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 $10,000/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 4%/year compounded monthly to meet her retirement goal? (Round your answer to the nearest cent.)

To solve this problem, we'll follow these steps:

1. **Determine the amount needed at the time of retirement** (future value of the withdrawals).
2. **Calculate the monthly contributions** required to accumulate that future value.

### Step 1: Calculate the Present Value of Withdrawals (at Retirement)

The goal is to find out how much Jennifer needs to have in her retirement account at the start of her retirement in order to withdraw $10,000 per month for 20 years.

This situation is an example of an **ordinary annuity** since she withdraws a fixed amount monthly. We will use the present value formula for an ordinary annuity:

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

where:
- $ PV $ = present value of the annuity (amount needed at retirement)
- $ PMT $ = payment per period ($10,000 per month)
- $ r $ = monthly interest rate = $\frac{4\%}{12} = 0.003333$
- $ n $ = total number of payments = $20 \text{ years} \times 12 \text{ months/year} = 240$

Plugging in the values:
\[
PV = 10,000 \times \frac{1 - (1 + 0.003333)^{-240}}{0.003333}
\]

Calculating this expression will give the amount she needs at retirement.

### Step 2: Calculate the Monthly Contribution

Now, we need to determine how much she should contribute monthly to accumulate the amount calculated in Step 1 over the 30 years.

The future value formula for a **regular annuity** is used here:

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

Rearranging this formula to solve for $ PMT $:

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

where:
- $ FV $ is the amount calculated from Step 1
- $ r $ = monthly interest rate = $\frac{4\%}{12} = 0.003333$
- $ n $ = total number of contributions = $30 \text{ years} \times 12 \text{ months/year} = 360$

Let's calculate these steps in detail.### Results:

1. **Amount Needed at Retirement:** Jennifer needs to have approximately \$1,650,218.58 in her retirement account at the time she retires.
   
2. **Monthly Contribution:** To reach this goal, she must contribute approximately \$2,377.67 per month for 30 years into her retirement account.

Would you like a more detailed breakdown of the calculations or any further assistance?

Here are some related questions to expand on this information:

1. What impact would a different interest rate have on Jennifer's monthly contribution?
2. How would the monthly contribution change if she plans to retire in 25 years instead of 30 years?
3. What if she wants to increase her monthly withdrawal to \$15,000? How would that affect her plan?
4. How does compounding frequency (e.g., monthly, quarterly) affect the total amount she needs to save?
5. How would a lump-sum initial investment affect her required monthly contributions?

**Tip:** Regularly reviewing and adjusting your retirement savings plan can help you stay on track with your financial goals, especially if there are changes in interest rates or other financial conditions.

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 $10,000/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 4%/year compounded monthly to meet her retirement goal?

Learn how to calculate monthly contributions for a retirement account earning 4% interest, with $10,000 monthly withdrawals over 20 years. This problem covers key finance concepts, including annuities, present value, and compound interest, with a detailed step-by-step solution.

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