Chapter 12, Problem 71P

To determine

The alternative to be selected.

Answer to Problem 71P

Choose Alternative A.

Explanation of Solution

Given:

After-tax minimum attractive rate of return is $12\%$.

Combined incremental income tax rate is $42\%$.

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 %)
$1$	$20\%$
$2$	$32\%$
$3$	$19.2\%$
$4$	$11.52\%$
$5$	$11.52\%$

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

Substitute $\$11000$ for $B$ and $20\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{1^{st}}}=\$11000\times \left(\frac{20}{100}\right)\\ =\$2200\end{array}$

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

Substitute $\$11000$ for $\text{Base}\text{book}\text{value}$ and $\$2200$ for $\text{Depreciation}\text{charge}\text{for}\text{year}{d}_t$ in Equation (II).

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{1^{\text{st}}}=\$11000-\$2200\\ =\$8800\end{array}$

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

Substitute $\$8800$ for $B$ and $32\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{2^{\text{nd}}}}=\$8800\times \left(\frac{32}{100}\right)\\ =\$2816\end{array}$

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

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{2^{\text{nd}}}=\$8800-\$2816\\ =\$5984\end{array}$

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

Substitute $\$5984$ for $B$ and $19.2\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{3^{\text{rd}}}}=\$5984\times \left(\frac{19.2}{100}\right)\\ =\$1148.93\end{array}$

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

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{3^{\text{rd}}}=\$5984-\$1128.93\\ =\$4855.07\end{array}$

Calculate the depreciation charge for $4^{\text{th}}$ year.

Substitute $\$4855.07$ for $B$ and $11.52\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{4^{\text{th}}}}=\$4855.07\times \left(\frac{11.52}{100}\right)\\ =\$559.30\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $4^{\text{th}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{4^{th}}=\$4855.07-\$559.30\\ =\$4295.77\end{array}$

Calculate the depreciation charge for $5^{\text{th}}$ year.

Substitute $\$4295.77$ for $B$ and $11.52\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{5^{\text{th}}}}=\$4295.77\times \left(\frac{11.52}{100}\right)\\ =\$494.87\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $5^{\text{th}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{5^{th}}=\$4295.77-\$494.87\\ =\$3800.9\end{array}$

Tabulate the values of annual depreciation charge and Net book value.

Year, $t$ (years)	Base book value(a)	Depreciation charge for year $d_t$ (b)	Net book value(a-b)
1	$\$11000$	$\$2200$	$\$8800$
2	$\$8800$	$\$2816$	$\$5984$
3	$\$5984$	$\$1148.93$	$\$4855.07$
$4$	$\$4855.07$	$\$559.30$	$\$4295.77$
$5$	$\$4295.77$	$\$494.87$	$\$3800.9$

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 $\$3000$ for $\text{Before-tax}\text{cash}\text{flow}$ and $\$2816$ for $\text{Depreciation}$ in Equation (III).

$\begin{array}{c}\text{Taxable}{\text{Incomes}}_{1^{\text{st}}}=\$3000-\$2816\\ =\$184\end{array}$

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

$\begin{array}{c}\text{Income}{\text{taxes}}_{1^{\text{st}}}=42\%\text{of}\text{Taxable}{\text{incomes}}_{1^{\text{st}}}\\ =\left(\frac{42}{100}\right)\times \left(\$184\right)\\ =\$77.28\end{array}$

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 $\$3000$ for $\text{Before-tax}\text{cash}\text{flow}$ and $\$77.28$ for $\text{Income}\text{taxes}$ in Equation (IV).

$\begin{array}{c}\text{After-tax}\text{cash}{\text{flow}}_{1^{\text{st}}}=\$3000-\left(\$77.28\right)\\ =\$2922.72\end{array}$

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 ($42\%$ rate) $\left(s\right)=0.42\times \left(r\right)$	After-tax cash flow $\left(t\right)=\left(p\right)-\left(s\right)$
$0$	$-\$11000$				$-\$11000$
$1$	$\$3000$	$\$2200$	$\$800$	$\$336$	$\$2664$
$2$	$\$3000$	$\$2816$	$\$184$	$\$77.28$	$\$2922.72$
$3$	$\$3000$	$\$1148.93$	$\$1851.07$	$\$777.45$	$\$2222.55$
$4$	$\$3000$	$\$559.30$	$\$2440.7$	$\$1025.09$	$\$1974.91$
$5$	$\$3000$	$\$494.87$	$\$2505.13$	$\$1052.15$	$\$1947.85$

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 five years is $F$.

Calculate present worth factor of annuity.

Substitute $-\$11000$ for $D$, $\$2922.72$ for $A$, $12\%$ for $i$, $5\text{years}$ for $n$ and $\$2000$ for $F$ in Equation (V).

$\begin{array}{c}PW=\left[-11000+\left(\$2922.72\right)\left(\frac{{\left(1+\frac{12}{100}\right)}^5-1}{\left(\frac{12}{100}\right){\left(1+\frac{12}{100}\right)}^5}\right)+\left(\$2000\right)\left(\frac{1}{{\left(1+\frac{12}{100}\right)}^5}\right)\right]\\ =\left[-\$11000+\left(\$2922.72\right)\left(3.6043\right)+\left(\$2000\right)\left(0.5674\right)\right]\\ =\left(-\$11000+\$10534.36+\$1134.8\right)\\ =\$669.16\end{array}$

Thus, the present worth value for Alternate A is $\$669.16$.

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 %)
$1$	$20\%$
$2$	$32\%$
$3$	$19.2\%$
$4$	$11.52\%$
$5$	$11.52\%$

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

Substitute $\$33000$ for $B$ and $20\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{1^{st}}}=\$33000\times \left(\frac{20}{100}\right)\\ =\$6600\end{array}$

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

Substitute $\$33000$ for $\text{Base}\text{book}\text{value}$ and $\$6600$ for $\text{Depreciation}\text{charge}\text{for}\text{year}{d}_t$ in Equation (II).

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{1^{\text{st}}}=\$33000-\$6600\\ =\$27000\end{array}$

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

Substitute $\$27000$ for $B$ and $32\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{2^{\text{nd}}}}=\$27000\times \left(\frac{32}{100}\right)\\ =\$8640\end{array}$

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

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{2^{\text{nd}}}=\$27000-\$8640\\ =\$18360\end{array}$

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

Substitute $\$18360$ for $B$ and $19.2\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{3^{\text{rd}}}}=\$18360\times \left(\frac{19.2}{100}\right)\\ =\$3525.12\end{array}$

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

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{3^{\text{rd}}}=\$18360-\$3525.12\\ =\$14834.88\end{array}$

Calculate the depreciation charge for $4^{\text{th}}$ year.

Substitute $\$14834.88$ for $B$ and $11.52\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{4^{\text{th}}}}=\$14834.88\times \left(\frac{11.52}{100}\right)\\ =\$1708.98\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $4^{\text{th}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{4^{th}}=\$14834.88-\$1708.98\\ =\$13125.9\end{array}$

Calculate the depreciation charge for $5^{\text{th}}$ year.

Substitute $\$13125.9$ for $B$ and $11.52\%$ for $r_t$ in Equation (I).

$\begin{array}{c}{d}_{t_{5^{\text{th}}}}=\$13125.9\times \left(\frac{11.52}{100}\right)\\ =\$1512.10\end{array}$

Calculate the $\text{Net}\text{book}\text{value}$ for $5^{\text{th}}$ year.

$\begin{array}{c}\text{Net}\text{book}{\text{value}}_{5^{th}}=\$13125.9-\$1512.1\\ =\$11613.8\end{array}$

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	$\$33000$	$\$6600$	$\$27000$
2	$\$27000$	$\$8640$	$\$18360$
3	$\$18360$	$\$3525.12$	$\$14834.88$
$4$	$\$14834.88$	$\$1708.98$	$\$13125.9$
$5$	$\$13125.9$	$\$1512.1$	$\$11613.8$

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

Substitute $\$9000$ for $\text{Before-tax}\text{cash}\text{flow}$ and $\$8640$ for $\text{Depreciation}$ in Equation (III).

$\begin{array}{c}\text{Taxable}{\text{Incomes}}_{1^{\text{st}}}=\$9000-\$8640\\ =\$360\end{array}$

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

$\begin{array}{c}\text{Income}{\text{taxes}}_{1^{\text{st}}}=42\%\text{of}\text{Taxable}{\text{incomes}}_{1^{\text{st}}}\\ =\left(\frac{42}{100}\right)\times \left(\$360\right)\\ =\$151.2\end{array}$

Calculate the after-tax cash flow.

Substitute $\$9000$ for $\text{Before-tax}\text{cash}\text{flow}$ and $\$151.2$ for $\text{Income}\text{taxes}$ in Equation (IV).

$\begin{array}{c}\text{After-tax}\text{cash}{\text{flow}}_{1^{\text{st}}}=\$9000-\left(\$151.2\right)\\ =\$8848.8\end{array}$

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 ($42\%$ rate) $\left(s\right)=0.42\times \left(r\right)$	After-tax cash flow $\left(t\right)=\left(p\right)-\left(s\right)$
$0$	$-\$33000$				$-\$33000$
$1$	$\$9000$	$\$6600$	$\$2400$	$\$1008$	$\$7992$
$2$	$\$9000$	$\$8640$	$\$360$	$\$151.2$	$\$8848.8$
$3$	$\$9000$	$\$3525.12$	$\$5474.88$	$\$2299.45$	$\$6700.55$
$4$	$\$9000$	$\$1708.98$	$\$7291.02$	$\$3062.22$	$\$5937.78$
$5$	$\$9000$	$\$1512.1$	$\$7487.9$	$\$3144.92$	$\$5855.08$

Calculate present worth factor of annuity.

Substitute $-\$33000$ for $D$, $\$8848.8$ for $A$, $12\%$ for $i$, $5\text{years}$ for $n$ and $\$3000$ for $F$ in Equation (V).

$\begin{array}{c}PW=\left[-33000+\left(\$8848.8\right)\left(\frac{{\left(1+\frac{12}{100}\right)}^5-1}{\left(\frac{12}{100}\right){\left(1+\frac{12}{100}\right)}^5}\right)+\left(\$3000\right)\left(\frac{1}{{\left(1+\frac{12}{100}\right)}^5}\right)\right]\\ =\left[-\$33000+\left(\$8848.8\right)\left(3.6043\right)+\left(\$3000\right)\left(0.5674\right)\right]\\ =\left(-\$33000+\$31893.73+\$1702.2\right)\\ =\$595.93\end{array}$

Thus, the present worth value for Alternate A is $\$595.93$.

Conclusion:

Alternative A will have much greater positive value of $\$669.16$.

Thus, choose Alternative A.