Final answer:
To ascertain the retirement savings needed at 63 for 25 years of $8,400 monthly withdrawals at 7% annual return, the present value of an ordinary annuity formula is used, considering yearly to monthly interest rate conversion and a total of 300 payments.
Step-by-step explanation:
To calculate the amount of retirement money needed by the age of 63 to support a monthly withdrawal of $8,400 for 25 years at an annual return rate of 7%, we need to use the formula for the present value of an annuity. An annuity is a series of equal payments made at consistent intervals over time. As you expect to make the first withdrawal one month after your 63rd birthday, this is an ordinary annuity scenario.
The formula for the present value of an ordinary annuity is P = PMT [((1 - (1 + r)^{-n}) / r)], where PMT is the monthly payment, r is the monthly interest rate (annual interest rate divided by 12), and n is the total number of payments. Since you expect to live for 25 years after retirement, and you'll be withdrawing monthly, n equals 25 times 12, which is 300.
In your case, PMT is $8,400, the annual interest rate (r) is 7%, so the monthly interest rate is 7%/12, and n is 300. Plugging these values into the formula:
P = $8,400 [((1 - (1 + 0.07/12)^{-300}) / (0.07/12))]
Calculating this will give you the total amount you need to have saved by your 63rd birthday to fund your retirement withdrawals till the age of 88.