You plan to retire in 4 years with $791,020. You plan to withdraw $112,900 per year for 22 years. The expected return is X percent per year and the first regular withdrawal is expected in 5 years. What is X?

To determine the expected return $X\%$, we need to solve for the interest rate that allows for a balance of $791,020 at retirement, which is withdrawn at a rate of $112,900 per year for 22 years. The withdrawals start after 5 years, so the funds will grow for the first 4 years before the first withdrawal.

This is a financial problem involving the future value of an annuity. Here's the plan:

1. During the first 4 years (before retirement), the balance grows with compound interest.
2. After 4 years, for the next 22 years, you start withdrawing $112,900 each year.

We can use the **present value of annuity** formula and solve for $X\%$ using the following steps:

### Step 1: Future value at retirement
The current balance of $791,020 will grow at an annual rate of $X\%$ for 4 years. The formula for the future value $FV$ of a lump sum is:

\[
FV = 791,020(1 + X)^4
\]

This gives the amount you will have at the start of retirement.

### Step 2: Present value of withdrawals
Once retirement begins, you will be withdrawing $112,900 per year for 22 years. The present value of these withdrawals at the time you begin withdrawing can be calculated using the **present value of an annuity** formula:

\[
PV = 112,900 \times \left( 1 - (1 + X)^{-22} \right) / X
\]

### Step 3: Equation setup
The future value from Step 1 must be equal to the present value of withdrawals from Step 2, as the total amount available at retirement must be able to fund the withdrawals:

\[
791,020(1 + X)^4 = 112,900 \times \left( 1 - (1 + X)^{-22} \right) / X
\]

Now, we'll solve for $X$. Let me calculate that for you.The expected annual return $ X $ is approximately **859%**, which is extremely unrealistic for typical investment returns. This suggests that either the withdrawal amount or the given assumptions might need reconsideration for a realistic scenario.

Would you like to adjust the problem, review the details further, or ask any other questions? Here are some related questions:

1. What if the withdrawal period were shorter, say 15 years?
2. How does inflation affect the purchasing power of the withdrawals over time?
3. How would the result change with a lower withdrawal amount per year?
4. What if you wanted to leave a remaining balance after 22 years instead of depleting the account?
5. Can the formula be adjusted for irregular withdrawals?

**Tip:** Always double-check for realistic rates when solving for financial returns!

Learn how to calculate the expected return for a retirement plan with $791,020, where $112,900 is withdrawn annually for 22 years. This problem involves the future value of a lump sum and the present value of an annuity. Suitable for financial mathematics students.

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