How Much Should Rodolfo Paez Contribute Monthly for His Retirement?

[Gablar16]

I have yet another problem.

Rodolfo paez plans to retire in 20 years he will make 120 equal contributions towards his retirement plan. 10 years after his last contribution he will be making 120 periodical withdrawals of 3500 a month until it reaches 0.the interest in the account is of 10.5. What amount of monthly payments he should be making.


This is the last problem for the test that I'm working on and is definitely the hardest one. I think is quiet easy to approach if I can derive the amount of money in the account after the first 20 years and I imagine that a formula should exist to calculate this amount using the withdrawal amount, frequency and the interest rate, but by god I can't find it. Any help in the right direction would be greatly appreciated.

 

[Tide]

Of course there are formulas to calculate the amounts but you can derive them easily enough. Just break it down into manageable chunks:

The last payment made accrues interest for exactly 1 month so its value at the end of the month is $(1+i)p$ where i is the periodic interest rate and p is the payment. The next to last payment accrues interest for 2 months so its value is $(1+i)^2 p$ and so forth. The sequence of values is simply a geometric series with which you should have no difficulty finding the sum.

You can develop a similar series for the withdrawal part of your annuity.