Question 1190392: You expect to retire in 25 years. After you retire, you want to be able to withdraw $3,000 from your account each month for 30 years.
If your account earns 4% interest compounded monthly, how much will you need to deposit each month until retirement to achieve your retirement goals? (Round to the nearest cent.)
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
This is the updated solution. I forgot about the present value aspect initially, but the error has been fixed.
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The goal is to have $3000 per month, over the course of 30*12 = 360 months.
Since we want this target value in the future, we need to figure out how much we effectively put into it in the present. So we'll be calculating the present value based on these targets.
Use the aptly named present value annuity formula
PV = C*(1-(1+i)^(-n))/i
PV = present value
C = amount of cash needed per period
i = interest rate per period
n = number of periods
We have
C = 3000 needed per month
i = 0.04/12, I'll leave it as a fraction
n = 12*30 = 360 months
The present value is
PV = C*(1-(1+i)^(-n))/i
PV = 3000*(1-(1+0.04/12)^(-360))/(0.04/12)
PV = 628,383.721362586
PV = 628,383.72
This means that if we had $628,383.72 today presently, then that is equivalent to having those $3000 payments spread out over the 360 months in the future. This is of course when involving interest (it would be a much simpler story if interest wasn't involved).
In other words, the goal is to aim for depositing equally spaced payments that accrue interest to land on $628,383.72
The next task is to figure out the monthly deposits (P) needed to get a future value of FV = $628,383.72
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We'll need to use the future value of annuity formula
FV = P*( (1+i)^n - 1 )/i
FV = future value
P = deposit per period
i = interest rate per period
n = number of periods
In this case,
FV = 628,383.72
P = what we want to find out
i = 0.04/12
n = 12*25 = 300 months
So,
FV = P*( (1+i)^n - 1 )/i
FV*i = P( (1+i)^n - 1 )
P = FV*i/( (1+i)^n - 1 )
P = (628,383.72)*(0.04/12)/( (1+0.04/12)^300 - 1 )
P = 1,222.22837259359
P = $1,222.23 is the amount you need to deposit each month, for 300 months (aka 25 years)
Disclaimer:
I didn't account for anything like taxes or inflation/deflation.
So please keep in mind that this isn't financial advice.
It's purely a hypothetical math problem to demonstrate the concepts of present value and future value.
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