the value of the investment at the end of the 9 years is 12,000 * (1 + .05/12) ^ (9 * 12) = 18802.15979
this becomes the present value of an annuity for 6 years at 5% compounded quarterly with payments made at the end of each quarter.
the formula to be used is:
ANNUITY FOR A PRESENT AMOUNT WITH END OF TIME PERIOD PAYMENTS
a = (p*r)/(1-(1/(1+r)^n))
a is the annuity.
p is the present amount.
r is the interest rate per time period.
n is the number of time periods.
this formula becomes:
a = (18802.159793065 * .05/4) / (1 - (1 / (1 + .05/4) ^ (6 * 4)))
solve for a to get:
a = 911.6537044.
this is confirmed through the use of a financial calculator that can be found at https://arachnoid.com/finance/index.html
the first calculation is the future value of 12,000 at 5% per year compounded monthly for 9 years.
inputs to the calculator are:
present value = 12,000
future value = 0
payments per time period = 0
percent rate = 5/12 = .41666666667
number of time periods = 9 * 12 = 108
calculator says that the future value is 18,802.16 which is the same value i calculated rounded to the nearest penny.
the second calculation is the quarterly payments from a present value at 5% per year compounded quarterly.
inputs to the calculator are:
present value = 18,802.15979
future value = 0
payments per time period = 0
percent rate = 5/4 = 1.25
number of time periods = 6 * 4 = 24
payments are made at the end of each time period
time periods are quarters of a year.
calculator says that the payment at the end of each quarter is 911.65.
this is the same value i got through use of the formula, rounded to the nearest penny.
here are the displays from the use of the calculator.
the calculator can be found at https://arachnoid.com/finance/index.html
if the money needs to be withdrawn at the beginning of each quarter, than the quarterly payment becomes 900.40.
since you normally pay the rent at the beginning of each time period, than 900.40 is probably the solution you want.
the reason the money is less is because the money doesn't sit in the account for the duration of the quarter being withdrawn, therefore losing the interest for that time period.
911.6537044 / 1.0125 = 900.3987204 which rounds to the nearest penny as 900.40.
i used the calculator to confirm by simply switching payment at the end of each time period to payments at the beginning of each time period.