Let's break down the problem step by step. We are given that the interest rate is 7.44% compounded monthly, and the process involves two phases:

1. Phase 1: Deposits for 30 years (360 months)

   • A person deposits $100 every month for 30 years (360 months).

2. Phase 2: Withdrawals for 15 years (180 months)

   • After the 30 years of deposits, the balance will be used to make monthly withdrawals for 15 years (180 months), reducing the balance to zero.

Step 1: Understanding the Interest Rate

The interest rate is compounded monthly, so the monthly interest rate is:

$r = \frac{7.44\%}{12} = 0.0744 / 12 = 0.0062 \text{ (monthly interest rate)}$

Step 2: Calculating the Future Value After 30 Years of Deposits

To find the amount accumulated after 30 years of monthly deposits, we will use the formula for the future value of an ordinary annuity:

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

Where:

• $P = 100$ is the monthly deposit,
• $r = 0.0062$ is the monthly interest rate,
• $n = 360$ is the number of months.

Substitute the values into the formula:

$FV = 100 \times \frac{(1 + 0.0062)^{360} - 1}{0.0062}$

Now, let's calculate the future value.

Step 3: Finding the Monthly Withdrawals

Once the amount accumulated in the first 30 years is calculated, we can determine the monthly withdrawals during the next 15 years (180 months) using the formula for the present value of an ordinary annuity, because the withdrawals are made at the beginning of each period:

$PV = W \times \frac{1 - (1 + r)^{-n}}{r}$

Where:

• $W$ is the monthly withdrawal,
• $PV$ is the future value after 30 years (the amount accumulated),
• $r = 0.0062$ is the monthly interest rate,
• $n = 180$ is the number of months.

We can now use the amount accumulated from Phase 1 as $PV$ and solve for $W$.

Let me now proceed to calculate these values. Given data P = 100 # monthly deposit in dollars r = 0.0062 # monthly interest rate n_deposits = 360 # number of months for deposits (30 years) n_withdrawals = 180 # number of months for withdrawals (15 years)

Step 2: Calculate the future value of the deposits after 30 years

FV = P * ((1 + r)**n_deposits - 1) / r

Step 3: Calculate the monthly withdrawals

W = FV * r / (1 - (1 + r)**-n_withdrawals)

FV, W # future value and monthly withdrawal amounts### (A) Results:

1. The future value of the deposits after 30 years is approximately $133,137.
2. The monthly withdrawals for the next 15 years (reducing the balance to zero) are approximately $1,229.66.

Interest Earned During the Entire Process

To calculate the interest earned over the entire 45-year process, we can subtract the total amount deposited from the total withdrawals.

• Total deposits over 30 years = $100 \times 360 = 36,000$.
• Total withdrawals over 15 years = $1,229.66 \times 180 = 221,340.00$.

Thus, the interest earned is:

$\text{Interest earned} = 221,340.00 - 36,000 = 185,340.00$

So, $185,340 in interest is earned during the entire 45-year process.

Would you like further details or any additional questions answered? Here are some related questions for you:

1. How would the monthly withdrawals change if the interest rate were higher or lower?
2. What happens if the deposit amount is increased to $200 per month?
3. How does the length of the withdrawal phase (15 years) affect the withdrawal amount?
4. What is the formula to calculate the total interest earned in an annuity process like this?
5. What would happen if the individual chose to make annual rather than monthly deposits?

Tip: When calculating annuities, always ensure the interest rate is converted to match the compounding period (monthly, quarterly, etc.).