Suppose you want to have $800,000 for retirement in 30 years. Your account earns 8% interest.

a) How much would you need to deposit in the account each month?

To determine how much you would need to deposit each month to accumulate $800,000 in 30 years, we can use the future value of an annuity formula, which is given by:

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

Where:
- $ FV $ is the future value, which is $800,000.
- $ P $ is the monthly payment (this is what we need to find).
- $ r $ is the monthly interest rate, which is the annual interest rate divided by 12.
- $ n $ is the total number of payments (months), which is the number of years times 12.

### Given:
- $ FV = 800,000 $
- Annual interest rate = 8% or 0.08
- $ r = \frac{0.08}{12} = 0.0066667 $
- $ n = 30 \times 12 = 360 $ months

### Rearrange the formula to solve for $ P $:

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

Substituting the given values:

\[
P = \frac{800,000 \times 0.0066667}{(1 + 0.0066667)^{360} - 1}
\]

Let's calculate this step-by-step.The amount you would need to deposit each month is approximately **$536.78** to accumulate $800,000 in 30 years at an 8% interest rate.

Would you like further details or have any questions? Here are some related questions to consider:

1. How would the monthly deposit change if the interest rate were 6% instead of 8%?
2. What is the total amount of money you would have deposited over the 30 years?
3. How much interest would you earn over the 30-year period?
4. What if you wanted to retire with $1,000,000 instead of $800,000? How much would you need to deposit monthly?
5. How would the required monthly deposit change if the investment period were 20 years instead of 30?

**Tip:** Always consider how changes in the interest rate or the investment period affect your monthly savings plan when planning for retirement.

Suppose you want to have $800,000 for retirement in 30 years. Your account earns 8% interest. How much would you need to deposit in the account each month?

Learn how to calculate the monthly deposit required to accumulate $800,000 in 30 years with an 8% interest rate. Understand the future value of annuity formula and compound interest. Suitable for financial planning and retirement savings scenarios.

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