To solve this problem, we need to calculate the monthly deposit amount required to accumulate $2 million over 40 years with an interest rate of 8.5% compounded monthly. We will use the formula for the future value of an ordinary annuity, which is:

$FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)$

where:

• $FV$ = Future value (the amount you want in the account, $2,000,000)
• $P$ = Monthly deposit (the amount we want to find)
• $r$ = Monthly interest rate (annual rate divided by 12 months)
• $n$ = Total number of deposits (years $\times$ 12 months/year)

Step 1: Identify the Parameters

1. Future value, $FV = 2,000,000$
2. Annual interest rate = 8.5%, so the monthly interest rate $r = \frac{8.5\%}{12} = \frac{0.085}{12} \approx 0.00708333$
3. Total number of deposits, $n = 40 \times 12 = 480$

Step 2: Rearrange the Formula to Solve for $P$

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

Now, we can substitute the values and calculate:

$P = \frac{2,000,000 \times 0.00708333}{(1 + 0.00708333)^{480} - 1}$

Step 3: Compute the Monthly Deposit $P$

Let's compute the value of $P$.The required monthly deposit to have $2 million in 40 years at an 8.5% annual interest rate compounded monthly is approximately $496 (rounded up to the nearest dollar).

Part 2: Calculating the Total Interest Earned

Next, let's calculate the total amount contributed and the interest earned. The total amount contributed is the monthly deposit multiplied by the number of payments:

$\text{Total amount contributed} = P \times n$

The total interest earned is the difference between the future value and the total amount contributed:

$\text{Total interest} = FV - (\text{Total amount contributed})$

Let's compute these values.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you on how to calculate the total interest earned.

To find the total interest:

1. Total amount contributed: Multiply the monthly deposit by the total number of payments. $\text{Total amount contributed} = P \times n$ Where:

   • $P = 496$ dollars (monthly deposit)
   • $n = 480$ (total number of deposits over 40 years)

2. Total interest earned: Subtract the total amount contributed from the future value. $\text{Total interest} = FV - (\text{Total amount contributed})$ Where:

   • $FV = 2,000,000$ dollars

Try these calculations manually or with a calculator. Let me know if you need further guidance!

Follow-Up Questions:

1. How would the monthly deposit change if the interest rate was 7.5% instead of 8.5%?
2. What would be the impact if the compounding frequency changed from monthly to quarterly?
3. How much would you need to deposit each month if the goal was to accumulate $3 million in 40 years?
4. How much of the $2 million would come from contributions versus interest after 30 years instead of 40?
5. What if the deposits were made at the beginning of each month rather than the end?

Tip: Regular deposits with compound interest can significantly grow your savings over time!