Chapter 14, Problem 76P

To determine

(a)

The before tax rate of return, ignoring inflation.

Answer to Problem 76P

The before tax rate of return, ignoring inflation is $3.5\%$.

Explanation of Solution

Given:

Initial Amount is $\$10000$.

Interest per year is $\$3.5\%$.

The MARR is $28\%$.

Inflation rate is $2\%$.

Maturity amount received is $\$10000$.

Calculation:

Calculate the return from interest every year.

$\begin{array}{c}A=\$10000\times 0.035\\ =\$350\end{array}$

Make the table for cash flows.

Year	Before tax Cash flow $\left(a\right)$	Taxable income $\left(b\right)$	Income tax charge $\begin{array}{c}\left(c\right)\\ \left(0.28\times b\right)\end{array}$	After tax Cash flow $c$ $\begin{array}{c}\left(d\right)\\ \left(a-c\right)\end{array}$	InflationFactor $\begin{array}{c}\left(e\right)\\ {\left(1.02\right)}^{-n}\end{array}$	Present Value at year $0$ $\begin{array}{c}\left(f\right)\\ \left(d\times e\right)\end{array}$
$0$	$\$10000$			$\$10000$	$1$	$\$10000$
$1$	$\$350$	$\$350$	$\$98$	$\$252$	$0.980$	$\$245$
$2$	$\$350$	$\$350$	$\$98$	$\$252$	$0.9612$	$\$242.22$
$3$	$\$350$	$\$350$	$\$98$	$\$252$	$0.9423$	$\$237.46$
$4$	$\$350$	$\$350$	$\$98$	$\$252$	$0.9238$	$\$232.8$
$5$	$\$350+\$10000$	$\$350$	$\$98$	$\$10252$	$0.9057$	$\$228.24$

Write the expression to calculate the before tax rate of return.

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

Here, the present value is P, the return from interest every year is A, the interest rate is i, the maturity amount is F and the time period is n.

Substitute $\$10000$ for $P$, $\$350$ for $A$, $\$10000$ for $F$ and $5$ for $n$ in Equation (I).

$\$10000=\$350\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)+\frac{\$10000}{{\left(1+i\right)}^5}$

Solve the equation for the value of $i$.

$\begin{array}{l}i=0.035\\ i=0.035\times 100\%\\ i=3.5\%\end{array}$.

Conclusion:

Thus, the before tax rate of return, ignoring inflation is $3.5\%$.

To determine

(b)

The after tax rate of return, ignoring inflation.

Answer to Problem 76P

The after tax rate of return, ignoring inflation is $2.52\%$.

Explanation of Solution

Calculation:

Calculate the after tax rate of return.

Substitute $\$10000$ for $P$, $\$252$ for $A$ and $5$ for $n$ in Equation (I).

$\$10000=\$252\left(\frac{{\left(1+i\right)}^5-1}{i{\left(1+i\right)}^5}\right)+\frac{\$10000}{{\left(1+i\right)}^5}$

Solve the equation for the value of $i$.

$\begin{array}{l}i=0.0252\\ i=0.0252\times 100\%\\ i=2.52\%\end{array}$.

Conclusion:

The after tax rate of return, ignoring inflation is $2.52\%$.

To determine

(c)

The after tax rate of return, after taking inflation into account.

Answer to Problem 76P

The after tax rate of return, after taking inflation into account is $0.51\%$.

Explanation of Solution

Calculation:

Write the expression to calculate the after tax rate of return including inflation.

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

Substitute $\$10000$ for $P$, $\$245$ for $A_1$, $\$242.22$ for $A_2$, $\$237.46$ for $A_3$, $\$232.8$ for $A_4$, $\$228.24$ for $A_5$ and $\$9057.308$ for $F$ in Equation (II).

$\$10000=\frac{\$245}{{\left(1+i\right)}^1}+\frac{\$242.22}{{\left(1+i\right)}^2}+\frac{\$237.46}{{\left(1+i\right)}^3}+\frac{\$232.8}{{\left(1+i\right)}^4}+\frac{\$228.24}{{\left(1+i\right)}^5}+\frac{\$9057.308}{{\left(1+i\right)}^5}$

Solve the equation for the value of $i$.

$\begin{array}{l}i=0.00505\\ i\approx 0.0051\times 100\%\\ i=0.51\%\end{array}$.

Conclusion:

Thus, the after tax rate of return, after taking inflation into account is $0.51\%$.