Chapter 14, Problem 87P

To determine

The better alternative.

Answer to Problem 87P

The alternative B should be preferred.

Explanation of Solution

Given:

Year	A	B
$0$	$-\$420$	$-\$300$
$1$	$\$200$	$\$150$
$2$	$\$200$	$\$150$
$3$	$\$200$	$\$150$

MARR is $8\%$.

Combined income tax rate is $38\%$.

Calculation:

Calculate the Straight line depreciation value.

$D=\frac{B-S}{N}$         ...... (I)

Here, book value is B, salvage value is S and useful life is N.

Substitute $\$420$ for B, $0$ for S and $3$ years for N in Equation (I).

$\begin{array}{c}D=\frac{\$420-\$0}{3}\\ =\$140\end{array}$

Calculate the IRR forAlternative A.

Year	Cash flow in Year $0$ dollars $\left(a\right)$	Cash flow in actual dollars $\begin{array}{c}\left(\mathrm{b}\right)\\ \left(a\times {\left(1+0.05\right)}^n\right)\end{array}$	Straight line depreciation $\left(c\right)$	Taxable income $\begin{array}{c}\left(d\right)\\ \left(\mathrm{b}-c\right)\end{array}$	$\text{Income}\text{tax}$ $\begin{array}{c}\left(e\right)\\ \left(d\times 38\%\right)\end{array}$	After tax cash flow(Actual dollars) $\begin{array}{c}\left(f\right)\\ \left(b-e\right)\end{array}$	After tax cash flow(in year zero dollars) $\begin{array}{c}\left(g\right)\\ \left(\frac{f}{{\left(1+0.05\right)}^n}\right)\end{array}$
$0$	$\$420$	$\$420$				$\$420$	$\$420$
$1$	$\$200$	$\$210$	$\$140$	$\$70$	$\$26.6$	$\$183.4$	$\$174.67$
$2$	$\$200$	$\$220.5$	$\$140$	$\$80.5$	$\$30.59$	$\$189.91$	$\$172.25$
$3$	$\$200$	$\$231.5$	$\$140$	$\$91.5$	$\$34.77$	$\$196.73$	$\$170$
						IRR	$11.22\%$

Calculate the IRR.

$A={\displaystyle \sum \frac{ATCF}{{\left(1+i\right)}^n}}$         ...... (II)

Here, initial cash flow is A, after tax cash flow is ATCF and rate of return is IRR.

Substitute $\$420$ for A and calculate the value of i in Equation (II).

$\begin{array}{l}\$420=\frac{174.67}{{\left(1+i\right)}^1}+\frac{172.5}{{\left(1+i\right)}^2}+\frac{170}{{\left(1+i\right)}^3}\\ i=11.22\%\end{array}$

Calculate the IRR for Alternative B.

Calculate the Straight line depreciation value.

Substitute $\$300$ for B, $0$ for S and $3$ years for N in Equation (I).

$\begin{array}{c}D=\frac{\$300-\$0}{3}\\ =\$100\end{array}$

Year	Cash flow in Year $0$ dollars $\left(a\right)$	Cash flow in actual dollars $\begin{array}{c}\left(\mathrm{b}\right)\\ \left(a\times {\left(1+0.05\right)}^n\right)\end{array}$	Straight line depreciation $\left(c\right)$	Taxable income $\begin{array}{c}\left(d\right)\\ \left(\mathrm{b}-c\right)\end{array}$	$\text{Income}\text{tax}$ $\begin{array}{c}\left(e\right)\\ \left(d\times 38\%\right)\end{array}$	After tax cash flow(Actual dollars) $\begin{array}{c}\left(f\right)\\ \left(b-e\right)\end{array}$	After tax cash flow(in year zero dollars) $\begin{array}{c}\left(g\right)\\ \left(\frac{f}{{\left(1+0.05\right)}^n}\right)\end{array}$
$0$	$\$300$	$\$300$				$\$300$	$\$300$
$1$	$\$150$	$\$157.5$	$\$100$	$\$57.5$	$\$21.85$	$\$135.65$	$\$129.2$
$2$	$\$150$	$\$165.4$	$\$100$	$\$65.4$	$\$24.85$	$\$140.55$	$\$127.48$
$3$	$\$150$	$\$173.6$	$\$100$	$\$73.6$	$\$27.97$	$\$145.63$	$\$125.8$
						IRR	$13.88\%$

Calculate the IRR.

Substitute $\$300$ for A and calculate the value of i in Equation (II).

$\begin{array}{l}\$300=\frac{\$129.2}{\left(1+i\right)}+\frac{\$127.48}{{\left(1+i\right)}^2}+\frac{\$125.48}{{\left(1+i\right)}^3}\\ i=13.22\%\end{array}$

Thus, IRR of Alternative B is more.

Conclusion:

The alternative B should be preferred.