A woman, with her employer's matching program, contributes $400 at the end of each month to her retirement account, which earns 7% interest, compounded monthly. When she retires after 47 years, she plans to make monthly withdrawals for 29 years. If her account earns 4% interest, compounded monthly, then when she retires, what is her maximum possible monthly withdrawal (without running out of money)?
To find the maximum possible monthly withdrawal, we need to calculate the amount of money in the retirement account at the time of retirement (after 47 years of contributing) and then use that amount to calculate the maximum monthly withdrawal for 29 years at 4% interest.
To calculate the retirement account balance after 47 years of contributing $400/month at 7% interest compounded monthly, we can use the formula:
A = P * ((1 + r/n)^(n*t) - 1) / (r/n)
where:
A = the ending account balance
P = the monthly contribution amount ($400)
r = the annual interest rate (7%)
n = the number of times interest is compounded per year (12 for monthly compounding)
t = the number of years contributing (47)
Plugging in the values, we get:
A = 400 * ((1 + 0.07/12)^(12*47) - 1) / (0.07/12)
A = $1,486,905.54
So, at the time of retirement, she will have $1,486,905.54 in her retirement account.
To calculate the maximum monthly withdrawal for 29 years at 4% interest, we can use the formula:
P = r * A / (1 - (1 + r)^(-n))
where:
P = the monthly withdrawal amount
r = the annual interest rate (4%)
A = the starting account balance ($1,486,905.54)
n = the number of months withdrawals will be made (29 years * 12 months/year = 348 months)
Plugging in the values, we get:
P = 0.04 * 1,486,905.54 / (1 - (1 + 0.04)^(-348))
P = $6,305.08 (rounded to the nearest cent)
So, the maximum possible monthly withdrawal without running out of money is $6,305.08.