This problem involves calculating the present value of an investment that grows to a future value under compound interest. The formula for present value ($PV$) in compound interest is:

$PV = \frac{FV}{(1 + r/n)^{n \cdot t}}$

Where:

• $FV$ = future value = $800,000
• $r$ = annual interest rate = 4% = 0.04
• $n$ = number of compounding periods per year = 12
• $t$ = time in years = 20

Step 1: Substitute the values

$PV = \frac{800,000}{(1 + 0.04/12)^{12 \cdot 20}}$

$PV = \frac{800,000}{(1 + 0.0033333)^{240}}$

$PV = \frac{800,000}{(1.0033333)^{240}}$

Step 2: Compute the compound factor

$(1.0033333)^{240} \approx 2.213645$

Step 3: Compute $PV$

$PV = \frac{800,000}{2.213645} \approx 361,431.35$

Final Answer:

You would need to deposit approximately $361,431.35 now to reach $800,000 in 20 years with 4% interest compounded monthly.

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Do you want a step-by-step explanation of how the compound factor was computed or have any questions?

Here are 5 related questions for further exploration:

1. What if the interest were compounded quarterly instead of monthly?
2. How much would you need to deposit if the interest rate were 5% instead of 4%?
3. What if you had only 15 years until retirement?
4. How much interest would you earn over the 20 years?
5. What is the difference between simple and compound interest in this scenario?

Tip: When planning for retirement, remember that consistent periodic contributions can also grow significantly due to compound interest.