45\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{2^{\text{nd}}}}=\$12000.6\times \left(\frac{44.45}{100}\right)\\ =\$5334.3\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $2^{\text{nd}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{2^{\text{nd}}}=\$12000.6-\$5334.3\\ =\$6666.3\end{array}$

Calculate the depreciation charge for $3^{\text{rd}}$ year.

Substitute $\$6666.3$ for $B$ and $14.81\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{3^{\text{rd}}}}=\$6666.3\times \left(\frac{14.81}{100}\right)\\ =\$987.3\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $3^{\text{rd}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{3^{\text{rd}}}=\$6666.3-\$987.3\\ =\$5679\end{array}$

Write the values of annual depreciation charge and Net book value in tabular form.

Year, $t$ (years)	Base book value(a)	Depreciation charge for year $d_t$ (b)	Net book value(a-b)
1	$\$18000$	$\$5999.4$	$\$12000.6$
2	$\$12000.6$	$\$5334.3$	$\$6666.3$
3	$\$6666.3$	$\$987.3$	$\$5679$

Calculate the taxable incomes for $1^{\text{st}}$ year.

Substitute $\$1000$ for $\text{Before-tax}\text{cash}\text{flow}$ and $\$5999.4$ for $\text{Depreciation}$ in Equation (III).

$\begin{array}{c}\text{Taxable}{\text{Incomes}}_{1^{\text{st}}}=\$1000-\$5999.4\\ =-\$4999.4\end{array}$

Calculate the $\text{Income}\text{taxes}$ for $1^{\text{st}}$ year.

$\begin{array}{c}\text{Income}{\text{taxes}}_{1^{\text{st}}}=45\%\text{of}\text{Taxable}{\text{incomes}}_{1^{\text{st}}}\\ =\left(\frac{45}{100}\right)\times \left(-\$4999.4\right)\\ =-\$2249.7\end{array}$

Calculate the After-tax cash flow.

Substitute $\$1000$ for $\text{Before-tax}\text{cash}\text{flow}$ and $-\$2249.7$ for $\text{Income}\text{taxes}$ in Equation (IV).

$\begin{array}{c}\text{After-tax}\text{cash}{\text{flow}}_{1^{\text{st}}}=\$1000-\left(-\$2249.7\right)\\ =\$3429.7\end{array}$

Calculate the After-tax cash flow for the remaining years in tabular form.

Period	Before-tax cash flow(p)	MACRS Depreciation(q)	Taxable Incomes $\left(r\right)=\left(p-q\right)$	Income taxes ($45\%$ rate) $\left(s\right)=0.45\times \left(r\right)$	After-tax cash flow $\left(t\right)=\left(p\right)-\left(s\right)$
$0$	$-\$18000$				$-\$18000$
$1$	$\$1000$	$\$5999.4$	$-\$4999.4$	$-\$2249.7$	$\$3249.7$
$2$	$\$1000$	$\$5334.3$	$-\$4334.3$	$-\$1950.4$	$\$2950.4$
$3$	$\$1000$	$\$987.3$	$\$12.7$	$\$5.72$	$\$994.3$
$4$	$\$1000$		$\$1000$	$\$450$	$\$550$
$5$	$\$1000$		$\$1000$	$\$450$	$\$550$

Calculate present worth factor of annuity.

Substitute $-\$18000$ for $D$, $\$3249.7$ for $A$, $25\%$ for $i$, $5\text{years}$ for $n$ and $\$10000$ for $F$ in Equation (V).

$\begin{array}{c}PW=\left[-18000+\left(\$3249.7\right)\left(\frac{{\left(1+\frac{25}{100}\right)}^5-1}{\left(\frac{25}{100}\right){\left(1+\frac{25}{100}\right)}^5}\right)+\left(\$10000\right)\left(\frac{1}{{\left(1+\frac{25}{100}\right)}^5}\right)\right]\\ =\left[-\$18000+\left(\$3249.7\right)\left(2.6893\right)+\left(\$10000\right)\left(0.3277\right)\right]\\ =\left(-\$18000+\$8739.4+\$3277\right)\\ =-\$5983.6\end{array}$

Thus, the present worth value for Alternate B is $-\$5983.6$.

Alternate C:

Determine the values of $r_t$ .throughout the recovery period.

A table would be most suitable to calculate the values of $r_t$.

Recovery year, $t$ (years)	MACRS percentage rate for the recovery year $t$, $r_t$ (in %)
$1$	$33.33\%$
$2$	$44.45\%$
$3$	$14.81\%$

Calculate the depreciation charge for $1^{\text{st}}$ year.

Substitute $\$10000$ for $B$ and $33.33\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{1^{st}}}=\$10000\times \left(\frac{33.33}{100}\right)\\ =\$3333\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $1^{\text{st}}$ year.

Substitute $\$10000$ for $\text{Base}\text{book}\text{value}$ and $\$3333$ for $\text{Depreciation}\text{charge}\text{for}\text{year}{d}_t$ in Equation (II).

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{1^{\text{st}}}=\$10000-\$3333\\ =\$6667\end{array}$

Calculate the depreciation charge for $2^{\text{nd}}$ year.

Substitute $\$6667$ for $B$ and $44.45\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{2^{\text{nd}}}}=\$6667\times \left(\frac{44.45}{100}\right)\\ =\$2963.5\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $2^{\text{nd}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{2^{\text{nd}}}=\$6667-\$2963.5\\ =\$3703.5\end{array}$

Calculate the depreciation charge for $3^{\text{rd}}$ year.

Substitute $\$3703.5$ for $B$ and $14.81\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{3^{\text{rd}}}}=\$3703.5\times \left(\frac{14.81}{100}\right)\\ =\$548.5\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $3^{\text{rd}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{3^{\text{rd}}}=\$3703.5-\$548.5\\ =\$3155\end{array}$

Write the values of annual depreciation charge and Net book value in tabular form.

Year, $t$ (years)	Base book value(a)	Depreciation charge for year $d_t$ (b)	Net book value(a-b)
1	$\$10000$	$\$3333$	$\$6667$
2	$\$6667$	$\$2963.5$	$\$3703.5$
3	$\$3703.5$	$\$548.5$	$\$3155$

Calculate the taxable incomes for $1^{\text{st}}$ year.

Substitute $\$5000$ for $\text{Before-tax}\text{cash}\text{flow}$ and $\$3333$ for $\text{Depreciation}$ in Equation (III).

$\begin{array}{c}TaxableIncome{s}_{1^{\text{st}}}=\$5000-\$3333\\ =-\$1667\end{array}$

Calculate the $\text{Income}\text{taxes}$ for $1^{\text{st}}$ year.

$\begin{array}{c}\text{Income}{\text{taxes}}_{1^{\text{st}}}=45\%\text{of}\text{Taxable}{\text{incomes}}_{1^{\text{st}}}\\ =\left(\frac{45}{100}\right)\times \left(\$1667\right)\\ =\$750.15\end{array}$

Calculate the After-tax cash flow.

Substitute $\$5000$ for $\text{Before-tax}\text{cash}\text{flow}$ and $\$750.15$ for $\text{Income}\text{taxes}$ in Equation (IV).

$\begin{array}{c}\text{After-tax}\text{cash}{\text{flow}}_{1^{\text{st}}}=\$5000-\left(\$750.15\right)\\ =\$4249.85\end{array}$

Calculate the After-tax cash flow for the remaining years in tabular form.

Period	Before-tax cash flow(p)	MACRS Depreciation(q)	Taxable Incomes(p-q)	Income taxes ($45\%$ rate)	After-tax cash flow
$0$	$-\$10000$				$-\$10000$
$1$	$\$5000$	$\$3333$	$-\$1667$	$\$750.15$	$\$4249.85$
$2$	$\$5000$	$\$2963.5$	$\$2036.5$	$\$916.42$	$\$4083.58$
$3$	$\$5000$	$\$548.5$	$\$4451.5$	$\$2003.2$	$\$2996.8$
$4$	$\$5000$		$\$5000$	$\$2250$	$\$2750$
$5$	$\$5000$		$\$5000$	$\$2250$	$\$2750$

Calculate present worth factor of annuity.

Substitute $-\$10000$ for $D$, $\$4249.85$ for $A$, $25\%$ for $i$, $5\text{years}$ for $n$ and $\$0$ for $F$ in Equation (V).

$\begin{array}{c}PW=\left[-10000+\left(\$4249.85\right)\left(\frac{{\left(1+\frac{25}{100}\right)}^5-1}{\left(\frac{25}{100}\right){\left(1+\frac{25}{100}\right)}^5}\right)+\left(\$0\right)\left(\frac{1}{{\left(1+\frac{25}{100}\right)}^5}\right)\right]\\ =\left[-\$10000+\left(\$4249.85\right)\left(2.6893\right)+\left(\$0\right)\left(0.3277\right)\right]\\ =\left(-\$10000+\$11429.12+\$0\right)\\ =\$1429.12\end{array}$

Thus, the present worth value for Alternate C is $\$1429.12$.

Conclusion:

Alternative C will have much greater positive value of $\$1429.12$.

Thus, choose Alternative C.