You plan to save $23,078 per year for 7 years, with your first savings contribution in 1 year. You then plan to withdraw $X per year for 6 years, with your first withdrawal expected in 7 years. What is X if the expected return per year is 14.81 percent per year?

This is a typical financial problem involving savings, investments, and withdrawals, where we need to calculate the withdrawal amount $ X $ given a constant annual return. Let's break it down into two phases:

1. **Accumulation Phase** (Years 1 to 7):
   You are saving $23,078 per year for 7 years. The first deposit is made in 1 year, and the funds grow at an annual return of 14.81%.

2. **Withdrawal Phase** (Years 7 to 12):
   After 7 years of savings, you withdraw an amount $ X $ per year for 6 years, with the first withdrawal at the end of year 7. The remaining funds continue to grow at 14.81% during the withdrawal period.

### Step 1: Future Value of Savings (Accumulation Phase)

We calculate the future value of the savings after 7 years. This is a future value of an ordinary annuity (since the first deposit occurs in one year). The formula for the future value $ FV $ of an ordinary annuity is:

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

Where:
- $ P = 23,078 $ (annual savings)
- $ r = 0.1481 $ (annual return rate)
- $ n = 7 $ (number of years)

\[
FV = 23,078 \times \frac{(1 + 0.1481)^7 - 1}{0.1481}
\]

### Step 2: Withdrawal Phase

Once the future value of the savings is calculated, we assume you will withdraw $ X $ for 6 years. The withdrawals form an ordinary annuity, and the present value $ PV $ of this annuity at year 7 is the future value from the accumulation phase.

The formula for the present value $ PV $ of an ordinary annuity is:

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

Where:
- $ r = 0.1481 $ (annual return rate)
- $ n = 6 $ (number of withdrawals)

Since the future value from the accumulation phase will be the present value at the start of the withdrawal phase, we can solve for $ X $:

\[
FV = X \times \frac{1 - (1 + 0.1481)^{-6}}{0.1481}
\]

### Step 3: Solve for $ X $

By solving these equations, we can find $ X $. Let me calculate this for you.The future value of your savings after 7 years will be approximately **$253,906.03**.

Based on this, the amount $ X $ you can withdraw each year for 6 years, starting at the end of year 7, is approximately **$66,748.37** per year.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How would the withdrawal amount change if the interest rate was lower?
2. What would be the impact of saving for a longer period, such as 10 years instead of 7?
3. How would the calculation change if withdrawals were to last for 8 years instead of 6?
4. What happens to the withdrawal amount if you make an initial lump sum investment in addition to the annual savings?
5. How does compound interest affect the future value in savings plans like this?

**Tip:** Regularly review your investment strategy, especially if market conditions or personal financial goals change.

You plan to save $23,078 per year for 7 years, with your first savings contribution in 1 year. You then plan to withdraw $X per year for 6 years, with your first withdrawal expected in 7 years. What is X if the expected return per year is 14.81 percent per year?

Learn how to calculate the annual withdrawal amount (X) after saving $23,078 per year for 7 years at an annual return of 14.81%. This problem covers key financial concepts such as the future value of an ordinary annuity and the present value of an annuity. Suitable for advanced high school or college-level students.

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