Questions: Carolyn has just retired, and has 500000 dollars in her retirement account. The account will earn interest at an annual rate of 10 percent, compounded monthly. At the end of each month, Carolyn will withdraw a fixed amount to cover her living expenses. Carolyn wants her savings to last exactly 25 years. How much money can she withdraw each month? monthly withdrawal: What is the maximum amount that Carolyn can withdraw each month if she wants her savings to last indefinitely? monthly withdrawal:

Solution Steps

To solve this problem, we need to use the formula for the future value of an annuity to determine the fixed monthly withdrawal amount that allows the retirement savings to last exactly 25 years. The formula accounts for the initial amount, interest rate, and the number of periods. For the second part, we need to find the maximum sustainable withdrawal amount, which is determined by the interest earned each month.

The annual interest rate is given as 10%, which is equivalent to 0.10 in decimal form. Since the interest is compounded monthly, we divide the annual rate by 12 to find the monthly interest rate: \[ r = \frac{0.10}{12} = 0.008333 \]

Carolyn wants her savings to last for 25 years. Since the interest is compounded monthly, the total number of compounding periods is: \[ n = 25 \times 12 = 300 \]

To find the fixed monthly withdrawal amount that allows the savings to last exactly 25 years, we use the formula for the future value of an annuity: \[ W = \frac{P \times r}{1 - (1 + r)^{-n}} \] where \( P = 500,000 \) is the initial amount, \( r = 0.008333 \) is the monthly interest rate, and \( n = 300 \) is the number of periods. Substituting these values, we get: \[ W = \frac{500,000 \times 0.008333}{1 - (1 + 0.008333)^{-300}} \approx 4543.50 \]

The maximum sustainable monthly withdrawal is determined by the interest earned each month, which is: \[ W_{\text{max}} = P \times r = 500,000 \times 0.008333 \approx 4166.67 \]

Final Answer