How much should you deposit at the end of each month into an investment account that pays 6.5 % compounded monthly to have $ 1 million when you retire in 39 ​years? How much of the $ 1 million comes from​ earnings?

To solve this problem, we need to use the **future value of an annuity formula** since you are making regular monthly deposits into an investment account. The formula is:

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

Where:
- $ FV $ is the future value of the annuity (in this case, $1,000,000),
- $ P $ is the monthly deposit (what we are trying to find),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of deposits.

### Step 1: Convert the annual interest rate to a monthly rate.
Given:
- Annual interest rate = 6.5%
- Monthly interest rate $ r = \frac{6.5\%}{12} = \frac{0.065}{12} = 0.00541667 $

### Step 2: Calculate the total number of payments.
Since the investment period is 39 years, and you make monthly deposits:

\[
n = 39 \times 12 = 468 \text{ months}
\]

### Step 3: Plug the values into the future value formula and solve for $ P $.

\[
1,000,000 = P \times \frac{(1 + 0.00541667)^{468} - 1}{0.00541667}
\]

Now, I'll solve for $ P $.The amount you need to deposit at the end of each month is approximately **$469.77**.

### Step 4: Calculate how much of the $1 million comes from earnings.

To find the total amount deposited, multiply the monthly deposit by the total number of payments:

\[
\text{Total Deposits} = P \times n
\]

Then, subtract this amount from $1,000,000 to find the earnings.

I'll calculate that next.### Summary of the Results:

- **Total Deposits:** Approximately **$219,853.28**
- **Total Earnings from Interest:** Approximately **$780,146.72**

This means that out of the $1 million, about **$780,146.72** comes from earnings, while the rest, about **$219,853.28**, comes from your own deposits.

---

Would you like more details or have any questions? Here are some related questions to explore:

1. How would the required monthly deposit change if the interest rate was higher?
2. What would the total earnings be if the deposit period was shorter, say 30 years instead of 39 years?
3. How does the frequency of compounding (e.g., quarterly or annually) affect the required monthly deposit?
4. How much would you need to deposit monthly if you wanted to reach $2 million instead?
5. What if the interest rate was variable over the investment period? How would that affect your strategy?

**Tip:** Starting early with consistent deposits allows compound interest to work in your favor, significantly boosting your earnings over time.

How much should you deposit at the end of each month into an investment account that pays 6.5% compounded monthly to have $1 million when you retire in 39 years? How much of the $1 million comes from earnings?

Learn how to calculate the monthly deposit required to reach $1 million in 39 years with a 6.5% interest rate compounded monthly. This guide covers key financial concepts such as the future value of annuities, compound interest, and detailed step-by-step calculations.

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