Andrea, a self-employed individual, wishes to accumulate a retirement fund of $800,000. How much should she deposit each month into her retirement account, which pays interest at a rate of 4.5%/year compounded monthly, to reach her goal upon retirement 20 years from now?

To solve this problem, we need to determine the monthly deposit Andrea should make into her retirement account to accumulate $800,000 in 20 years, given an annual interest rate of 4.5% compounded monthly.

### Step-by-Step Solution:

1. **Identify the variables:**
   - Future value (FV): $800,000 (the amount Andrea wants to have in her retirement fund)
   - Interest rate (annual): 4.5%
   - Compounding frequency: monthly
   - Number of years (t): 20 years
   - Monthly deposit amount (PMT): this is what we are solving for
   - Number of compounding periods per year (n): 12 (monthly compounding)

2. **Convert the annual interest rate to a monthly interest rate:**
   \[
   r = \frac{4.5\%}{12} = \frac{0.045}{12} = 0.00375
   \]
   So, the monthly interest rate $r = 0.00375$.

3. **Calculate the total number of payments:**
   \[
   N = 20 \times 12 = 240
   \]

4. **Use the formula for the future value of an ordinary annuity to find the monthly deposit $PMT$:**
   \[
   FV = PMT \times \frac{(1 + r)^N - 1}{r}
   \]
   Rearranging to solve for $PMT$:
   \[
   PMT = \frac{FV \times r}{(1 + r)^N - 1}
   \]

5. **Plug in the known values:**
   \[
   PMT = \frac{800,000 \times 0.00375}{(1 + 0.00375)^{240} - 1}
   \]

6. **Calculate the values step-by-step:**

   - Compute $(1 + r)^N$:
   \[
   (1 + 0.00375)^{240}
   \]

   - Then compute the entire expression for $PMT$.

Let's calculate $PMT$:

\[
PMT = \frac{800,000 \times 0.00375}{(1 + 0.00375)^{240} - 1}
\]

\[
PMT = \frac{800,000 \times 0.00375}{(1.00375)^{240} - 1}
\]

Now let's compute this value using a calculator.

Andrea should deposit approximately **$2,061.20** each month into her retirement account to accumulate $800,000 in 20 years with an annual interest rate of 4.5% compounded monthly.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the monthly deposit change if the interest rate were higher, say 6%?
2. What if Andrea wanted to retire in 15 years instead of 20 years? How would this affect her monthly deposit?
3. How much total interest would Andrea earn over the 20 years?
4. What is the difference in the final amount if the interest is compounded annually instead of monthly?
5. How can Andrea adjust her contributions if she wants to account for inflation over the next 20 years?

**Tip:** When saving for retirement or any long-term goal, starting earlier allows you to take advantage of compound interest, reducing the amount you need to save each month.

Learn how to calculate the monthly deposit needed for Andrea to accumulate $800,000 in her retirement fund in 20 years, with a 4.5% annual interest rate compounded monthly. Step-by-step solution and calculations provided.

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