Chapter 12, Problem 22P

To determine

If the investment is desirable or not.

Answer to Problem 22P

The investment is not desirable.

Explanation of Solution

Concept used:

Write the expression to calculate the depreciation by double declining method.

$d_1=\frac{2}{N}\left(B{V}_{t-1}\right)$ ....... (I)

Here, the depreciation is $d_1$, the total number of year is $N$ and the book value of previous year is $B{V}_{t-1}$.

Write the expression to calculate cumulative depreciation.

$CD=d_t+d_{t-1}$ ....... (II)

Here, the cumulative depreciation is $CD$, the depreciation of the current year is $d_t$ and the depreciation o the previous year is $d_{t-1}$.

Write the expression to calculate the book value.

$B{V}_t=B{V}_0-C{D}_t$ ....... (III)

Here, the book value of the current year is $B{V}_t$, the book value at $0$ year is $B{V}_0$ and the depreciation for current year is $D_t$.

Write the expression to calculate the depreciation by straight line method.

$d_t=\frac{BV-S}{N}$ ....... (IV)

Here, the depreciation is $d_t$, the book value is $BV$, the salvage value is $S$ and the total number of year is $N$.

Write the expression to calculate the depreciation recaparture.

$DC=SV-BV$ ....... (V)

Here, the depreciation recapture is $DC$, the book value is $BV$ and the salvage value is $SV$.

Calculations:

All values are taken in $\left(\$1000\right)$.

Calculate the depreciation for year $1$.

Substitute $\$12000$ for $B{V}_{t-1}$ and $6$ for $N$ in Equation (I).

$\begin{array}{l}{d}_1=\frac{2}{6}\left(\$12000\right)\\ d_1=\$4000\end{array}$

Calculate the cumulative depreciation for year $1$.

Substitute $\$4000$ for $d_t$ and $\$0$ for $C{D}_{t-1}$ in Equation (II).

$\begin{array}{l}C{D}_1=\$4000+\$0\\ C{D}_1=\$4000\end{array}$

Calculate the book value for year $1$.

Substitute $\$12000$ for $B{V}_0$ and $\$4000$ for $C{D}_1$.

$\begin{array}{c}B{V}_1=\$12000-\$4000\\ =\$8000\end{array}$

Calculate the depreciation for year $2$.

Substitute $\$8000$ for $B{V}_{t-1}$ and $6$ for $N$ in Equation (I).

$\begin{array}{l}{d}_2=\frac{2}{6}\left(\$8000\right)\\ d_2=\$2667\end{array}$

Calculate the cumulative depreciation for year $2$.

Substitute $\$2667$ for $d_t$ and $\$4000$ for $C{D}_{t-1}$ in Equation (II).

$\begin{array}{l}C{D}_2=\$4000+\$2667\\ C{D}_2=\$6667\end{array}$

Calculate the book value for year $2$.

Substitute $\$12000$ for $B{V}_0$ and $\$6667$ for $C{D}_2$.

$\begin{array}{c}B{V}_2=\$12000-\$6667\\ =\$5333\end{array}$

Calculate the depreciation for year $3$.

Substitute $\$5333$ for $B{V}_{t-1}$ and $6$ for $N$ in Equation (I).

$\begin{array}{l}{d}_3=\frac{2}{6}\left(\$5333\right)\\ d_3=\$1778\end{array}$

Calculate the cumulative depreciation for year $3$.

Substitute $\$1778$ for $d_t$ and $\$6667$ for $C{D}_{t-1}$ in Equation (II).

$\begin{array}{l}C{D}_3=\$6667+\$1778\\ C{D}_3=\$8445\end{array}$

Calculate the book value for year $3$.

Substitute $\$12000$ for $B{V}_0$ and $\$8445$ for $C{D}_3$

$\begin{array}{c}B{V}_3=\$12000-\$8445\\ =\$3555\end{array}$

Calculate the depreciation for year $4$.

Substitute $\$3555$ for $B{V}_{t-1}$ and $6$ for $N$ in Equation (I)

$\begin{array}{l}{d}_4=\frac{2}{6}\left(\$3555\right)\\ d_4=\$1185\end{array}$

Calculate the cumulative depreciation for year $4$.

Substitute $\$1185$ for $d_t$ and $\$8445$ for $C{D}_{t-1}$ in Equation (II).

$\begin{array}{l}C{D}_4=\$1185+\$8445\\ C{D}_4=\$9630\end{array}$

Calculate the book value for year $4$.

Substitute $\$12000$ for $B{V}_0$ and $\$9630$ for $$ \begin{array}{c}B{V}_4=\$12000-\$9630\\ =\$2370\end{array}$$

CD4

The depreciation for year $5$ and $6$ has to be calculated by straight line method.

Calculate the depreciation for year $5$.

Substitute $\$2370$ for $BV$, $\$700$ for $S$ and $2$ for $N$ in Equation (IV).

$\begin{array}{c}{d}_5=\frac{\$2370-\$700}{2}\\ =\$835\end{array}$

Calculate the cumulative depreciation for year $5$.

Substitute $\$835$ for $d_t$ and $\$9630$ for $C{D}_{t-1}$ in Equation (II).

$\begin{array}{l}C{D}_5=\$9630+\$835\\ C{D}_5=\$10465\end{array}$

Calculate the book value for year $5$.

Substitute $\$12000$ for $B{V}_0$ and $\$10465$ for $C{D}_5$

$\begin{array}{c}B{V}_t=\$12000-\$10465\\ =\$1535\end{array}$

Substitute $\$2370$ for $BV$, $\$700$ for $S$ and $2$ for $N$ in Equation (IV).

$\begin{array}{c}{d}_6=\frac{\$2370-\$700}{2}\\ =\$835\end{array}$

Calculate the cumulative depreciation for year $6$.

Substitute $\$835$ for $d_t$ and $\$10465$ for $C{D}_{t-1}$ in Equation (III).

$\begin{array}{l}C{D}_6=\$10465+\$835\\ C{D}_6=\$11300\end{array}$

Calculate the book value for year $6$.

Substitute $\$12000$ for $B{V}_0$ and $\$11300$ for $C{D}_6$.

$\begin{array}{c}C{D}_1=\$12000-\$11300\\ =\$700\end{array}$

Tabulate the obtained above values.

Year	Deprecation	Cumulative Depreciation	Book Value
$1$	$\$4000$	$\$4000$	$\$8000$
$2$	$\$2667$	$\$6667$	$\$5333$
$3$	$\$1778$	$\$8445$	$\$3555$
$4$	$\$1185$	$\$9360$	$\$2370$
$5$	$\$835$	$\$10465$	$\$1535$
$6$	$\$835$	$\$11300$	$\$700$

The book value obtained after the end of $6\text{years}$ is $\$700$ while the salvage value is $\$1000$. This results into depreciation recaparture.

Substitute $\$1000$ for $SV$ and $\$700$ for $BV$ in Equation (V).

$\begin{array}{c}DC=\$1000-\$700\\ =\$300\end{array}$

Tabulate the values for before tax and after cash flows.

Year	Before taxCash flow $\left(a\right)$	Depreciation $\left(b\right)$	Taxable income $c=\left(a-b\right)$	Income taxes $d=-\left(c\times 0.34\right)$	After taxCash flow $e=\left(a+d\right)$
$0$	$-\$12000$				$-\$12000$
$1$	$\$1727$	$\$4000$	$-\$2273$	$\$773$	$\$2500$
$2$	$\$2414$	$\$2667$	$-\$253$	$\$86$	$\$2500$
$3$	$\$2872$	$\$1778$	$\$1094$	$-\$372$	$\$2500$
$4$	$\$3177$	$\$1185$	$\$1992$	$-\$677$	$\$2500$
$5$	$\$3358$	$835$	$\$2523$	$-\$858$	$\$2500$
$6$	$\$1997$	$\$835$	$\$1162$	$-\$395$	$\$1602$
$6$	$\$1000$		$\$300$	$-\$102$	$\$898$

Now calculate the after tax return.

Write the expression to calculate the rate when $A$ and $P$ given.

$P=A\left(\frac{{\left(1+i\right)}^n-1}{i\left(1+i\right)}\right)$ ....... (V)

Here the present value is $P$, the annual investment is $A$, the number of years is $n$ and the rate of return is $i$.

Substitute $\$12000$ for $P$, $6$ for $n$ and $\$2500$ for $A$ in Equation (V).

$\begin{array}{l}12000=2500\left(\frac{{\left(1+i\right)}^6-1}{i{\left(1+i\right)}^6}\right)\\ 4.8i{\left(1+i\right)}^6-{\left(1+i\right)}^6+1=0\end{array}$ ........ (VI)

Solve Equation (VI) for the value of $i$.

$\begin{array}{l}i=0.0677\\ i=\left(0.0677\times 100\right)\%\\ i=6.77\%\end{array}$.

Conclusion:

The after tax rate of return is less than expected rate of return.

Hence the investment is not desirable.