Chapter 9A, Problem 1P

To determine

(a)

The amount of money invested each year.

Answer to Problem 1P

The amount of money invested each year is $\$20056$.

Explanation of Solution

Given:

Annual income is $\$65000$.

Number of year is $30$.

Future Value is $\$2000000$.

Calculation:

Write the expression for future value.

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

Here, the future value is $F$, the annual investment is $A$, the interest rate is $i$ and the number of period is $n$.

The interest rate for stock fund is $7.3\%$.

Substitute $\$2000000$ for $F$ and $0.073$ for $i$ and $30$ for $n$ in Equation (I).

$\begin{array}{l}\$2000000=A\frac{\left({\left(1+0.073\right)}^{30}-1\right)}{0.073}\\ A=\frac{\$2000000}{99.72}\\ A=\$20056\end{array}$.

Conclusion:

Thus, the amount of money invested each year is $\$20056$.

To determine

(b)

The amount of money each should invest annually.

Answer to Problem 1P

The amount invested by the employer annually is $\$1950$.

The amount invested by the employee annually is $\$18106$.

Explanation of Solution

Given:

Annual income is $\$65000$.

Number of year is $30$.

Future Value is $\$2000000$.

Saving is $3\%$.

Calculation:

Calculate the amount contributed each year by the employer.

$\begin{array}{c}{A}_E=\$65000\times 3\%\\ =\$65000\times 0.03\\ =\$1950\end{array}$

Calculate the amount invested by the employer.

$A_{EE}=A-A_E$ ...... (II)

Here, the annual amount to be invested is $A$, the annual amount invested by the employer is $A_E$ and the annual amount invested by the employee is $A_{EE}$.

Substitute $\$20056$ for $A$, $\$1950$ for $A_E$ in Equation (II).

$\begin{array}{c}{A}_{EE}=\$20056-\$1950\\ =\$18106\end{array}$.

Conclusion:

Thus, the amount invested by the employer annually is $\$1950$.

The amount invested by the employee annually is $\$18106$.