You contract with a financial company to manage a retirement account for you. Th

Question

You contract with a financial company to manage a retirement account for you. The contract specifyes the following:  At the end of each month, starting in month 1, you will deposit a fixed amount A into the account.  The account will earn continuous interest with a nominal annual rate r.  The financial company gets paid for its services by withdrawing a fraction f of the principal each month. This fraction is calculated on the amount of principal before you make your monthly deposit. Let Pn be the amount of principal in the account at the end of the n'th month after your deposit of A and after the nancial company has withdrawn its fraction f of the principal. Develop a model for Pn and solve this model for Pn as a function of n, A, r and f. All sums in the solution must be evaluated for the solution to be complete. If you solve the model by guessing a solution, substitute this guess into the model to confirm that the guess is correct.

Explanation / Answer

Net payment each month = (1-f)*A

Thus, amount accumulated :
Pn = (1-f)*A*{(1+r)n/12 + (1+r)(n-1)/12 + ... + (1+r)1/12} , where r = rate of return
Pn = (1-f)*A*[(1+r)1/12*{(1+r)1/12*n - 1}/{(1+r)1/12 - 1}]
Pn = (1-f)*A*[(1+r)1/12*{(1+r)n/12 - 1}/{(1+r)1/12 - 1}]

Thus, Pn = f(n,A,r,f) = (1-f)*A*[(1+r)1/12*{(1+r)n/12 - 1}/{(1+r)1/12 - 1}]

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