Chapter 14, Problem 81P

To determine

If the device should be purchased or not.

Answer to Problem 81P

The device should be purchased.

Explanation of Solution

Given:

Purchase amount is $\$10000$.

MACRS $5$ year depreciation.

Net Benefit is $\$2500$ at the end of year $1$.

Increase in benefit per year is $\$1250$.

Inflation rate is $10\%$.

Combined interest rate is $40\%$.

After tax rate of return is $15\%$.

Calculation:

Make the table to calculate the MARCS $5$ year depreciation.

Year	BTCF $\left(a\right)$	MACR rate $\left(b\right)$	Depreciation $\begin{array}{c}\left(c\right)\\ \left[\left(BTC{F}_0-d_{n-1}\right)\times b_n\right]\end{array}$	Cumulative depreciation $\left(d\right)$
$0$	$-\$10000$
$1$	$\$2500$	$20\%$	$\$2000$	$\$2000$
$2$	$\$3750$	$32\%$	$\$2560$	$\$4560$
$3$	$\$5000$	$19\%$	$\$1033.6$	$\$5593.6$
$4$	$\$6250$	$11.52\%$	$\$507.62$	$\$6101.22$
$5$	$\$7500$	$11.52\%$	$\$449.14$	$\$6550.36$
$6$	$\$8750$	$5.76\%$	$\$198.7$	$6749.06

Draw the table to calculate the after tax cash flow.

Year	BTCF $\left(a\right)$	Depreciation $\left(b\right)$	TaxableIncome $\begin{array}{c}\left(c\right)\\ \left(a\right)-\left(b\right)\end{array}$	Income taxes $\begin{array}{c}\left(d\right)\\ -\left(c\times 0.4\right)\end{array}$	ATCF $\begin{array}{c}\left(e\right)\\ \left(a+d\right)\end{array}$
$0$	$-\$10000$				$-\$10000$
$1$	$\$2500$	$\$2000$	$\$500$	$-\$200$	$\$2300$
$2$	$\$3750$	$\$2560$	$\$1190$	$-\$476$	$\$3274$
$3$	$\$5000$	$\$1033.6$	$\$3966.4$	$-\$1586.4$	$\$3413.6$
$4$	$\$6250$	$\$507.62$	$\$5742.38$	$-\$2297.12$	$\$3952.88$
$5$	$\$7500$	$\$449.14$	$\$7050.86$	$-\$2820$	$\$4680$
$6$	$\$8750$	$\$198.7$	$\$8551.3$	$-\$3420.52$	$\$5329.48$

Make the table to compute the required data.

Year	ATCF $\left(a\right)$	ATCF Year $0$ $\begin{array}{c}\left(b\right)\\ \left(a\times {1.07}^{-n}\right)\end{array}$
$0$	$-\$10000$	$-\$10000$
$1$	$\$2300$	$\$2149.5$
$2$	$\$3274$	$\$2860$
$3$	$\$3413.6$	$\$2786.5$
$4$	$\$3952.88$	$\$3015.63$
$5$	$\$4680$	$\$3336.77$
$6$	$\$5329.48$	$\$3551.25$

Calculate after tax rate of return.

$P=\frac{A_1}{{\left(1+i\right)}^1}+\frac{A_2}{{\left(1+i\right)}^2}+\frac{A_3}{{\left(1+i\right)}^3}+\frac{A_4}{{\left(1+i\right)}^4}+\frac{A_5}{{\left(1+i\right)}^5}+\frac{A_6}{{\left(1+i\right)}^6}$ ...... (I)

Here, the after tax cash flow for year $1$ is $A_1$, for year $2$ is $A_2$, for year $3$ is $A_3$, for year $4$ is $A_4$, for year $5$ is $A_5$, for year $6$ is $A_6$ and the after tax rate of return is $i$.

Substitute, $\$10000$ for $P$

$\$2149.5$ for $A_1$, $\$2860$ for $A_2$, $\$2786.5$ for $A_3$, $\$3015.63$ for $A_4$, $\$3336.77$ for $A_5$ and $\$3551.25$ for $A_6$ in Equation (I).

$\$10000=\frac{\$2149.5}{{\left(1+i\right)}^1}+\frac{\$2860}{{\left(1+i\right)}^2}+\frac{\$2786.5}{{\left(1+i\right)}^3}+\frac{\$3015.63}{{\left(1+i\right)}^4}+\frac{\$3336.77}{{\left(1+i\right)}^5}+\frac{\$3551.25}{{\left(1+i\right)}^6}$

Solve for the value of $i$.

$\begin{array}{l}i=0.1764\\ i=0.1764\times 100\%\\ i=17.64\%\end{array}$.

Conclusion:

Since the after tax rate of return is greater than $15\%$.

Hence, the device should be purchased.