Question

You invest $25/month at a rate of 3.25% APR compounded monthly for 30 years.
What is the total interest earned?

Answer 1

To calculate the total amount saved with regular monthly contributions, we can use the Future Value of a series formula for compound interest. The formula for the future value of an annuity is:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \( FV \) is the future value of the annuity (total amount accumulated),
- \( P \) is the amount of each payment (monthly investment),
- \( r \) is the monthly interest rate (annual interest rate divided by 12),
- \( n \) is the total number of payments (number of months).

Given:
- Monthly investment \( P = 25 \) dollars,
- Annual interest rate \( 3.25\% \) or \( 0.0325 \),
- Monthly interest rate \( r = \frac{0.0325}{12} \approx 0.00270833 \),
- Time period \( 30 \) years or \( 30 \times 12 = 360 \) months.

Now we can substitute the values into the formula:

\[
FV = 25 \times \frac{(1 + 0.00270833)^{360} - 1}{0.00270833}
\]

First, calculate \( (1 + r)^{n} \):

\[
(1 + 0.00270833)^{360} \approx 2.619121
\]

Now substitute this value back into the formula:

\[
FV = 25 \times \frac{2.619121 - 1}{0.00270833}
\]

Calculate \( 2.619121 - 1 = 1.619121 \) and then divide:

\[
\frac{1.619121}{0.00270833} \approx 597.3367
\]

Now multiply by \( 25 \):

\[
FV \approx 25 \times 597.3367 \approx 14933.42
\]

So, the future value of the investment after 30 years is approximately \( 14933.42 \) dollars.

Next, to calculate the total amount contributed, we find:

\[
\text{Total Contributions} = P \times n = 25 \times 360 = 9000 \text{ dollars}
\]

Finally, to find the total interest earned, we subtract the total contributions from the future value:

\[
\text{Total Interest Earned} = FV - \text{Total Contributions} = 14933.42 - 9000 \approx 5933.42
\]

Thus, the total interest earned is approximately **$5933.42**.