Marcus plans to retire in 20 years. He will make 10 years (120 months) of equal monthly payments to his retirement account. Ten years after his last contribution, he will begin the first of 120 months of withdrawals of $3500 per month.
Assume that the retirement account earns interest of 7.2% compounded monthly for the duration of his contributions, the 10 years in between his contributions and the beginning of his withdrawals, and the 10 years of withdrawals. How large must Marcus's monthly contributions be in order to accomplish his goal?

We know that Marcus will make 120 monthly payments into his retirement account and then wait 10 years (120 months) before starting to withdraw from the account. With an interest rate of 7.2% per year compounded monthly, the monthly interest rate (r) is 7.2% / 12 = 0.006.

First, we'll calculate the future value of his contributions after 10 years:

FV_contributions = P * [(1 + 0.006)^120 - 1] / 0.006

Then, this future value becomes the present value for the withdrawals phase. We need to find the present value of the withdrawals, which would equate to the future value of the contributions accounted for interest (FV_contributions):

PV_withdrawals = FV_contributions / (1 + 0.006)^120

Using the formula for the present value of an ordinary annuity, we can then solve for the monthly payment (P):

PV_withdrawals = P * [(1 - (1 + 0.006)^-120) / 0.006

Solving for P, we can find the monthly contribution Marcus needs to make.