An ordinary annuity pays 6.84% compounded monthly (A) A person deposits $200 monthly for 30 years and then makes equal monthly withdrawals for the next 15 years reducing the balance to zero. What are the monthly withdrawals? How much interest is earned during the entire 45-year process? (B) If the person wants to make withdrawals of 2,000 per month for the last 15 years , how much must be deposited monthly for the first 30 years?

This problem deals with two phases of an annuity: the accumulation phase and the withdrawal phase, both under compound interest. We will solve this in two parts.

### Part A: Monthly Withdrawals and Interest Earned
#### 1. Accumulation Phase (First 30 years):
The formula to find the future value of an ordinary annuity is:

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

Where:
- $P = 200$ (monthly deposit)
- $r = \frac{6.84\%}{12} = 0.0057$ (monthly interest rate)
- $n = 30 \times 12 = 360$ (total number of months)

We first calculate the future value of the annuity after 30 years.

#### 2. Withdrawal Phase (Next 15 years):
The formula to find the present value of an ordinary annuity (since we’re drawing down the balance) is:

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

Where:
- $PV$ is the future value from the first phase
- $W$ is the monthly withdrawal (what we need to find)
- $r = 0.0057$
- $n = 15 \times 12 = 180$

This gives us the monthly withdrawal amount for the next 15 years.

#### 3. Interest Earned:
The total interest earned is the difference between the total amount of deposits and withdrawals and the amount of interest accumulated.

We can now calculate this.

### Part B: Monthly Deposits for Desired Withdrawals
In this case, we are given the withdrawal amount for the second phase (\$2,000 per month) and need to calculate the monthly deposit for the first 30 years to support this withdrawal.

The steps are:
1. Calculate the present value of the withdrawals using the formula for the present value of an ordinary annuity, but with a withdrawal amount of \$2,000.
2. Set this present value equal to the future value of the deposits from the first phase and solve for $P$, the monthly deposit.

I'll compute these values now.### Part A:
1. **Monthly Withdrawals:** After 30 years of depositing \$200 per month, the monthly withdrawals for the next 15 years will be **\$2,104.00**.
2. **Interest Earned:** The total interest earned over the entire 45-year period is **\$306,720.22**.

### Part B:
To support monthly withdrawals of \$2,000 for the last 15 years, the person must deposit **\$190.11** per month for the first 30 years.

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Do you want further details or have any questions? Here are some related questions to deepen your understanding:

1. How does the interest rate affect both the accumulation and withdrawal phases of the annuity?
2. What happens if the monthly deposit during the first 30 years is increased?
3. How would the solution change if the interest were compounded quarterly instead of monthly?
4. What if the person wanted to withdraw over 20 years instead of 15 years? How would the deposit amount change?
5. How can the formula for future value be derived from basic principles of compound interest?

**Tip:** The earlier you start saving in an annuity, the more your investment benefits from compound interest, especially when compounded monthly.

An ordinary annuity pays 6.84% compounded monthly. (A) A person deposits $200 monthly for 30 years and then makes equal monthly withdrawals for the next 15 years reducing the balance to zero. What are the monthly withdrawals? How much interest is earned during the entire 45-year process? (B) If the person wants to make withdrawals of 2,000 per month for the last 15 years, how much must be deposited monthly for the first 30 years?

Learn how to calculate the monthly withdrawals and interest earned for a 45-year annuity paying 6.84% compounded monthly. Includes solutions for monthly deposits and withdrawals over different periods using future and present value formulas. Suitable for finance students and those studying business mathematics.

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