This is a financial problem involving compound interest and monthly savings. The goal is to determine the monthly deposit required to accumulate $300,000 in 20 years with an annual interest rate of 9%, compounded monthly.

Formula:

The future value of an annuity formula is: $FV = PMT \cdot \frac{(1 + r/n)^{n \cdot t} - 1}{r/n}$ Where:

• $FV$ is the future value ($300,000),
• $PMT$ is the monthly deposit (what we’re solving for),
• $r$ is the annual interest rate (0.09),
• $n$ is the number of compounding periods per year (12),
• $t$ is the number of years (20).

Rearrange to solve for $PMT$:

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

Substituting values:

• $FV = 300,000$,
• $r = 0.09$,
• $n = 12$,
• $t = 20$.

Let’s calculate step-by-step.

Step 1: Calculate $(1 + r/n)$:

$1 + \frac{0.09}{12} = 1.0075$

Step 2: Raise it to $n \cdot t$:

$(1.0075)^{12 \cdot 20} = (1.0075)^{240} \approx 6.84885$

Step 3: Subtract 1:

$6.84885 - 1 = 5.84885$

Step 4: Multiply by $r/n$:

$\frac{0.09}{12} = 0.0075$ $5.84885 \cdot 0.0075 = 0.043866375$

Step 5: Divide $FV$ by the result:

$PMT = \frac{300,000 \cdot 0.0075}{5.84885} \approx \frac{2250}{5.84885} \approx 384.86$

Final Answer:

You would need to deposit approximately $384.86 per month.

Would you like a breakdown of each step in further detail? Let me know!

Related Questions:

1. What happens to the required monthly deposit if the interest rate increases to 10%?
2. How much would you need to deposit if the target was $500,000 instead of $300,000?
3. What is the total amount of interest earned over 20 years?
4. How does compounding frequency (e.g., yearly vs. monthly) affect the monthly deposit required?
5. What if you deposit a lump sum now instead of monthly—how much would you need?

Tip:

When planning for retirement, always factor in inflation, as the purchasing power of your savings might decrease over time. Adjust contributions accordingly!