This is a classic retirement problem involving calculating the rate of return (X) that will allow you to meet your withdrawal goals over time. To solve this, we need to apply the concept of the Present Value of an Annuity combined with the compound interest formula. Here's the breakdown:

Given:

• Initial amount (current value of retirement savings) = $377,491
• Annual withdrawal amount = $41,800
• Number of withdrawals = 18 years
• First withdrawal in 6 years (so, no withdrawals for the first 5 years)
• Unknown interest rate = $X$ (per year)

Steps:

1. Grow the initial amount for 5 years: During the first 5 years, the account will grow without withdrawals. The formula for the future value after 5 years is:

$FV = PV \times (1 + r)^5$ Where:

• $PV = 377,491$
• $r$ is the annual return rate (X%)

2. Present value of withdrawals after the first 5 years: After the 5 years, the account will start the annual withdrawals. The withdrawals will form an ordinary annuity (a series of equal annual withdrawals), and the present value of this annuity must equal the future value at the end of year 5.

The formula for the present value of an annuity is: $PV_{\text{annuity}} = W \times \left[1 - \frac{1}{(1 + r)^{18}} \right] \div r$ Where:

• $W = 41,800$ (withdrawal amount)
• $r$ is the annual return rate
• $18$ is the number of withdrawals

3. Equating the future value of the investment to the present value of the withdrawals: The future value calculated in step 1 will be equal to the present value of the withdrawals.

Now let's solve for $r$ using these relationships.

I'll calculate it for you.The expected annual return rate $X$ that allows you to meet your withdrawal goals is approximately 5.11%.

Would you like further clarification or details about any part of the calculation?

Here are 5 related questions you may find useful:

1. How does changing the withdrawal amount affect the required return rate?
2. What if the number of withdrawals (retirement years) increases or decreases?
3. How does the timing of withdrawals (starting sooner or later) impact the interest rate needed?
4. Can inflation adjustments be added to this model, and how would that affect the result?
5. What happens if the initial savings were higher or lower?

Tip: In financial planning, accounting for inflation can help adjust your withdrawal strategy to maintain your purchasing power over time.