How much money can you withdraw each month during retirement if you deposit $250 per month for 40 years and you want to have $1,500,000 left over to leave to your grandkids after 25 years of retirement?  Assume you will earn 9% while you are working and 6% after you retire.  This is an extra credit question worth 5 points.  Answer choices below have been rounded to the nearest dollar.

To solve this problem, we need to break it down into two phases: 

1. **Accumulation Phase** (working years: 40 years of contributions)
2. **Withdrawal Phase** (retirement years: 25 years of withdrawals)

### Step 1: Accumulation Phase (40 years of contributions)

You are contributing $250 per month for 40 years at an interest rate of 9% annually. To find how much money will accumulate at the end of 40 years, we will use the **future value of an annuity formula**:

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

Where:
- $ P = 250 $ (monthly contribution),
- $ r = \frac{9\%}{12} = 0.0075 $ (monthly interest rate),
- $ n = 40 \times 12 = 480 $ (number of months).

Let's calculate the future value of your contributions after 40 years.At the end of 40 years, you will have approximately **$1,170,330** from your monthly contributions of $250.

### Step 2: Withdrawal Phase (25 years of retirement)

Now, during retirement, you want to withdraw a certain amount every month while leaving $1,500,000 for your grandchildren after 25 years. You will earn 6% annually during this time.

We can use the **present value of annuity formula** with a future value component (since you want to leave $1,500,000):

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

Where:
- $ PV $ is the initial amount you start retirement with (the value after accumulation),
- $ W $ is the monthly withdrawal,
- $ r = \frac{6\%}{12} = 0.005 $ (monthly interest rate),
- $ n = 25 \times 12 = 300 $ (number of months),
- $ FV = 1,500,000 $ (the amount you want to leave).

We need to find $ W $ (monthly withdrawal). Rewriting the formula to solve for $ W $:

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

Let's calculate the monthly withdrawal amount $ W $.The monthly withdrawal during retirement is approximately **$5,376**.

### Answer:
You can withdraw approximately **$5,376** each month during retirement while leaving $1,500,000 for your grandchildren after 25 years.

Would you like further details on this solution or have any questions? Here are some related questions to deepen your understanding:

1. How would the result change if you contributed more than $250 per month?
2. What happens if the interest rate during retirement is lower than 6%?
3. How does changing the retirement duration from 25 to 30 years affect your monthly withdrawal?
4. What if you didn’t want to leave any money to your grandchildren? How much could you withdraw then?
5. How would the future value change if you invested a lump sum instead of monthly contributions?

**Tip:** Using a financial calculator or spreadsheet can greatly simplify solving such time value of money problems.

How much money can you withdraw each month during retirement if you deposit $250 per month for 40 years and you want to have $1,500,000 left over to leave to your grandkids after 25 years of retirement? Assume you will earn 9% while you are working and 6% after you retire. This is an extra credit question worth 5 points.

Discover how to calculate your monthly withdrawal during retirement if you contribute $250 per month for 40 years, earn 9% during working years, and want to leave $1,500,000 after 25 years of retirement. This problem explores the time value of money using the future value of annuity and present value of annuity formulas. Suitable for college-level or advanced high school students.

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