Assume you are 25 and earn ​$31 comma 800 per​ year, never expect to receive a​ raise, and plan to retire at age 55. If you invest 5 percent of your salary in a​ 401(k) plan returning 12 percent​ annually, and the company provides a​ $0.50 per​ $1.00 match on your contributions up to 3 percent of​ salary, what is the estimated future value of your​ 401(k) account? Once you​ retire, how much can you withdraw monthly if you want to deplete your account over 25 ​years?  

Let's break this problem down into two parts:

1. **Calculate the Future Value of the 401(k) Account at Retirement:**
    - You earn $31,800 per year.
    - You contribute 5% of your salary to the 401(k).
    - The company matches $0.50 per $1.00 on your contributions up to 3% of your salary.
    - The plan returns 12% annually.
    - You will invest for $ 55 - 25 = 30 $ years.

2. **Calculate the Monthly Withdrawal After Retirement:**
    - You will withdraw over 25 years.

### Part 1: Future Value of the 401(k) Account

1. **Your Contribution:**
    \[
    \text{Annual contribution} = 0.05 \times 31,800 = 1,590 \text{ USD}
    \]

2. **Company Match:**
    The company matches $0.50 per $1.00 on contributions up to 3% of your salary:
    \[
    \text{Company match} = 0.5 \times (0.03 \times 31,800) = 0.5 \times 954 = 477 \text{ USD}
    \]
    - Total annual contribution (including company match):
    \[
    \text{Total contribution} = 1,590 + 477 = 2,067 \text{ USD}
    \]

3. **Future Value Calculation:**
    The future value $ FV $ of your 401(k) can be calculated using the formula for the future value of an annuity:
    \[
    FV = P \times \frac{(1 + r)^n - 1}{r}
    \]
    Where:
    - $ P = 2,067 $ USD (total annual contribution)
    - $ r = 0.12 $ (annual return rate)
    - $ n = 30 $ years

    Let's calculate this.

### Part 2: Monthly Withdrawal

After finding the future value, we will use the following formula to calculate the monthly withdrawal amount $ PMT $:

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

Where:
- $ r = 0.12 / 12 = 0.01 $ (monthly interest rate)
- $ t = 25 \times 12 = 300 $ months

Let’s compute these values.### Results:

1. **Estimated Future Value of the 401(k) Account at Retirement:**
   \[
   \text{Future Value} \approx 498,835 \text{ USD}
   \]

2. **Monthly Withdrawal Amount After Retirement:**
   \[
   \text{Monthly Withdrawal} \approx 5,254 \text{ USD}
   \]

You can withdraw approximately $5,254 per month for 25 years, depleting your account by the end of this period.

Would you like any more details or have any questions?

### Relative Questions:
1. What if the return rate was different, say 10% annually?
2. How would the future value change if you decided to retire at 60 instead of 55?
3. What happens if the company match was dollar-for-dollar instead of $0.50 per $1.00?
4. How does the total future value change if you increased your contribution to 7%?
5. What impact does inflation have on the purchasing power of your future withdrawals?

### Tip:
**Regularly reviewing and increasing your retirement contributions, even by a small percentage, can significantly boost your future retirement savings due to the power of compound interest.**

Assume you are 25 and earn $31,800 per year, never expect to receive a raise, and plan to retire at age 55. If you invest 5 percent of your salary in a 401(k) plan returning 12 percent annually, and the company provides a $0.50 per $1.00 match on your contributions up to 3 percent of salary, what is the estimated future value of your 401(k) account? Once you retire, how much can you withdraw monthly if you want to deplete your account over 25 years?

Calculate the estimated future value of your 401(k) account at retirement and determine how much you can withdraw monthly over 25 years. This example uses a salary of $31,800, a 12% annual return, and a company match of $0.50 per $1.00 up to 3% of salary. Suitable for advanced high school or college-level students.

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