Chapter 14, Problem 84P

To determine

If inflation decrease or increase the annual cost.

Answer to Problem 84P

The rate of return is increased in with inflation method.

Explanation of Solution

Given:

The first cost is $\$18,000$.

The maintenance cost per year is $\$1300$.

The salvage value is $\$2000$.

MARR is $8\%$.

Inflation rate is $5\%$.

Calculation:

Calculate the MACRS depreciation for year $1$.

$MACRS{\left(d_1\right)}_1=\left(\text{Maintenance}\text{cost}\right)\times \left(\text{Depreciation}\text{convention}\right)$ ...... (I)

Refer the table for MACRS depreciation for personal property.

Substitute $\$1300$ for maintenance cost and $10\%$ for depreciation convention in Equation (I).

$\begin{array}{c}MACRS{\left(d_1\right)}_1=\$1300\times \text{0}\text{.10}\\ =\$130\end{array}$

Calculate the depreciation.

Year	MACRS Rate(a)	Before Tax Cash Flow(b)	Depreciation(c) $\left(a\times b\right)$
$1$	$10\%$	$\$1,300$	$\$130$
$2$	$18\%$	$\$1,300$	$\$234$
$3$	$14.4\%$	$\$1,300$	$\$187$
$4$	$11.52\%$	$\$1,300$	$\$150$
$5$	$9.22\%$	$\$1,300$	$\$120$
$6$	$7.37\%$	$\$1,300$	$\$96$
$7$	$6.55\%$	$\$1,300$	$\$85$
$8$	$6.55\%$	$\$1,300$	$\$85$
$9$	$6.56\%$	$\$1,300$	$\$85$
$10$	$6.55\%$	$\$1,300$	$\$85$

Calculate the before and after tax cash flows.

Year	Before tax cash flow $\left(\text{a}\right)$	MACRS Depreciation $\left(\mathrm{b}\right)$	Taxable Income $\left(\text{c}\right)$ $\left(\text{a}-\mathrm{b}\right)$	Income taxes $35\%$ $\begin{array}{c}\left(\text{d}\right)\\ \left(\text{c}\times 35\%\right)\end{array}$	After tax cash flowActual dollars $\left(\text{c}-\text{d}\right)$
$0$	$\$18,000$				$\$18,000$
$1$	$\$1,300$	$\$130$	$\$1,170$	$\$409$	$\$891$
$2$	$\$1,300$	$\$234$	$\$1066$	$\$373$	$\$927$
$3$	$\$1,300$	$\$187$	$\$1113$	$\$390$	$\$910$
$4$	$\$1,300$	$\$150$	$\$1150$	$\$402$	$\$898$
$5$	$\$1,300$	$\$120$	$\$1180$	$\$413$	$\$887$
$6$	$\$1,300$	$\$96$	$\$1204$	$\$421$	$\$879$
$7$	$\$1,300$	$\$85$	$\$1215$	$\$425$	$\$875$
$8$	$\$1,300$	$\$85$	$\$1215$	$\$425$	$\$875$
$9$	$\$1,300$	$\$85$	$\$1215$	$\$425$	$\$875$
$10$	$\$1,300$	$\$85$	$\$1215$	$\$425$	$\$2,875$

Determine the rate of return after tax.

$\begin{array}{c}\text{First}\text{cost}=\left[\begin{array}{l}\left({\text{ATCF}}_2+\text{Sum}\text{of}\text{MACRS}\text{depreciation}\right)\left(\frac{P}{A},i,n\right)\\ +\text{Salvage}\text{value}\left(\frac{P}{F},i,n\right)\end{array}\right]\\ =\left[\begin{array}{c}\left({\text{ATCF}}_2+\text{Sum}\text{of}\text{MACRS}\text{depreciation}\right)\left(\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right)\\ +\text{Salvage}\text{value}\left(\frac{1}{{\left(1+i\right)}^n}\right)\end{array}\right]\end{array}$ ...... (II)

Use trial and error method.

Try $i=5\%$.

Substitute $\$18,000$ for first cost, $\$927$ for ${\text{ATCF}}_{\text{2}}$, $\$1257$ for sum of MACRS depreciation and $5\%$ for $i$ in Equation (II).

$\begin{array}{c}\$1800=\left[\left(\$927+\$1257\right)\left(\frac{{\left(1+0.05\right)}^{10}-1}{0.05{\left(1+0.05\right)}^{10}}\right)+\$2000\left(\frac{1}{{\left(1+0.05\right)}^{10}}\right)\right]\\ \$1800=\left(\$2184\left(7.714\right)+1228.5\right)\\ \$1800\approx \$18,076\end{array}$

Thus, the rate of return after tax without inflation is $5\%$.

Calculate the inflation applied cash flows.

Determine the before tax cash flow for year $2$.

${\text{BTCF}}_2=\left({\text{BTCF}}_1\right)+\left(\text{Maintenance}\text{cost}\times f\right)$ ...... (III)

Here, f is the inflation rate.

Substitute $\$1300$ for ${\text{BTCF}}_1$, $\$1300$ for maintenance cost and $5\%$ for i in Equation (III).

$\begin{array}{c}{\text{BTCF}}_2=\$1300+\left(\$1300\times 0.05\right)\\ =\$1365\end{array}$

Determine the before tax cash flow for year $2$.

Substitute $\$1365$ for ${\text{BTCF}}_2$, $\$1365$ for maintenance cost and $5\%$ for i in Equation (III).

$\begin{array}{c}{\text{BTCF}}_3=\$1365+\left(\$1365\times 0.05\right)\\ =\$1433.25\end{array}$

Substitute $\$1300$ for maintenance cost and $10\%$ for depreciation convention in Equation (I).

$\begin{array}{c}MACRS{\left(d_1\right)}_1=\$1300\times \text{0}\text{.10}\\ =\$130\end{array}$

Calculate the MACRS Depreciation for year $2$.

Substitute $\$1365$ for ${\text{BTCF}}_2$ and $18\%$ for depreciation convention in Equation (I).

$\begin{array}{c}MACRS{\left(d_1\right)}_2=\$1365\times \text{0}\text{.18}\\ =\$246\end{array}$

Calculate the depreciation when inflation is applied.

Year	MACRS Rate(a)	Before Tax Cash Flow(b)	Depreciation(c) $\left(a\times b\right)$
$1$	$10\%$	$\$1,300$	$\$130$
$2$	$18\%$	$\$1,365$	$\$246$
$3$	$14.4\%$	$\$1,433$	$\$206$
$4$	$11.52\%$	$\$1,505$	$\$173$
$5$	$9.22\%$	$\$1,580$	$\$146$
$6$	$7.37\%$	$\$1,659$	$\$122$
$7$	$6.55\%$	$\$1,742$	$\$114$
$8$	$6.55\%$	$\$1,829$	$\$120$
$9$	$6.56\%$	$\$1,921$	$\$126$
$10$	$6.55\%$	$\$2017$	$\$132$

Calculate the before and after tax cash flows with inflation.

Year	Before tax cash flow $\left(\text{a}\right)$	MACRS Depreciation $\left(\mathrm{b}\right)$	Taxable Income $\left(\text{c}\right)$ $\left(\text{a}-\mathrm{b}\right)$	Income taxes $35\%$ $\begin{array}{c}\left(\text{d}\right)\\ \left(\text{c}\times 35\%\right)\end{array}$	After tax cash flowActual dollars $\left(\text{c}-\text{d}\right)$
$0$	$\$18,000$				$\$18,000$
$1$	$\$1,300$	$\$130$	$\$1,170$	$\$409$	$\$891$
$2$	$\$1,365$	$\$246$	$\$1119$	$\$392$	$\$973$
$3$	$\$1,433$	$\$206$	$\$1227$	$\$429$	$\$1004$
$4$	$\$1,505$	$\$173$	$\$1332$	$\$466$	$\$1039$
$5$	$\$1,580$	$\$146$	$\$1434$	$\$502$	$\$1078$
$6$	$\$1,659$	$\$122$	$\$1537$	$\$540$	$\$1119$
$7$	$\$1,742$	$\$114$	$\$1628$	$\$570$	$\$1172$
$8$	$\$1,829$	$\$120$	$\$1709$	$\$598$	$\$1231$
$9$	$\$1,921$	$\$126$	$\$1795$	$\$628$	$\$1293$
$10$	$\$2017$	$\$132$	$\$1885$	$\$660$	$\$3357$

Use trial and error method.

Try $i=10\%$.

Substitute $\$18,000$ for first cost, $\$1293$ for ${\text{ATCF}}_{\text{2}}$, $\$1515$ for sum of MACRS depreciation and $10\%$ for $i$ in Equation (II).

$\begin{array}{c}\$1800=\left[\left(\$1293+\$1515\right)\left(\frac{{\left(1+0.10\right)}^{10}-1}{0.10{\left(1+0.10\right)}^{10}}\right)+\$2000\left(\frac{1}{{\left(1+0.10\right)}^{10}}\right)\right]\\ \$1800=\left(\$2808\left(6.14\right)+771\right)\\ \$1800\approx \$18,017\end{array}$

Thus, the rate of return after tax without inflation is $10\%$.

Conclusion:

The rate of return is increased in with inflation method.