Jennifer is the owner of a video game and entertainment software retail store. She is currently planning to retire in 30 years and wishes to withdraw $13,000/month for 20 years from her retirement account starting at that time. How much must she contribute each month for 30 years into a retirement account earning interest at the rate of 3%/year compounded monthly to meet her retirement goal? (Round your answer to the nearest cent.)

To determine how much Jennifer must contribute each month, we need to break the problem into two parts:

1. **Calculate the amount needed at the time of retirement** to support her monthly withdrawals for 20 years.
2. **Determine the monthly contributions required** to accumulate that amount over 30 years.

### Step 1: Calculate the amount needed at retirement

Jennifer wants to withdraw $13,000/month for 20 years. This is an annuity problem where we can use the formula for the **present value of an annuity**:

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

Where:
- $PV$ = Present value (the amount needed at retirement)
- $PMT$ = $13,000 (monthly withdrawal)
- $r$ = Monthly interest rate = $\frac{3\%}{12} = 0.0025$
- $n$ = Total number of withdrawals = $20 \times 12 = 240$ months

Substituting the values:

\[
PV = 13,000 \times \left( \frac{1 - (1 + 0.0025)^{-240}}{0.0025} \right)
\]

Let's calculate this:

\[
PV = 13,000 \times \left( \frac{1 - (1.0025)^{-240}}{0.0025} \right)
\]

\[
PV = 13,000 \times \left( \frac{1 - 0.548811636}{0.0025} \right)
\]

\[
PV = 13,000 \times \left( \frac{0.451188364}{0.0025} \right)
\]

\[
PV = 13,000 \times 180.4753456 = 2,346,179.49
\]

So, Jennifer needs **$2,346,179.49** at the time of retirement to meet her withdrawal goal.

### Step 2: Calculate the monthly contributions required

Now, we need to determine how much Jennifer should contribute monthly for 30 years to accumulate $2,346,179.49, with the retirement account earning 3% interest compounded monthly. This is a **future value of an annuity** problem, and we use the formula:

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

Where:
- $FV$ = Future value = $2,346,179.49
- $PMT$ = Monthly contribution (unknown)
- $r$ = Monthly interest rate = 0.0025
- $n$ = Total number of contributions = $30 \times 12 = 360$ months

Rearranging the formula to solve for $PMT$:

\[
PMT = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)}
\]

Substituting the known values:

\[
PMT = \frac{2,346,179.49}{\left( \frac{(1.0025)^{360} - 1}{0.0025} \right)}
\]

Calculating the denominator:

\[
(1.0025)^{360} = 2.457619
\]

\[
\frac{2.457619 - 1}{0.0025} = 583.0476
\]

Now, calculate $PMT$:

\[
PMT = \frac{2,346,179.49}{583.0476} = 4,023.64
\]

### Final Answer:
Jennifer must contribute **$4,023.64** per month for 30 years to meet her retirement goal.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does changing the interest rate affect the monthly contributions?
2. What if Jennifer wants to withdraw for 25 years instead of 20?
3. What would happen if the contributions were made quarterly instead of monthly?
4. How would a higher withdrawal amount ($15,000 per month) change her savings plan?
5. Can Jennifer reach her goal sooner with a higher contribution?

**Tip:** Regularly reviewing and adjusting contributions ensures you stay on track with financial goals, especially if interest rates fluctuate.

Learn how to calculate the monthly contributions needed for Jennifer's retirement plan, including present value and future value annuity calculations. The problem involves compound interest, with step-by-step solutions to help you understand the process. Suitable for advanced high school and college students.

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