Nerd hour

So the last few days have given me the chance to ponder how I'll be managing my money in the future, seeing how I'll be joining the working class in a few months (with sizeable education loans to boot). The question I had was that given a certain amount of extra money every month, how do I optimally distribute it among different investment opportunities and loan obligations with varying interest rates (my first two years of medical school loans were fortunately consolidated at very low rates). To those for whom this answer is obvious, please humor me a bit.

In the most simple scenario, the quantity you want to look at is your net change in wealth over a billing cycle or whatever they call the period over which interest accrues. It can be shown that for billing cycle i, your net wealth during the next cycle is:

W(i+1) = P1(i)* R1 + P2(i)*R2 + P3(i) *R3 + ... + PN(i)*RN + constant(i)

where P and R represent, respectively, payments and (1+r) for different investments or loan balances, with r being the interest rates. If you want, you can vary P from cycle to cycle (that's why I wrote P(i)). The constant takes into account the balances in these different investments and loans and has components growing exponentially as a function of i (increasing for the investments, and decreasing for the loans). What is interesting about this is that as long as your loan payments are at least enough to make sure that the constant factor is not increasing (this can be done by paying off at least the interest on those loans every cycle), then you can be guaranteed to maximize your growth of wealth from cycle to cycle as long as you can maximize the first few terms. Of note, those terms are identical in form to each other regardless of whether they represent interest rates on investments or loans.

The consequence of all this is that if you have a fixed amount of money, you're always better from cycle to cycle to throw it all behind the highest interest rate. This is because the gradient of W(i+1) as a function of payments always points predominantly in the direction of the higher interest rate. When the payments are all contstrained by a limited amount of money, that always leads to setting to the max the one associated with the higher interest rate (again, you need to still pay off accumulating interest on debts). Even if you propagate the process out a thousand times and try to find what would lead to the most wealth at W(1000), optimizing your payments at each cycle along the way would lead to the same conclusion. That may seem like a no-brainer, but I just wanted to check it for myself.