Chapter 8, Problem 34QP

Summary Introduction

To identify: Nominal dollar amount of last withdrawal.

Real Cash Flows:

Real cash flow is the cash flow which is modified due to increase or decrease in money value so as to present and evaluate any changes that arise in the value of money over a period of time.

Explanation of Solution

Calculation of monthly withdrawal:

Given,

Amount invested per month in stock accounts is $900.

Amount invested per month in bond account is $300.

Annual return on stock is 12%.

Annual return on bond is 7%.

Effective rate of account where money is combined is 8%.

Inflation rate is 4%.

Formula to calculate monthly withdrawal:

$\text{Present Value of an <x-custom-btb-me data-me-id="2167" class="microExplainerHighlight">Annuity</x-custom-btb-me>}\left(\text{PVA}\right)=\text{Coupon}\times \frac{\left[1-\frac{1}{{\left(1+\text{r}\right)}^{\text{t}}}\right]}{\text{r}}$

Where,

• r is interest rate.
• t is number of time period.

Substitute $1,369,505.19 for present value of annuity as computed, 0.315% for r as computed and $300\left(25\text{ years}\times \text{12 months}\right)$ months for t,

$\begin{array}{c}\text{Present Value of an Annuity}\left(\text{PVA}\right)=\text{Coupon}\times \frac{\left[1-\frac{1}{{\left(1+0.315\%\right)}^{300}}\right]}{0.00315}\\ \$1,369,505.19=\text{Coupon}\times 193.886\\ \text{Coupon}=\text{\$7,063}\text{.45}\end{array}$

Formula to calculate total future value:

$\text{Total Future Value}=\left(\begin{array}{l}\text{Future value}\left(\text{stock account}\right)\\ +\text{Future value}\left(\text{bond account}\right)\end{array}\right)$

Substitute $1,198,233 for future value (stock account) and $171,272.19 for future value (bond account),

$\begin{array}{c}\text{Total Future Value}=\$1,198,233+\$171,272.19\\ =\$1,369,505.19\end{array}$

Given below is the calculation of last withdrawal in nominal terms:

Formula to calculate last withdrawal:

$\text{FV}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}$

Substitute $7063.45 for PV, 0.04 for r and $55\left(30+25\right)$ for t,

$\begin{array}{c}\text{FV}=\$7,063.45{\left(1+.04\right)}^{\left(55\right)}\\ =\$61,073.18\end{array}$

Working Note:

Monthly Withdrawal:

Calculation of Effective Annual Rate (EAR):

$\begin{array}{c}\left(\text{1}+\text{Nominal rate}\right)=\left(1+\text{Real return}\right)\left(1+\text{Inflation rate}\right)\\ \left(1+0.08\right)=\left(1+\text{r}\right)\left(1+0.04\right)\\ \text{r}=\frac{1.08-1.04}{1.04}\times 100\\ \text{r}=\text{3}\text{.85\%}\end{array}$

Calculation of Annual percentage rate (APR):

$\begin{array}{c}\text{EAR}={\left[\text{1}+\left(\text{APR /m}\right)\right]}^{\text{m}}-\text{1}\\ \text{or, APR}=\text{m}{\left[\left(\text{1 + EAR}\right)\right]}^{\frac{1}{\text{m}}}-\text{1]}\\ =\text{12}\left[{\left(1+3.85\%\right)}^{\frac{1}{12}}-1\right]\\ =3.78\%\end{array}$

Calculation of monthly interest rate:

$\begin{array}{c}\text{Monthly rate}=\frac{\text{APR}}{\text{Number of months}}\times \text{100}\\ =\frac{3.78\%}{12}\times 100\\ =0.315\%\end{array}$

Total Future Value:

Calculation of future value of the stock account:

$\begin{array}{c}\text{Future value }\left(\text{stock account}\right)=\text{C}\left[\frac{{\left(1+\text{r}\right)}^{\text{t}}-1}{\text{r}}\right]\\ =\$900\left\{\frac{\left[{\left(1+0.62\%\right)}^{360}-1\right]}{0.62\%}\right\}\\ =\$900\left\{\frac{8.254}{0.62\%}\right\}\\ =\$1,198,233\end{array}$

Calculation of future value of the bond account:

$\begin{array}{c}\text{Future value }\left(\text{stock account}\right)=\text{C}\left[\frac{{\left(1+\text{r}\right)}^{\text{t}}-1}{\text{r}}\right]\\ =\$300\left\{\frac{\left[{\left(1+0.24\%\right)}^{360}-1\right]}{0.24\%}\right\}\\ =\$300\left\{\frac{1.370}{0.24\%}\right\}\\ =\$171,272.19\end{array}$

Stock Account:

Calculation of Effective Annual Rate (EAR):

$\begin{array}{c}\left(\text{1}+\text{Nominal rate}\right)=\left(1+\text{Real return}\right)\left(1+\text{Inflation rate}\right)\\ \left(1+0.12\right)=\left(1+\text{r}\right)\left(1+0.04\right)\\ \text{r}=\frac{1.12-1.04}{1.04}\times 100\\ \text{r}=7.69\text{\%}\end{array}$

Calculation of Annual percentage rate (APR):

$\begin{array}{c}\text{EAR}={\left[\text{1}+\left(\text{APR /m}\right)\right]}^{\text{m}}-\text{1}\\ \text{or, APR}=\text{m}{\left[\left(\text{1 + EAR}\right)\right]}^{\frac{1}{\text{m}}}-\text{1]}\\ =\text{12}\left[{\left(1+7.69\%\right)}^{\frac{1}{12}}-1\right]\\ =7.43\%\end{array}$

Calculation of monthly interest rate:

$\begin{array}{c}\text{Monthly rate}=\frac{\text{APR}}{\text{Number of months}}\times \text{100}\\ \text{=}\frac{7.43\%}{12}\times \text{100}\\ \text{=0}\text{.62\%}\end{array}$

Bond Account:

Calculation of Effective Annual Rate (EAR):

$\begin{array}{c}\left(\text{1}+\text{Nominal rate}\right)=\left(1+\text{Real return}\right)\left(1+\text{Inflation rate}\right)\\ \left(1+0.07\right)=\left(1+\text{r}\right)\left(1+0.04\right)\\ \text{r}=\frac{1.07-1.04}{1.04}\times 100\\ \text{r}=2.88\text{\%}\end{array}$

Calculation of Annual percentage rate (APR):

$\begin{array}{c}\text{EAR}={\left[\text{1}+\left(\text{APR /m}\right)\right]}^{\text{m}}-\text{1}\\ \text{or, APR}=\text{m}{\left[\left(\text{1 + EAR}\right)\right]}^{\frac{1}{\text{m}}}-\text{1]}\\ =\text{12}\left[{\left(1+2.88\%\right)}^{\frac{1}{12}}-1\right]\\ =2.85\%\end{array}$

Calculation of monthly interest rate:

$\begin{array}{c}\text{Monthly rate}=\frac{\text{APR}}{\text{Number of months}}\times \text{100}\\ \text{=}\frac{2.85\%}{12}\times \text{100}\\ \text{=0}\text{.24\%}\end{array}$

Conclusion

Hence, monthly withdrawal is computed to be $7,063.45.