Chapter 12, Problem 14P

To determine

(a)

The prospective rate of return before income tax.

Answer to Problem 14P

The prospective rate of return before income tax is, $15.4\%$.

Explanation of Solution

Given:

Annual receipts is $\$650000$.

Annual disbursements is $\$225000$.

Total asset value is of $\$2000000$.

Life time is $9\text{years}$

Combined tax is $40\%$.

Calculation:

Calculate the net revenue.

$\begin{array}{c}\text{Net}\text{revenue}=\$650000-\$225000\\ =\$425000\end{array}$

Calculate rate of return before tax as shown below.

$P=A\left(\frac{P}{A},i,n\right)$ ....... (I)

Here, initial investment is $P$, annual amount is $A$, rate of return is $i$, and time period is $n$.

Substitute $\$2000000$ for P and $\$425000$ for A and $9\text{years}$ for n in Equation (I).

$\begin{array}{l}\$2000000=\$425000\left(\frac{P}{A},i,9\right)\\ 4.70=\left(\frac{P}{A},i,9\right)\end{array}$

Look through the compound interest table to find the value of $\left(\frac{P}{A},i,n\right)$.

Interest rate	$\left(\frac{P}{A},i,n\right)$
$15\%$	$4.77$
$i\%$	$4.70$
$20\%$	$4.03$

Table-(1)

The rate of return, which is between $15\%$ and $20\%$, may indeed be calculated by linear interpolation.

$IRR=i_{low}\%+\left(i{\%}_{high}-i_{low}\%\right)\left[\frac{F_{high}-P}{F_{high}-F_{low}}\right]$ ....... (II)

Here, lower interest rate is $i_{low}\%$, higher interest rate is $i{\%}_{high}$, factor of lower interest rate is $F_{low}$, and factor of higher interest rate is $F_{high}$.

Substitute $15\%$ for $i_{low}\%$, $20\%$ for $i{\%}_{high}$, $4.77$ for $F_{high}$, $4.70$ for $P$ and $4.03$ for $F_{low}$ in Equation (II).

$\begin{array}{c}IRR=15\%+\left(20\%-15\%\right)\left[\frac{4.77-4.70}{4.77-4.03}\right]\\ =15\%+0.472\%\\ =15.472\%\\ \approx 15.4\%\end{array}$.

Conclusion:

Therefore, the prospective rate of return before income tax is, $15.4\%$.

To determine

(b)

The prospective rate of return after income tax.

Answer to Problem 14P

The prospective rate of return after income tax is, $9.75\%$.

Explanation of Solution

Concept used:

Write the formula to calculate the taxable income.

$\text{TI}=\text{DI}-\text{BTCF}$

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

Write the formula to calculate the taxable amount.

$\text{TAX}=\text{TI}\times \text{Tax}\text{rate}$

Here, the taxable amount is $\text{TAX}$.

Write the formula to calculate the after tax cash flow.

$\text{ATCF}=\text{TAX}-\text{BTCF}$

Here, the after tax cash flow is $\text{ATCF}$.

Calculation:

Write the formula to calculate the straight line depreciation.

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

Here, the asset value is $B$, salvage value of the machinery is $S$, and life of the machinery is $N$.

Substitute $\$2000000$ for $B$, $0$ for $S$, and $9$ for $N$ in Equation (III).

$\begin{array}{c}\text{Depreciation}=\frac{\$2000000-0}{9}\\ =\$222222.22\end{array}$

Calculate the After cash flow of as shown in table below.

Year	BTCF	DI	TI	Tax	ATCF
$0$	$-2000000$	-	$0$		$-2000000$
$1$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$2$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$3$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$4$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$5$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$6$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$7$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$8$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$9$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$

Table-(2)

Calculate the rate of return after tax.

$P=A\left(\frac{P}{A},i,n\right)$

Substitute $\$2000000$ for P and $\$343888.89$ for A and $9\text{years}$ for n.

$\begin{array}{l}\$2000000=\$343888.89\left(\frac{P}{A},i,9\right)\\ 5.81=\left(\frac{P}{A},i,9\right)\end{array}$

Look through the compound interest table to find the value of $\left(\frac{P}{A},i,n\right)$.

Interest rate	$\left(\frac{P}{A},i,n\right)$
$9\%$	$5.99$
$i\%$	5.81
$10\%$	$5.75$

Table-(3)

The rate of return, which is between $15\%$ and $20\%$, may indeed be calculated by linear interpolation.

Substitute $9\%$ for $i_{low}\%$, $10\%$ for $i{\%}_{high}$, $5.99$ for $F_{high}$, $5.81$ for $P$ and $5.75$ for $F_{low}$ in Equation (II).

$\begin{array}{c}IRR=9\%+\left(10\%-9\%\right)\left[\frac{5.99-5.81}{5.99-5.75}\right]\\ =9\%+0.75\%\\ =9.75\%\end{array}$.

Conclusion:

Therefore, the prospective rate of return after income tax is, $9.75\%$.

To determine

(c)

The prospective rate of return after income tax.

Answer to Problem 14P

The prospective rate of return after income tax is $10.989\%$.

Explanation of Solution

Given:

The life time is $20\text{years}$.

Calculation:

Write the expression to calculate the depreciation using Straight line depreciation method.

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

Here, asset value is $B$, salvage value of machinery is $S$, and life of machinery is $N$.

Substitute $\$2000000$ for $B$, $\$0$ for $S$, and $20$ for $N$ in Equation (IV).

$\begin{array}{c}\text{Depreciation}=\frac{\$2000000-\$0}{20}\\ =\$100000\end{array}$

Calculate the After cash flow of as shown in table below.

Year	BTCF	DI	TI	Tax	ATCF
$0$	$-2000000$		$0$		$-2000000$
$1$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$2$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$3$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$4$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$5$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$6$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$7$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$8$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$9$	$\$425000$	$\$222222.22$	$202777.78$	$81111.11$	$343888.89$
$10$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$11$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$12$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$13$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$14$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$15$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$16$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$17$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$18$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$19$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$
$20$	$0$	$\$100000$	$\$-100000$	$\$-40000$	$\$40000$

Table-(4)

Calculate the rate of return after tax for $20\text{years}$.

$\text{Asset}\text{value}=\left(\begin{array}{l}\left(\text{Average}{\text{ATCF}}_{\text{1}\text{to}\text{9}}-\text{Average}{\text{ATCF}}_{\text{9}\text{to}\text{20}}\right)\left(\frac{{\left(1+\text{RR}\right)}^{\text{N}}-\text{1}}{\text{RR}{\left(\text{1}+\text{RR}\right)}^{\text{N}}}\right)\\ +\text{Average}{\text{ATCF}}_{\text{9}\text{to}\text{20}}\left(\frac{{\left(1+\text{RR}\right)}^{\text{N}}-\text{1}}{\text{RR}{\left(\text{1}+\text{RR}\right)}^{\text{N}}}\right)\end{array}\right)$ ....... (V)

Calculate average ATCF for $1$ to $9$.

$\begin{array}{c}\text{Average}{\text{ATCF}}_{\text{1}\text{to}\text{9}}=\frac{9\times \$343888.89}{9}\\ =\$343888.89\end{array}$

Calculate average ATCF for $9$ to $20$.

$\begin{array}{c}\text{Average}{\text{ATCF}}_{\text{9}\text{to}\text{20}}=\frac{12\times \$40000}{12}\\ =\$40000\end{array}$

Substitute $\$2000000$ for asset value, $\$343888.89$ for average ATCF $1$ to $9$, $\$40000$ for average ATCF $9$ to $20$, and, $9$ and $20$ for N in Equation (V) and solve the right hand side.

$\begin{array}{c}\text{RHS}=\left(\begin{array}{l}\left(\$343888.89-\$40000\right)\left(\frac{{\left(\text{1}+\text{RR}\right)}^9-\text{1}}{\text{RR}\left({\left(\text{1}+\text{RR}\right)}^9\right)}\right)\text{+}\\ \$40000\left(\frac{{\left(\text{1}+\text{RR}\right)}^{20}-\text{1}}{\text{RR}\left({\left(\text{1}+\text{RR}\right)}^{20}\right)}\right)\end{array}\right)\\ =\left(\begin{array}{l}\$303888.89\left(\frac{{\left(\text{1}+\text{RR}\right)}^9-\text{1}}{\text{RR}\left({\left(\text{1}+\text{RR}\right)}^9\right)}\right)\text{+}\\ \$40000\left(\frac{{\left(\text{1}+\text{RR}\right)}^{20}-\text{1}}{\text{RR}\left({\left(\text{1}+\text{RR}\right)}^{20}\right)}\right)\end{array}\right)\end{array}$

Use the trial and error method to verify the interest rate.

Substitute rate of return to $10\%$.

$\begin{array}{c}\text{RHS}=\left(\begin{array}{l}\$303888.89\left(\frac{{\left(\text{1}+10\%\right)}^9-\text{1}}{10\%\left({\left(\text{1}+10\%\right)}^9\right)}\right)\text{+}\\ \$40000\left(\frac{{\left(1+10\%\right)}^{20}-\text{1}}{10\%\left({\left(1+10\%\right)}^{20}\right)}\right)\end{array}\right)\\ =\left(\$303888.89\times 5.75\right)+\left(\$40000\times 8.51\right)\\ =1747361.118+340542.54\\ =2087903.58\end{array}$

The value is greater than $\$2,000,000$.

Substitute rate of return to $11\%$.

$\begin{array}{c}\text{RHS}=\left(\begin{array}{l}\$303888.89\left(\frac{{\left(1+11\%\right)}^9-\text{1}}{11\%\left({\left(1+11\%\right)}^9\right)}\right)\text{+}\\ \$40000\left(\frac{{\left(1+11\%\right)}^{20}-\text{1}}{11\%\left({\left(\text{1}+11\%\right)}^{20}\right)}\right)\end{array}\right)\\ =\left(\$303888.89\times 5.53\right)+\left(\$40000\times 7.96\right)\\ =1680505.562+318533.12\\ =1999038.662\end{array}$

The value is lower than $\$2,000,000$.

The rate of return, which is between $10\%$ and $11\%$, may indeed be calculated by linear interpolation.

Substitute $10\%$ for $i_{low}\%$, $11\%$ for $i{\%}_{high}$, $2087903.58$ for $F_{high}$, $2000000$ for $P$ and $1999038.662$ for $F_{low}$ in Equation (II).

$\begin{array}{c}IRR=10\%+\left(11\%-10\%\right)\left[\frac{2087903.53-2000000}{2087903.53-1999038.662}\right]\\ =10\%+0.989\%\\ =10.989\%\end{array}$.

Conclusion:

Therefore, the prospective rate of return after income tax is, $10.989\%$.