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

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?

• Welcome to math.SE. Please provide some more information about what you have tried already, and where you get stuck. You're likely to get a better answer that way, rather than just posting the (homework?) problem and hoping someone will answer the whole thing. Also, you tagged a lot of different methodologies, and I suspect this problem could be solved in a number of different ways. Is there a particular way you are trying to model this problem (e.g., as an LP or a dynamic programming problem)? – LarrySnyder610 May 20 at 22:31
• Not a homework problem! Just a problem that me and my friend came up with while discussing theoretical financial models. Have no idea what techniques would be useful. I guess a DP algorithm can solve a particular case pretty efficiently but I'm wondering if a general mathematical solution exists. I would be happy if someone could suggest a suitable methodology! – Derekster May 20 at 22:44
• OK fair enough. :) So you are looking for an analytical solution, as opposed to an algorithm. I'm more of an algorithm person so if it were me I would formulate it as something like a DP and then see whether there's a pattern that can lead to a proof of an optimal policy...but others will probably have better suggestions. – LarrySnyder610 May 20 at 23:00
• Also, what would be a good categorization of this problem? Discrete math? Game theory? – Derekster May 21 at 19:32

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 $$mn$$.

• Will ridding of the simplifying assumption (constant period) make the problem significantly harder? – Derekster May 21 at 19:31
• The next level of complexity might be to optimize over a set of different period lengths $m_1,\ldots, m_n$ between transfers subject to a horizon constraint $T = \sum_{j=1}^n m_j$. Now you have a multivariate nonlinear optimization problem. – RRL May 22 at 4:49