Suppose you want to have $500,000 for retirement in 20 years. Your account earns 10% interest. How much would you need to deposit in the account each month?

To determine how much you need to deposit each month to reach $500,000 in 20 years with an interest rate of 10% per year compounded monthly, we can use the formula for the future value of an annuity:

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

where:
- $ FV $ is the future value ($500,000),
- $ P $ is the monthly deposit amount (which we need to find),
- $ r $ is the annual interest rate (0.10),
- $ n $ is the number of compounding periods per year (12, since it is monthly),
- $ t $ is the number of years (20).

Rearranging the formula to solve for $ P $:

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

Substitute the given values into the formula:

\[
P = \frac{500,000 \times \frac{0.10/12}{(1 + 0.10/12)^{12 \times 20} - 1}}
\]

Let's calculate this step by step:

1. Calculate the monthly interest rate:

\[
\frac{r}{n} = \frac{0.10}{12} = 0.008333
\]

2. Calculate the total number of compounding periods:

\[
nt = 12 \times 20 = 240
\]

3. Calculate the future value factor:

\[
(1 + 0.008333)^{240} - 1 = (1.008333)^{240} - 1
\]

Let's compute $ (1.008333)^{240} $ first:

\[
(1.008333)^{240} \approx 7.328
\]

So,

\[
7.328 - 1 = 6.328
\]

4. Now calculate the denominator:

\[
\frac{0.008333}{6.328} \approx 0.001316
\]

5. Finally, calculate the monthly deposit $ P $:

\[
P = \frac{500,000 \times 0.001316}{1}
\]

\[
P \approx 658.39
\]

So, you would need to deposit approximately **$658.39** each month to have $500,000 in 20 years.

If you have any questions or need further clarification, feel free to ask.

Here are 8 related questions:
1. What is the difference between a lump sum investment and regular deposits in terms of future value?
2. How would the required monthly deposit change if the interest rate were lower?
3. What if the compounding were quarterly instead of monthly?
4. How does the time period affect the required monthly deposit amount?
5. What is the formula for calculating the present value of an annuity?
6. How can you determine the total interest earned over the 20 years?
7. What happens if you skip a monthly deposit?
8. How does inflation affect the purchasing power of your future savings?

**Tip:** When saving for long-term goals, consider the impact of inflation and adjust your target amount accordingly.

Learn how to calculate the monthly deposit needed to achieve $500,000 for retirement in 20 years with a 10% annual interest rate. This problem covers compound interest, future value of annuities, and monthly deposit calculations. Suitable for adults looking to plan their retirement savings.

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