Maximizing interest question

Question

My initial investment is $\$100,$ and I earn $1\%$ interest per day. I can opt for any number of compoundings per day (if twice per day, then the interest rate per compounding period is $0.5\%,$ and so on), but I have to pay $\$0.01$ each time my interest is compounded. After $365$ days I will close the account.

What would this equation look like, and how should I include this to maximize my total deposit? How to generalize and figure out a good or optimal maximization?

Answer 1

Let $m$ be the number of compound periods per day.

Then $T_{n+1}=T_n\left(1+\frac{0.01}m\right)-0.01\;\;(n\geq0).$

This is a first-order linear recurrence relation; derive the closed-form expression for $T_n$ using the information here.

The task is to finally determine the positive integer $m$ at which $T_{365m}$ is greatest (among the positive integer values of $m$).

(Alternatively, the answer is easy to determine by simply varying $m$ in the above equation and observing the effect.)

Additionally, the closed-form formula above for $T_n$ also reveals that

1. $m=100\implies$ you will collect exactly $\$100$ when closing the account;
2. $100<m<103\implies$ you will collect less than your $\$100$ principal when closing the account;
3. $m\geq103\implies$ you will be owing money when closing the account.