Hi /sci/, I have a problem for you. I named it continuous compound interest with fee.

Let's suppose we have a deposit of P dollars at the beginning. Also we have r percent per, let's say, day (aka nominal interest rate). In opposite to casual compound interest problem where we have finite number of compounds, our compounding process is continuous or in other words we may reinvest our interest at every moment. That problem solved Bernoulli centuries ago and by the end of a day we will have A*e dollars on our account.

BUT, what if we would still have an opportunity to continiously reinvest our interest but it would cost us some fixed number of money (let it be a fee) f. Then profit of our reinvestment depends on how frequenly we actually reinvest the interest. If we would reinvest too frequently, the fee will even make our money to get lesser. In opposite, if we reinvest too rarely, then we loose key benefits of compound interest. So some optimum reinvestment rate must exist (and of course it also depends on current amount of money on our account).

So the question is: how to calculate this optimum rate?

Exact solution with proofs is very appreciated.
Approximate numerical solution with particular numbers is welcome.
Links to the articles I probably missed in Google are wellcome as well (I believe someone before me had the same question).