# Optimizing interest for a set of debt payments

Suppose I've got $n$ debts with principals $P_1, P_2, ..., P_n$, with corresponding interest rates $r_1, r_2, ..., r_n$, compounded monthly.

Further, assume I have a constant $A$ dollars per month to split between all $n$ debt payments.

EDIT: There are monthly minimum interest payments as well.

Given these constraints, how can I minimize the total interest paid?

## 1 Answer

Pay off the highest interest rate until it is done, then pay off the next. If you consider moving 1 dollar from the highest to some other loan, clearly the interest goes up.

• However, depending on the principal, you may be able to completely pay off some of the lower-interest loans first, then re-allocate some of that money to the higher interest loans. I'm not totally convinced. – tlehman May 5 '14 at 21:37
• No, as long as there aren't minimum monthly interest charges, you always want to pay off the high interest ones. – Ross Millikan May 5 '14 at 23:38
• Say I have debts with principals of \$10 and \$2, with rates 10% and 20%, respectively. I must pay at least the interest on each, each pay period. In addition, I have an extra dollar to spend each month. Then, your claim is that I should simply pay the one with the 50% rate first? – Zduff Sep 24 '19 at 20:49
• @Zduff: It would be even better to pay off the high interest one and only pay the other when the high interest one is paid. In your example, assume the rates are per month, so the first has 1 interest per month and the second 0.40. If you have 2.40 to spend you can pay off the 20% one while the other goes to 11, then pay off the 11 over the coming months. If you insist on paying the 1 interest on the 10% loan, the other will still have a balance of 1 after the first month and you will pay 0.20 on that dollar instead of the 0.1 you could have. Minimums and penalties make sure – Ross Millikan Sep 24 '19 at 21:27
• @Zduff: you pay the interest on each loan usually, though. I do have one card that (at least until I carry a balance) will let me pay only 1% as a minimum even though the monthly interest is higher than that. – Ross Millikan Sep 24 '19 at 21:28