A woman, with her employer's matching program, contributes $400 at the end of each month to her retirement account, which earns 9% interest, compounded monthly. When she retires after 49 years, she plans to make monthly withdrawals for 31 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 determine her maximum possible monthly withdrawal, we need to calculate the balance of her retirement account at the time of retirement and then use that balance to make calculations for her monthly withdrawals.
To calculate the balance of her retirement account at retirement, we can use the formula for future value of an annuity:
FV = PMT x [(1 + (r/n))^(n*t) - 1] / (r/n)
where PMT = $400, r = 9% (interest rate), n = 12 (number of compounding periods per year), and t = 49 (number of years).
FV = $400 x [(1 + (0.09/12))^(12*49) - 1] / (0.09/12)
FV = $2,699,515.33
So her retirement account balance at retirement is estimated to be $2,699,515.33.
Next, we need to calculate the monthly withdrawal amount she can take for 31 years from this retirement account balance. To do this, we can use the formula for fixed periodic payments:
PMT = (r/n) x PV / [1 - (1 + r/n)^(-n*t)]
where PMT = the monthly withdrawal amount, r = 4% (interest rate), n = 12 (number of compounding periods per year), t = 31 (number of years), and PV = $2,699,515.33 (present value or starting balance).
PMT = (0.04/12) x $2,699,515.33 / [1 - (1 + 0.04/12)^(-12*31)]
PMT = $9,606.24
So her maximum possible monthly withdrawal amount, given these assumptions, is approximately $9,606.24.
To find the maximum possible monthly withdrawal amount without running out of money, we need to calculate the future value of the retirement account after 49 years with the 9% compounded interest and then determine the monthly withdrawal that can be sustained for 31 years with the 4% compounded interest.
Step 1: Find the future value of the retirement account after 49 years
The formula to calculate the future value of an investment with compound interest is:
FV = P * (1 + r/n)^(n*t)
Where:
FV = future value of the investment
P = principal amount (monthly contribution)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, we have:
P = $400
r = 9% (0.09)
n = 12 (monthly compounding)
t = 49 years
Using the formula, we can calculate the future value:
FV = 400 * (1 + 0.09/12)^(12*49)
FV ≈ $3,935,192.43
So, the retirement account will have a future value of approximately $3,935,192.43 after 49 years.
Step 2: Determine the maximum monthly withdrawal for 31 years
Now, we need to calculate the maximum monthly withdrawal that can be sustained for 31 years with the 4% compounded interest.
Using the same formula as before, but with different values:
P = $3,935,192.43 (future value)
r = 4% (0.04)
n = 12 (monthly compounding)
t = 31 years
Let's calculate the monthly withdrawal:
FV = 3935192.43
r = 4% (0.04)
n = 12
t = 31
The formula becomes:
3935192.43 = X * (1 + 0.04/12)^(12*31)
Solving for X:
X = 3935192.43 / (1 + 0.04/12)^(12*31)
X ≈ $11,157.47
Therefore, the maximum possible monthly withdrawal without running out of money is approximately $11,157.47.
To find the maximum possible monthly withdrawal without running out of money, we need to calculate the accumulated balance of the retirement account over 49 years with the contributions and interest earned. Then, we will calculate the monthly withdrawals for 31 years using a 4% interest rate to see how much she can withdraw each month without depleting the account.
Step 1: Calculate the accumulated balance after 49 years of contributions and compounded interest at 9%.
First, let's calculate the total number of contributions made over 49 years:
Months = 49 years * 12 months/year = 588 months
Next, we'll calculate the future value of each monthly contribution of $400 at a 9% annual interest rate compounded monthly over 49 years using the future value of a monthly payment (annuity) formula:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value
P = Monthly Payment ($400)
r = Monthly Interest Rate (9% / 12)
n = Number of Months (588 months)
Plugging in the values:
FV = 400 * ((1 + (0.09 / 12))^588 - 1) / (0.09 / 12)
FV ≈ $2,938,976.25
So, after 49 years of contributions and compounded interest at 9%, the accumulated balance in the retirement account would be approximately $2,938,976.25.
Step 2: Calculate the maximum monthly withdrawal for 31 years using a 4% interest rate.
Since the future value has already been calculated, we can use it to calculate the maximum monthly withdrawal for 31 years using the future value of an annuity formula:
P = (r * FV) / ((1 + r)^n - 1)
Where:
P = Monthly Withdrawal
r = Monthly Interest Rate (4% / 12)
n = Number of Months (31 years * 12 months/year = 372 months)
Plugging in the values:
P = (0.04 / 12) * 2,938,976.25 / ((1 + (0.04 / 12))^372 - 1)
P ≈ $10,081.92
Therefore, with an accumulated balance of approximately $2,938,976.25, the woman can make maximum monthly withdrawals of approximately $10,081.92 for 31 years without running out of money.