ou plan to save $X per year for 6 years, with your first savings contribution in 1 year. You then plan to withdraw $22,101 per year for 9 years, with your first withdrawal expected in 6 years. What is X if the expected return is 10.02 percent per year?

To solve this problem, we need to calculate the value of $ X $, the annual savings contribution, given that you are saving for 6 years and then withdrawing for 9 years. The interest rate is 10.02% per year, and the first withdrawal happens in year 6.

We can break down the problem into two stages:

1. **Stage 1: Savings for 6 years**
   You plan to save an amount $ X $ at the end of each year for 6 years. Each of these savings earns interest at 10.02% annually. After the 6th year, the total amount of savings accumulated will be used for withdrawals in the next stage.

2. **Stage 2: Withdrawals for 9 years**
   After the 6 years of saving, you will begin withdrawing $22,101 per year for 9 years. The withdrawals will also happen at the end of each year, and they will be supported by the accumulated value from your savings plus any interest that continues to accrue on the remaining balance during the withdrawal period.

### Step-by-step Approach:

1. **Accumulation of savings:**
   The total future value of the savings after 6 years is given by the future value of an ordinary annuity formula:

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

   Where:
   - $ X $ is the annual savings amount (what we need to find),
   - $ r = 0.1002 $ (the annual interest rate),
   - $ n = 6 $ (the number of savings periods).

2. **Present value of withdrawals:**
   The withdrawals can be viewed as an annuity. The present value (PV) of these 9 withdrawals starting from year 6 can be calculated using the present value of an ordinary annuity formula:

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

   Where:
   - $ C = 22,101 $ (the annual withdrawal),
   - $ r = 0.1002 $ (the annual interest rate),
   - $ n = 9 $ (the number of withdrawals).

3. **Setting the future value equal to the present value:**
   The future value of the savings after 6 years must be equal to the present value of the withdrawals, because the accumulated savings must be sufficient to cover the withdrawals. This gives us the equation to solve for $ X $.

Let's compute this step-by-step:

1. **Future value of savings after 6 years:**

   \[
   FV = X \times \frac{(1 + 0.1002)^6 - 1}{0.1002}
   \]

2. **Present value of withdrawals over 9 years:**

   \[
   PV = 22,101 \times \frac{1 - (1 + 0.1002)^{-9}}{0.1002}
   \]

3. Set the future value equal to the present value:

   \[
   X \times \frac{(1 + 0.1002)^6 - 1}{0.1002} = 22,101 \times \frac{1 - (1 + 0.1002)^{-9}}{0.1002}
   \]

Now, let's solve for $ X $.The annual savings amount, $ X $, that you need to save for 6 years is approximately **$16,475.03**.

Would you like more details on the calculation or further clarification on any part?

Here are 5 follow-up questions to expand on this:

1. How would the required savings change if the interest rate were different?
2. What happens if you want to start saving immediately instead of waiting one year?
3. How would the total amount saved after 6 years compare to the total amount withdrawn?
4. What is the total interest earned over the 6 years of saving?
5. How does the present value of the withdrawals influence your savings plan?

**Tip**: Small changes in the interest rate can significantly impact both the required savings and the total withdrawals over time, due to compounding effects.

You plan to save $X per year for 6 years, with your first savings contribution in 1 year. You then plan to withdraw $22,101 per year for 9 years, with your first withdrawal expected in 6 years. What is X if the expected return is 10.02 percent per year?

Learn how to calculate the annual savings required to withdraw $22,101 per year for 9 years, starting in year 6, with a 10.02% interest rate. This problem covers financial concepts like annuities and the time value of money, providing detailed step-by-step solutions suitable for finance students.

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