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 $14,000/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 2%/year compounded monthly to meet her retirement goal? (Round your answer to the nearest cent.)


To solve this problem, we can break it down into two stages:

1. **Calculate the amount needed at the time of retirement (future value):** This is the amount Jennifer will need in her retirement account when she starts withdrawing $14,000 per month for 20 years.

2. **Calculate the monthly contribution required:** Based on the amount needed at retirement, we'll determine how much Jennifer needs to contribute each month over the next 30 years to reach that goal.

### Step 1: Calculate the Future Value Needed at Retirement

Jennifer wants to withdraw $14,000 per month for 20 years from her retirement account. The interest rate is 2% per year, compounded monthly. We will use the present value of an annuity formula to find out how much she needs at retirement:

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

Where:
- $ PV $ is the present value of the annuity (the amount needed at retirement)
- $ R = 14,000 $ (monthly withdrawal)
- $ i = \frac{0.02}{12} $ (monthly interest rate)
- $ n = 20 \times 12 = 240 $ (total number of withdrawals)

Now, let's calculate $ PV $:

\[
i = \frac{0.02}{12} = 0.0016667
\]

\[
PV = \frac{14,000 \times \left(1 - (1 + 0.0016667)^{-240}\right)}{0.0016667}
\]

\[
PV = \frac{14,000 \times \left(1 - (1.0016667)^{-240}\right)}{0.0016667}
\]

\[
PV \approx \frac{14,000 \times \left(1 - 0.699313\right)}{0.0016667} 
\]

\[
PV \approx \frac{14,000 \times 0.300687}{0.0016667}
\]

\[
PV \approx 1,122,530.98
\]

So, Jennifer will need approximately $1,122,530.98 at retirement.

### Step 2: Calculate the Monthly Contribution

Now, we calculate the monthly contribution required to accumulate $1,122,530.98 over 30 years, with the same interest rate of 2% per year compounded monthly.

We use the future value of an ordinary annuity formula:

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

Where:
- $ FV = 1,122,530.98 $ (the amount needed at retirement)
- $ i = \frac{0.02}{12} = 0.0016667 $ (monthly interest rate)
- $ n = 30 \times 12 = 360 $ (total number of contributions)
- $ P $ is the monthly contribution

Now, solve for $ P $:

\[
1,122,530.98 = \frac{P \times \left((1 + 0.0016667)^{360} - 1\right)}{0.0016667}
\]

\[
P = \frac{1,122,530.98 \times 0.0016667}{(1.0016667^{360} - 1)}
\]

\[
P \approx \frac{1,122,530.98 \times 0.0016667}{(2.0311 - 1)}
\]

\[
P \approx \frac{1,870.88}{1.0311}
\]

\[
P \approx 1,814.46
\]

Thus, Jennifer must contribute approximately **$1,814.46** per month for 30 years to meet her retirement goal.

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How would the monthly contribution change if the interest rate were higher, say 3% per year?
2. What if Jennifer wanted to retire in 25 years instead of 30 years?
3. How does the frequency of compounding affect the monthly contribution required?
4. What if Jennifer's monthly withdrawal needs to be $16,000 instead of $14,000?
5. How much would Jennifer need to contribute monthly if she could start with an initial lump sum?

**Tip:** Compounding frequency can significantly impact the growth of investments, so it's important to understand how it works when planning for long-term goals like retirement.

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 $14,000/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 2%/year compounded monthly to meet her retirement goal? (Round your answer to the nearest cent.)

Learn how to calculate the monthly contributions needed for retirement savings using compound interest and annuity formulas. This problem involves planning a 30-year contribution period for a retirement goal of $14,000 monthly withdrawals over 20 years, with a 2% interest rate compounded monthly.

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