You have $375,000 saved for retirement.
Your retirement account earns 4% interest.
How much will you be able to withdraw each month
if you want to be able to take withdrawals for 20 years?
We will answer this question
using the payout annuity formula shown below
where P sub-zero is the account balance at the beginning,
PMT is the regular withdrawal amount,
r is the annual interest rate as a decimal,
n is the number of compounds per year,
which will be the number of withdrawals
per year unless given otherwise,
and t is the number of years of withdrawals.
So because you have $375,000 saved for retirement,
P sub-zero is 375,000,
giving us the equation 375,000 equals,
and on the right side, we're trying to solve for PMT.
So in the numerator, we have PMT,
the regular withdrawal amount, times,
and then in parentheses, we have one minus,
and then in parentheses,
we have one plus r divided by n
raised to the power of negative nt.
So we have open parenthesis one plus r
is 4% as a decimal, which is 0.04.
And then for N, the number of compounds per year
or the number of withdrawals per year is going to be 12
because you want to withdraw an amount each month,
and there are 12 months in a year.
And the sum is raised to the power of negative n times t
where again, n is 12, so we have negative 12 times t
is the number of years of withdrawals,
and you want withdrawals for 20 years, therefore, t is 20.
Then we have close parenthesis.
In the denominator, we have r divided by n,
which is 0.04 divided by 12.
I now need to solve this equation for PMT,
which is the regular withdrawal amount.
To do this, we will begin by evaluating
this fraction here on the graphing calculator.
We enter this quotient just like it looks.
We have open parenthesis one minus
open parenthesis one plus 0.04
divided by 12, close parenthesis.
This is raised to the power of negative 12 times 20,
so we press the exponent key, negative 12 times 20,
and now we press the right arrow to get out
from the position of the exponent,
close parenthesis, and then divided by,
in parentheses, we have 0.04 divided by 12.
And enter.
So now, on the right side of the equation,
we have PMT times 165.0218582.
Let's go ahead and write this down.
The equation is now 375,000
equals PMT times, again, 165.0218582.
And now, to solve for PMT,
we divide both sides of the equation by 165.0218582.
On the right side of the equation,
this quotient is equal to one.
PMT times one is PMT.
PMT, the regular withdrawal amount,
is equal to this quotient,
which we will now evaluate and round to the nearest cent.
We have 375,000 divided by 165.0218582.
Enter.
To the nearest cent, we have $2272.43.
So we now know you can withdraw $2272.43
each month for 20 years.
I hope you found this helpful.