How much must be set aside each month at 12% annual growth compounded monthly for 30 years in order to be able to retire on $4,500 per month for 15 years at 3% annual growth compounded monthly?
x(1.01^360 - 1)/.01 = 4500(1- 1.0025^-180)/.0025
x(3494.964133) = 651624.6215
x = 186.4467..
x = $186.45
check:
amount of 186.45 deposited monthly for 360 months at .01
= 186.45(1.01^360 - 1).01
= 651636.06
if I use 186.4476.. I get 651624.6214 , the fact that your deposit gets rounded off to the nearest penny explains the $11.40 discrepancy.
To calculate how much needs to be set aside each month, we need to use the formula for the future value of an ordinary annuity:
FV = PMT * (1 + r)^n - 1) / r
where:
FV = Future Value (the desired retirement fund)
PMT = Monthly payment or set-aside amount
r = Interest rate per period (monthly interest rate)
n = Number of periods
Now let's break down the information provided:
Desired retirement fund:
Monthly payment = $4,500
Number of payment periods = 15 years * 12 months = 180 months
Annual growth rate = 3% = 0.03
Monthly growth rate = 0.03 / 12 = 0.0025
To calculate the future value of the retirement fund, we can use the given formula:
FV = PMT * ((1 + r)^n - 1) / r
Plugging in the values:
$4,500 = PMT * ((1 + 0.0025)^180 - 1) / 0.0025
Now we need to solve for PMT. Rearranging the formula:
PMT = ($4,500 * 0.0025) / ((1 + 0.0025)^180 - 1)
Using a calculator or spreadsheet, the monthly payment (PMT) comes out to be approximately $894.76.
Therefore, to retire on $4,500 per month for 15 years with a 3% annual growth rate, you would need to set aside approximately $894.76 each month for 30 years at a 12% annual growth rate, compounded monthly.