I'm working in the crypto currency space. I have the ability to deposit some cash into an account and obtain interest from it. To get the interest, I have to claim it. I'd like compound the interest back into the account to increase earning power.
If we assume that I can claim the interest as often as I want and it costs a fixed fee of $1 to do so, how would I compute the optimal frequency of claiming and compounding given that I dont want to waste money on fees.
Let's use the values, $25$% APR, $1$ fee, and $100,000$ principle. It seems clear to me that it's a function of the principle, where the frequency increases as the principle gets larger.
The (years) time line $[0,\infty)$ is partitioned in the discrete grid $(\Delta_t^n)_{n \in \mathbb{N}_0}$. $n$ is an index. These are the intervals at which you claim the interest, which increases your account. In $[\Delta_t^{n},\Delta_t^{n+1})$ your account is a fixed sum (we ignore inflation). $r$ is the annual rate. The interest you can claim at $\Delta_t^{n+1}$ is $r(\Delta_t^{n+1}-\Delta_t^n)\cdot W_{\Delta_t^n}$ and you pay a fee $c \in \mathbb{R}^+$. So $$\begin{matrix} \textrm{Wealth} & \textrm{Time (years)} \\ W_0 & [0,\Delta^1_t) \\ W_0(1+r \Delta_t^1)-c & [\Delta_t^1,\Delta_t^2) \\ (W_0(1+r\Delta_t^1)-c)(1+r({\Delta_t^2}-\Delta_t^1))-c & [\Delta_t^2,\Delta_t^3) \\ (...) & (...) \end{matrix}$$ So we have the recursion $$W_{\Delta_t^n}=W_{\Delta_t^{n-1}}(1+r(\Delta^n_t-\Delta^{n-1}_t))-c \ \ \ \ \ \ n \in \mathbb{N}$$ $$W_{\Delta_t^n}-W_{\Delta_t^{n-1}}=W_{\Delta_t^{n-1}}\cdot r(\Delta^n_t-\Delta^{n-1}_t)-c $$ If we set $W_{\Delta_t^n}-W_{\Delta_t^{n-1}}>a$ we get $$W_{\Delta_t^{n-1}}\cdot r(\Delta^n_t-\Delta^{n-1}_t)-c>a$$ $$\Delta^n_t-\Delta^{n-1}_t > \frac{a+c}{W_{\Delta_t^{n-1}}\cdot r}$$ So the bound to guarantee a difference greater than $a$ changes with time. As wealth grows, we have to wait less to claim interest, because we cover the fees more quickly.