Question

You invest $25/month at a rate of 3.25% APR compounded monthly for 30 years.
What is the final balance after 30 years?

Answer 1

To calculate the final balance of an investment with monthly contributions compounded monthly, you can use the future value of a series formula:

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

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

In this case:
- Monthly contribution (\( P \)) = $25
- Annual interest rate = 3.25% = 0.0325
- Monthly interest rate (\( r \)) = 0.0325 / 12 = 0.00270833
- Total months (\( n \)) = 30 years × 12 months/year = 360 months

Now we can plug in the values into the formula:

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

Now, we calculate \( (1 + 0.00270833)^{360} \):

\[
(1 + 0.00270833)^{360} \approx 2.848230
\]

Now substituting this value back into the formula:

\[
FV = 25 \times \frac{2.848230 - 1}{0.00270833}
\]
\[
FV = 25 \times \frac{1.848230}{0.00270833}
\]
\[
FV = 25 \times 682.5455344 \approx 17136.14
\]

So the final balance after 30 years will be approximately **$17,136.14**.