Your aunt has $500,000 invested at 5.5%, and she now wants to retire.She wants to withdraw $45,000 at the beginning of each year, beginning immediately.When she makes her last withdrawal (at the beginning of a year), she also wants to have enough left in the account so that you can make a final withdrawal of $50,000 at the end of that year (her last withdrawal is at the beginning of the year, your withdrawal is at the end of that same year).What is the maximum number of $45,000 withdrawals that she can make and still have enough in the account so that you can make a $50,000 withdrawal at the end of the year of her last withdrawal? (Hint: If your solution for N is not an integer, round down to the nearest whole number.)

Question

Quizplus · Accepted Answer

We can use the formula for the present value of an annuity to solve this problem:
Present Value = Payment x ((1 - (1 + interest rate)^-n) / interest rate)
Where payment is the withdrawal amount, interest rate is the annual interest rate (5.5%), and n is the number of withdrawals.

We can first calculate the present value of the $50,000 withdrawal at the end of the final year:
PV = $50,000 / (1 + 0.055) = $47,393.36

Next, we can use trial and error to find the number of withdrawals that will result in a remaining balance of at least $47,393.36 after the last $45,000 withdrawal. Starting with 15 withdrawals, we get:
PV = $45,000 x ((1 - (1 + 0.055)^-15) / 0.055) = $525,772.43
Remaining balance = $525,772.43 x 1.055 - $45,000 = $484,419.51
This is more than the required $47,393.36, so we can try with 14 withdrawals:
PV = $45,000 x ((1 - (1 + 0.055)^-14) / 0.055) = $480,529.69
Remaining balance = $480,529.69 x 1.055 - $45,000 = $436,101.66
This is less than the required $47,393.36, so we need to try with more than 14 withdrawals. Continuing in this way, we find that 15 withdrawals result in a remaining balance of $484,419.51, which is sufficient for the final $50,000 withdrawal. Therefore, the maximum number of $45,000 withdrawals that she can make is 15.

Your Aunt Has $500,000 Invested at 5