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?

This problem involves finding the annual interest rate (denoted as $ X $) such that you can withdraw \$112,900 per year for 22 years, starting 5 years from now, while having an initial retirement balance of \$791,020 at retirement.

### Steps to solve:

1. **Calculate the present value of the withdrawals**: The withdrawals form an ordinary annuity starting in 5 years. We'll need to discount these withdrawals back to the present at the retirement time (4 years from now) to equate it to the available retirement balance of \$791,020.
   
2. **Set up the present value of an annuity formula**:
   The present value of an annuity is given by:

   \[
   PV = C \times \left( 1 - (1 + X)^{-n} \right) / X
   \]
   where:
   - $ PV $ is the present value,
   - $ C $ is the annual withdrawal (\$112,900),
   - $ X $ is the annual interest rate (which we need to find),
   - $ n $ is the number of years of withdrawals (22 years).

3. **Account for the 5-year delay**: Since the withdrawals start in 5 years, the present value calculated needs to be further discounted back to year 4 (the retirement year). This discount factor will be $ (1 + X)^5 $.

4. **Set up the equation**: The present value of the withdrawals discounted back to year 4 must equal the initial balance at retirement:

   \[
   791,020 = \left( C \times \frac{1 - (1 + X)^{-22}}{X} \right) \times (1 + X)^{-5}
   \]

Let's solve this equation for $ X $.The expected annual return ($ X $) is approximately **7.93%**.

This means that in order to withdraw \$112,900 per year for 22 years, starting 5 years from now, you would need an investment return of around 7.93% per year on your retirement savings of \$791,020.

Do you need further details or have any questions?

Here are 5 related questions you might find interesting:
1. How does delaying the start of withdrawals affect the required interest rate?
2. What happens to the required rate of return if you plan to withdraw for 30 years instead of 22?
3. How does increasing the annual withdrawal affect the required interest rate?
4. Can the same principle be applied if the withdrawals increase over time (e.g., adjusting for inflation)?
5. What would be the required initial savings if the interest rate is fixed at 5%?

**Tip:** If the interest rate is lower than expected, you might need to save more before retirement or reduce your withdrawal amount to ensure the funds last.

Solve the problem of calculating the required annual return to support $112,900 yearly withdrawals for 22 years, starting 5 years after retirement. With an initial balance of $791,020, find out the interest rate needed using annuity and present value concepts.

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