Optimal Strategy in a money compounding model where one's interest is only consolidated at a fee

Question

Say I have my main account with $ \$ 10000$ that gains interest at a rate of .1% a day. The interest collects in a separate account and I have to pay a certain fee, say $\$1$, to consolidate this interest with my main account. What is the optimal strategy for long term gain? How does it change based on starting amount, interest rate, and fee?

Answer 1

Let the intitial balance be $B_0 = 10000$. Simple interest accrues daily in the separate account at the rate $r = 0.1 \%$ until you decide to pay the fee $F$ (assumed fixed) and transfer into the main account. Thereafter the interest accrues again in the separate account but is computed on the new balance in the main account. The process is repeated after you make another transfer.

To get you started, make the simplifying assumption that the interest is transferred at the end of every $m$-day period.

After the first transfer the balance is

$$B_1 = B_0(1 + mr)-F$$

After the second transfer the balance is

$$B_2 = ( B_0(1 + mr)-F)(1+mr) -F$$

After the $n$ transfers the balance is

$$B_n = B_0(1+mr)^n - F\sum_{k=0}^{n-1}(1+mr)^k $$

Summing the finite geometric series we get

$$B_n = B_0(1+mr)^n - F \frac{(1+mr)^n-1}{mr}$$

You can work with this formula to explore the optimal parameter $m$ for maximizing the terminal balance assuming, for example, a given horizon $mn$.