Question
What is the future value of a savings account that earns 5% interest compounded monthly if I deposit $100 every month for 10 years?
Answers
Answered by
GPT-4o mini
To calculate the future value of a savings account with monthly deposits compounded monthly, we can use the formula for the future value of a series of cash flows:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) = future value of the investment/loan, including interest
- \( P \) = payment amount per period (monthly deposit)
- \( r \) = interest rate per period
- \( n \) = total number of payments (periods)
Given:
- Monthly deposit \( P = 100 \)
- Annual interest rate = 5% = 0.05
- Monthly interest rate \( r = \frac{0.05}{12} \approx 0.00416667 \)
- Total number of months \( n = 10 \times 12 = 120 \)
Now, we can plug the values into the formula:
1. Calculate \( (1 + r)^n \):
\[
(1 + 0.00416667)^{120} \approx 1.647009
\]
2. Plug this into the future value formula:
\[
FV = 100 \times \frac{(1.647009 - 1)}{0.00416667}
\]
\[
FV = 100 \times \frac{0.647009}{0.00416667} \approx 100 \times 155.2854 \approx 15528.54
\]
Therefore, the future value of the savings account after 10 years, with monthly deposits of $100 at an interest rate of 5% compounded monthly, will be approximately **$15,528.54**.
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) = future value of the investment/loan, including interest
- \( P \) = payment amount per period (monthly deposit)
- \( r \) = interest rate per period
- \( n \) = total number of payments (periods)
Given:
- Monthly deposit \( P = 100 \)
- Annual interest rate = 5% = 0.05
- Monthly interest rate \( r = \frac{0.05}{12} \approx 0.00416667 \)
- Total number of months \( n = 10 \times 12 = 120 \)
Now, we can plug the values into the formula:
1. Calculate \( (1 + r)^n \):
\[
(1 + 0.00416667)^{120} \approx 1.647009
\]
2. Plug this into the future value formula:
\[
FV = 100 \times \frac{(1.647009 - 1)}{0.00416667}
\]
\[
FV = 100 \times \frac{0.647009}{0.00416667} \approx 100 \times 155.2854 \approx 15528.54
\]
Therefore, the future value of the savings account after 10 years, with monthly deposits of $100 at an interest rate of 5% compounded monthly, will be approximately **$15,528.54**.