Chapter 13, Problem 53P

To determine

The after-tax annual costs and compare and decide whether Machine A should be retained or replaced by Machine B.

Answer to Problem 53P

The machineAshould be retained.

Explanation of Solution

Given for Machine A :

Cost of machine A is $\$54000$.

The overhauled cost of Machine A is $\$9000$.

Life of Machine A is $12$ years.

Tax expense is $\$9000$.

Selling price of Machine is $\$30000$.

Salvage value after $12\text{years}$ is zero.

Cost of machine B is $\$42000$.

Life of Machine B is $12$ years.`

Combined tax rate is $40\%$.

Concept used:

Write the formula to calculate the taxable income.

$TI=DI-BTCF$ ...... (I)

Here, the taxable income is $TI$, the depreciation is $DI$ and the before tax cash flow is $BTCF$.

Write the formula to calculate the taxable amount.

$TAX=TI\times \left(\text{0}\text{.40}\right)$ ...... (II)

Here, the taxable amount is $TAX$.

Write the formula to calculate the after tax cash flow.

$ATCF=TAX-BTCF$ ...... (III)

Here, the after tax cash flow is $ATCF$.

Calculation:

Calculate the book value of the Machine A.

$\begin{array}{c}\text{Book}\text{value}=\text{Cost}-\left(\text{Depreciation}\text{till}\text{date}\right)\\ =\text{\$54000}-\frac{9}{12}\left(\$54000-0\right)\\ =\$13500\end{array}$

Calculate Recaptured depreciation if the machine is sold.

$\begin{array}{c}\text{Recaptured Depreciation}=\$30000-\$13500\\ =\$16500\end{array}$

Calculate the annual depreciation of machine by using straight line depreciation.

$\text{Depreciation}=\frac{B-S}{N}$ ...... (IV)

Here, cost of machine is B, salvage value is S, and number of year is, N.

Calculate the annual depreciation of machine A.

Substitute $\$54000$ for B, $0$ for S, and $12$ years for N in Equation (IV).

$\begin{array}{c}\text{Depreciation}=\frac{\$54000-0}{12}\\ =\$4500\end{array}$

Calculate the annual depreciation of machine B.

Substitute $\$42000$ for B, $0$ for S, and $12$ years for N in Equation (IV).

$\begin{array}{c}\text{Depreciation}=\frac{\$42000-0}{12}\\ =\$3500\end{array}$

Calculate the after-tax cash flow as shown in table below.

Year	BTCF	Depreciation(DI)	Taxable income(TI)	Income tax(TAX)	ATCF
$0$	$-\$30000$		$-\$16500$	$\$6600$	$-\$23400$
$1$	$\$0$	$\$4500$	$-\$4500$	$\$1800$	$\$1800$
$2$	$\$0$	$\$4500$	$-\$4500$	$\$1800$	$\$1800$
$3$	$\$0$	$\$4500$	$-\$4500$	$\$1800$	$\$1800$
$4-12$	$\$0$		$\$0$	$\$0$	$\$0$

Here, calculate the taxable income, income tax and after tax cash flow from Equations (I), (II) and (III) respectively.

If machine A were sold the year $0$ entries would be as shown below.

Year	BTCF	Depreciation(DI)	Taxable income(TI)	Income tax(TAX)	ATCF
$0$	$\$30000$		$\$16500$	$\$6600$	$\$23400$

If machine A is kept, the entries are just the reverse.

Calculate the after-tax annual cost.

$\text{after-tax annual cost}=\left[\$23400-\$1800\left(\frac{P}{A},10\%,3\right)\right]\left(\frac{A}{P},10\%,12\right)$ ...... (V)

Write the equation for factor $\left(\frac{A}{P},i\%,n\right)$.

$\left(\frac{A}{P},i\%,n\right)=\left[\frac{i{\left(1+i\right)}^n}{{\left(1+i\right)}^n-1}\right]$ ...... (VI)

Here, annual cost is A, present worth is P, rate of interest is i, and number of year is n.

Calculate the factor $\left(\frac{A}{P},10\%,12\right)$

Substitute $10\%$ i, and $12$ years for n in Equation (VI).

$\begin{array}{c}\left(\frac{A}{P},10\%,12\right)=\left[\frac{0.10{\left(1+0.10\right)}^{12}}{{\left(1+0.10\right)}^{12}-1}\right]\\ =0.1468\end{array}$

Calculate the factor $\left(\frac{P}{A},10\%,3\right)$

$\left(\frac{P}{A},i\%,n\right)=\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]$ ...... (VII)

Substitute $10\%$ i, and $3$ years for n in Equation (VII).

$\begin{array}{c}\left(\frac{P}{A},10\%,3\right)=\left[\frac{{\left(1+0.10\right)}^3-1}{0.10{\left(1+0.10\right)}^3}\right]\\ =2.487\end{array}$

Substitute $0.1468$ for $\left(\frac{A}{P},10\%,12\right)$, and $2.487$ for $\left(\frac{P}{A},10\%,3\right)$ in Equation (V).

$\begin{array}{c}\text{After-tax Annual cost}=\left[\$23400-\$1800\times 2.487\right]0.1468\\ =\$2778\end{array}$

The cash flow in year $0$ reflects the loss of income after Recaptured Depreciation tax from not selling Machine A. this is the preferred way to handle the current market value of the defender.

Calculate the after-tax cash flow for Machine B as shown in table below.

Year	BTCF	Depreciation	Taxable income	Income tax	ATCF
$0$	$-\$42000$				$-\$42000$
$1-12$	$\$2500$	$\$3500$	$-\$1100$	$\$400$	$\$2900$

Calculate the after-tax Annual cost.

$\text{after-tax Annual cost}=\left[\$4200000\left(\frac{A}{P},10\%,12\right)-\$2900\right]$ ...... (VIII)

Substitute $0.1468$ for $\left(\frac{A}{P},10\%,12\right)$ in Equation (VIII).

$\begin{array}{c}\text{After-tax Annual cost}=\left[\$4200000\times 0.1468-\$2900\right]\\ =\$3266\end{array}$.

Conclusion:

The after-tax annual cost of B is more than the A. Hence, keep machine A.