Question
You invest $25/month at a rate of 3.25% APR compounded monthly for 30 years.
Answers
GPT-4o mini
To calculate the future value of an investment where you contribute a fixed amount monthly, you can use the formula for the future value of a series of cash flows (an annuity). The formula is:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) = future value of the investment
- \( P \) = amount contributed each period (monthly contribution)
- \( r \) = interest rate per period (monthly interest rate)
- \( n \) = total number of contributions (number of periods)
Given:
- Monthly contribution, \( P = 25 \)
- Annual interest rate, \( APR = 3.25\% \) (0.0325 as a decimal)
- Monthly interest rate, \( r = \frac{0.0325}{12} = 0.00270833 \)
- Investment duration, 30 years, which means total months, \( n = 30 \times 12 = 360 \)
Now substituting these values into the formula:
1. Calculate the monthly interest rate:
\[
r = \frac{0.0325}{12} = 0.00270833
\]
2. Calculate the total number of contributions (months):
\[
n = 30 \times 12 = 360
\]
3. Plug the values into the future value formula:
\[
FV = 25 \times \frac{(1 + 0.00270833)^{360} - 1}{0.00270833}
\]
4. Calculate \( (1 + 0.00270833)^{360} \):
\[
(1 + 0.00270833)^{360} \approx 2.63925
\]
5. Now substitute this back into the future value formula:
\[
FV = 25 \times \frac{2.63925 - 1}{0.00270833}
\]
6. Calculate the future value:
\[
FV = 25 \times \frac{1.63925}{0.00270833} \approx 25 \times 604.55598 \approx 15113.90
\]
Thus, the future value of your investment after 30 years, making monthly contributions of $25 at an interest rate of 3.25% compounded monthly, would be approximately **$15,113.90**.
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) = future value of the investment
- \( P \) = amount contributed each period (monthly contribution)
- \( r \) = interest rate per period (monthly interest rate)
- \( n \) = total number of contributions (number of periods)
Given:
- Monthly contribution, \( P = 25 \)
- Annual interest rate, \( APR = 3.25\% \) (0.0325 as a decimal)
- Monthly interest rate, \( r = \frac{0.0325}{12} = 0.00270833 \)
- Investment duration, 30 years, which means total months, \( n = 30 \times 12 = 360 \)
Now substituting these values into the formula:
1. Calculate the monthly interest rate:
\[
r = \frac{0.0325}{12} = 0.00270833
\]
2. Calculate the total number of contributions (months):
\[
n = 30 \times 12 = 360
\]
3. Plug the values into the future value formula:
\[
FV = 25 \times \frac{(1 + 0.00270833)^{360} - 1}{0.00270833}
\]
4. Calculate \( (1 + 0.00270833)^{360} \):
\[
(1 + 0.00270833)^{360} \approx 2.63925
\]
5. Now substitute this back into the future value formula:
\[
FV = 25 \times \frac{2.63925 - 1}{0.00270833}
\]
6. Calculate the future value:
\[
FV = 25 \times \frac{1.63925}{0.00270833} \approx 25 \times 604.55598 \approx 15113.90
\]
Thus, the future value of your investment after 30 years, making monthly contributions of $25 at an interest rate of 3.25% compounded monthly, would be approximately **$15,113.90**.