Chapter 17, Problem 1PA

Summary Introduction

To determine:

The future value after contributing $1,000 at the end of each year for 40 years

Introduction:

Future value is the amount received at the end of a certain period with a certain money input in the beginning.

Explanation of Solution

Given,

For the maturity period of 40 years the value can be calculated using the following values:

Present value is $1,000.

Duration is 40 years.

Interest rate is 9%.

Formula to calculate future value,

$FV=PMT\times \left(\frac{{(1+R)}^n-1}{R^{}}\right)$

Substitute present value as $1,000, N as 40 years and R as 9%.

$\begin{array}{c}FV=PMT\times \left(\frac{{(1+R)}^n-1}{R^{}}\right)\\ =\$1,000\times \frac{\left({\left(1+.09\right)}^{40}\right)}{.09}\\ =\$337,882.45\end{array}$

Hence, the future value for the amount deposited $1,000 at the end of period for 40 years compounded annually at the rate of 9% will be $337,882.45.

When there is no contribution after the end of 30 years then the receivable amount will be calculated as follows:

Given,

PMT is $1,000.

N is 30 years.

R is 9%.

Formula to calculate future value,

$FV=PMT\times \left(\frac{{(1+R)}^n-1}{R^{}}\right)$

Substitute the values of PMT as $1,000, N as 30 years and R as 9%.

$\begin{array}{c}FV=PMT\times \left(\frac{{(1+R)}^n-1}{R^{}}\right)\\ =\$1,000\times \frac{\left({\left(1+.09\right)}^{30}\right)}{.09}\\ =\$136,307.54\end{array}$

The amount which will be received after 30 years will produce only the compound interest at 9% which is $136,307.54.

Now to calculate the future value of $136,307.54 for the rest of the 10 years at the rate of 9% the following could be calculated as follows:

Given,

PMT is $136,307.54.

N is 10 years.

r is 9%.

Formula to calculate future value,

$FV=PMT\times \left(\frac{{(1+R)}^n-1}{R^{}}\right)$

Substitute the above values of PMT as $136,30754, N as 10 years and R as 9%.

$\begin{array}{c}FV=PMT\times \left(\frac{{(1+R)}^n-1}{R^{}}\right)\\ =\$136,307.54\times \frac{\left({\left(1+.09\right)}^{10}\right)}{.09}\\ =\$332,689.52\end{array}$

Conclusion

Thus the amount to be received after 40 years would be $332,689.52.