Chapter 4, Problem 57QP

Summary Introduction

To determine: The amount to be saved from 11 to 30 years in each month.

Annuity:

Annuity refers to the payments of equal amount to be made after certain periods. These payments are made monthly, semi-annually or annually.

Effective Annual Rate:

The effective annual rate is the rate, which is incurred or received on various investment or loans. The effective annual rate is affected by the increase in compounding years.

Explanation of Solution

Solution:

Given,

The time period will be 240 months $\left(12\times 20\right)$

The effective annual rate before retirement is 8%.

The effective annual rate after retirement is 11%.

Calculation of the value of the monthly savings after 11^th year till 30^th year:

The formula to calculate the monthly savings is,

$\text{Monthly}\text{savings}=\text{Future}\text{value}\times \frac{\text{Rate}\text{of}\text{interest}}{{\left(1+\text{Rate}\text{of}\text{interest}\right)}^{\text{Time}}-1}$

Substitute $2,021,528 for future value, 0.00873 for the rate of interest and 240 months for the time (refer working note) in the above formula.

$\begin{array}{c}\text{Monthly}\text{savings}=\$2,021,528\times \frac{0.00873}{{\left(1+0.00873\right)}^{240}-1}\\ =\$2,021,528\times \frac{0.00873}{7.062}\\ =\$2,021,528\times 0.0012\\ =\$2,499\end{array}$

The monthly savings are $2,499.

Working note:

Calculation of the annual percentage rate before retirement:

$\begin{array}{c}\text{Annual}\text{percentage}\text{rate}=\left[\begin{array}{l}\left\{{\left(1+\text{Effective}\text{annual}\text{rate}\right)}^{\frac{1}{\text{Corresponding}\text{period}}}-1\right\}\\ \times \text{Number}\text{of}\text{months}\end{array}\right]\\ ={\left(1+0.11\right)}^{\frac{1}{12}}-1\times 12\\ =\left(1.873-1\right)\times 12\\ =10.48\%\end{array}$

The annual percentage rate is 10.48%.

Calculation of the annual percentage rate after retirement:

$\begin{array}{c}\text{Annual}\text{percentage}\text{rate}=\left[\begin{array}{l}\left\{{\left(1+\text{Effective}\text{annual}\text{rate}\right)}^{\frac{1}{\text{Corresponding}\text{period}}}-1\right\}\\ \times \text{Number}\text{of}\text{months}\end{array}\right]\\ ={\left(1+0.8\right)}^{\frac{1}{12}}-1\times 12\\ =\left(1.643-1\right)\times 12\\ =7.72\%\end{array}$

The annual percentage rate is 7.72%.

Calculation of the monthly interest rate before retirement:

$\begin{array}{c}\text{Monthly}\text{interest}\text{rate}=\frac{\text{Annual}\text{percentage}\text{rate}}{\text{Number}\text{of}\text{months}}\\ =\frac{10.48\%}{12\text{months}}\\ =0.873\%\end{array}$

The monthly interest rate before retirement is 0.873%.

Calculation of the monthly interest rate after retirement:

$\begin{array}{c}\text{Monthly}\text{interest}\text{rate}=\frac{\text{Annual}\text{percentage}\text{rate}}{\text{Number}\text{of}\text{months}}\\ =\frac{7.72\%}{12\text{months}}\\ =0.643\%\end{array}$

The monthly interest rate before retirement is 0.643%.

Calculation of the future value after monthly savings of ten years:

$\begin{array}{c}\text{Future}\text{value}=\text{Monthly}\text{payments}\times \frac{{\left(1+\text{Rate}\text{of}\text{interest}\right)}^{\text{Time}}-1}{\text{Rate}\text{of}\text{interest}}\\ =\$2,100\times \frac{{\left(1+0.00873\right)}^{120}-1}{0.00873}\\ =\$2,100\times 210.538\\ =\$442,130\end{array}$

The future value is $442,130.

Calculation of the remaining amount of future value:

$\begin{array}{c}\text{Remaining}\text{amount}=\text{Future}\text{value}\text{of}\text{monthly}\text{savings}-\text{Purchase}\text{price}\text{of}\text{cabin}\\ =\text{\$442,130}-\text{\$350,000}\\ =\text{\$92,130}\end{array}$

The amount remaining after the purchase of cabin is $92,130.

Calculation of the future value of the remaining amount after the repurchase of the cabin:

$\begin{array}{c}\text{Future}\text{value}=\text{Balance}\text{amount}\times {\left(1+\text{Rate}\text{of}\text{interest}\right)}^{\text{Time}}\\ =\$92,130\times {\left(1+0.00873\right)}^{240}\\ =\$92,130\times 8.062\\ =\$742,752\end{array}$

The future value is $742,752.

Calculation of the value of annuity after 30 years:

$\begin{array}{c}\text{Annuity}\text{after}\text{30}\text{years}=\text{Monthly}\text{payment}\times {\text{PVIFA}}_{0.643\%,240\text{months}}\\ =\$20,000\times 122.089\\ =\$2,441,780\end{array}$

The annuity after 30 years is $2,441,780.

Calculation of the value of inheritance after 30 years:

$\begin{array}{c}\text{Annuity}\text{after}\text{30}\text{years}=\text{Inheritance}\text{value}\times \frac{1}{{\left(1+\text{Annual}\text{interest}\text{rate}\right)}^{\text{Time}\text{period}}}\\ =\$1,500,000\times \frac{1}{{\left(1+0.08\right)}^{20}}\\ =\$1,500,000\times 0.215\\ =\$322,500\end{array}$

The annuity after 30 years is $322,500.

Calculation of the total required funds at requirement:

$\begin{array}{c}\text{Total}\text{funds}=\left[\begin{array}{l}\text{Retirement}\text{income}+\text{Inheritance}\text{value}\\ -\text{Future}\text{value}\text{amount}\text{remaining}\\ \text{after}\text{purchase}\text{of}\text{cabin}\end{array}\right]\\ =\$2,441,780+\$322,500-\$742,752\\ =\$2,021,528\end{array}$

The total funds required are $2,021,528.

Conclusion

Thus, the amount to be saved from 11 years to 30 years each month is $2,499.