A woman, with her employer's matching program, contributes $600 at the end of each month to her retirement account, which earns 5% interest, compounded monthly. When she retires after 40 years, she plans to make monthly withdrawals for 25 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)?
We can start by using the future value formula to find the balance in the retirement account after 40 years:
PV = 0 (since she is starting with no money in the account)
PMT = 600 (her monthly contributions)
i = 0.05/12 (monthly interest rate)
n = 40*12 (number of months in 40 years)
FV = PMT * ((1 + i)^n - 1) / i
FV = 600 * ((1 + 0.05/12)^(40*12) - 1) / (0.05/12)
FV = $1,059,695.32
So after 40 years of contributing $600 per month, with 5% monthly compounded interest, her retirement account will have a balance of $1,059,695.32.
Now we need to find the maximum monthly withdrawal she can make for 25 years without running out of money. We can use the present value of an annuity formula to do this:
PV = FV * i / ((1 + i)^n - 1)
PV = 1,059,695.32 * 0.04/12 / ((1 + 0.04/12)^(25*12) - 1)
PV = $4355.87
So the maximum monthly withdrawal she can make for 25 years, with 4% monthly compounded interest, is $4355.87.
amount after 40 years
= 600(1.0041666..^240 - 1)/.0041666.. = 915,612.09
not the answer the bot got
This becomes the PV of an annuity with
i = .04/12 or .00333...
n = 300
Paym = ?
paym(1 - 1.003333..^-300 )/.00333... = 915.612.09
I get payment = $4,832.944
I apologize for the mistake in my initial response. Thank you for correcting me.
I agree with your calculations, and the correct maximum monthly withdrawal she can make for 25 years, with 4% monthly compounded interest, is $4,832.94.
To find the maximum possible monthly withdrawal without running out of money, we need to calculate the future value of the retirement account after 40 years and then calculate the monthly withdrawals for 25 years with a 4% interest rate.
Here's the step-by-step process to find the answer:
Step 1: Calculate the future value of the retirement account after 40 years.
To calculate the future value, you can use the formula for compound interest:
FV = P * (1 + r/n)^(nt)
Where:
FV = Future Value
P = Principal amount (initial contribution)
r = Annual interest rate (5% in this case)
n = Number of times interest is compounded per year (12 for monthly compounding)
t = Number of years
Here, P = $600, r = 0.05, n = 12, and t = 40.
FV = 600 * (1 + 0.05/12)^(12*40)
Using a calculator, you can compute the future value, which comes out to be approximately $1,503,428.76.
Step 2: Calculate the monthly withdrawals for 25 years.
Now that we have the future value at retirement, we can calculate the monthly withdrawals. The formula for calculating the monthly withdrawal amount is:
Withdrawal = (Future Value * (r/n)) / ((1 + r/n)^(nt) - 1)
Where:
Withdrawal = Monthly Withdrawal Amount
Future Value = $1,503,428.76 (calculated in step 1)
r = Annual interest rate (4% in this case)
n = Number of times interest is compounded per year (12 for monthly compounding)
t = Number of years for withdrawal (25 years)
Using the above formula, the monthly withdrawal amount can be calculated:
Withdrawal = (1,503,428.76 * (0.04/12)) / ((1 + 0.04/12)^(12*25) - 1)
Using a calculator, you can compute the monthly withdrawal amount, which comes out to be approximately $5,386.07.
Therefore, the maximum possible monthly withdrawal without running out of money would be approximately $5,386.07. This amount should be sustainable for 25 years, given the specified interest rates and time periods.
Note: All calculations here are approximate due to rounding.