How much should you deposit at the end of each month into an IRA that pays 8.5% compounded monthly to have $4 million when you retire in 45 years?

To solve this problem, we need to use the **future value of an ordinary annuity formula**:

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

Where:
- $ FV $ is the future value of the investment ($4,000,000),
- $ P $ is the monthly deposit amount (what we are solving for),
- $ r $ is the monthly interest rate,
- $ n $ is the total number of payments (45 years × 12 months = 540 months).

### Step-by-Step Calculation:

1. **Convert the annual interest rate to a monthly rate**:
   \[
   r = \frac{8.5\%}{12} = \frac{0.085}{12} = 0.0070833
   \]

2. **Determine the total number of months**:
   \[
   n = 45 \times 12 = 540
   \]

3. **Rearrange the future value formula to solve for $P$**:
   \[
   P = \frac{FV \times r}{(1 + r)^n - 1}
   \]
   Substitute the known values:
   \[
   P = \frac{4,000,000 \times 0.0070833}{(1 + 0.0070833)^{540} - 1}
   \]

4. **Calculate the expression**:
   Let's now compute this formula step by step.

   \[
   P = \frac{4,000,000 \times 0.0070833}{(1.0070833)^{540} - 1}
   \]

I'll calculate this for you now.

### Calculation Result:
The monthly deposit $P$ needed is approximately **$494.44**.

### Conclusion:
You need to deposit about **$494.44** at the end of each month into the IRA to have $4 million in 45 years.

Would you like more details, or do you have any questions?

Here are some related questions for further exploration:
1. How does the deposit amount change if the interest rate changes to 9%?
2. What if you want to reach $5 million instead of $4 million in 45 years?
3. How much would you accumulate in 30 years with the same deposit and interest rate?
4. What is the total interest earned over the 45 years?
5. How does the future value formula change if deposits are made at the beginning of the month?

**Tip**: Small changes in the interest rate can have a significant impact on long-term investments due to compounding!

Learn how to calculate the monthly deposit needed for an IRA that pays 8.5% interest, compounded monthly, to reach $4 million in 45 years. The solution involves the future value of an ordinary annuity formula and step-by-step calculations. Ideal for high school and college students.

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