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Right off the bat, I do hope this question doesn't attract a bunch of derisive comments about my personal affairs. I give the lengthy personal anecdote because I don't have the mathematical training necessary to know how to ask my question(s) formally, and so I hope the additional information will make answerer's job easier.

Also, I know in mathematics you're taught to be formal and concise, but I am not a mathematician. I'm good at math, got an A in (college) Calculus while working full-time, but I've forgotten how to find limits and perform integration a long time ago. I do still have a firm grasp of College Algebra, geometry, and everything else covered in high school. If you go into a topic more advanced than my training, if you can add a little teaching, some hints about keywords I should look up on Wikipedia/Google, or even a topic I should research so I can ask this question again in a better way, I would much appreciate it. In the end, I just want to learn.

So here we go, some background: I have large number of debts I've acquired over time-- for better or worse, I started early in life applying liberally for credit. I didn't overuse it or get into trouble; handling the money hasn't been a problem, it was the unexpectedly time-consuming and tedious process of staying on top of each account. When I get around to paying each creditor, I usually pay some random amount since the minimum payment on the statement can sometimes be significantly lower than the interest charges, leading to a larger balance that slowly accumulates more and more every month. Yeah, obviously I should pay as much as I can, but it's a difficult balance to strike between paying down debt and having cash on hand. There are some things you must pay with cash or a checking account (though they dwindle every year).

Anyway, I finally decided that it was time to actually run the numbers and see if there wasn't a better way to understand my debts individually and as a whole... and frankly, see if I could find a way to make the payments I send to each creditor not just a number I pulled out of you-know-where.

So I pulled out a pad and pen and with Wikipedia, card statements, and algebra I actually found out that when I plug my situation into the big formulas and put them through the calculator, there are some quick and easy tricks that I'm sure you'll all find immediately obvious:

  • Estimate reasonable payment: For balances of \$1000-\$5000, balance divided by 10 (10%) will pay off the debt in 12 months.
  • Estimate lower-bound: Again, for \$1000-\$5000, just half the above amount is 5%, and puts you on track to payoff in ~24 months. (This one is doubly useful, as it indicates that I'm probably overextending myself.)
  • Interest-only Payment: Balance divided by 100, then doubled is 2%, which is more than the monthly interest charge for ~20% APR cards/loans.

Perhaps it's just me, but I was quite surprised at just how simple these estimations are. Multiplication and division by factors of ten and two can tell me roughly how much I should pay and the timeline in which the debt will be paid at that rate. When you've just finished working twelve hours and want to pay your bill and go to bed, the 10% guideline is quick, easy, and effective. And I always know I can cut it in half (the 5% guideline) if it's one of those months.

Those were a little to easy to find, so I didn't stop there. The 10% guideline doesn't work for the large number of low-limit cards I don't use or very rarely use or gave to a sibling so they would always have means or I don't know the balance, only the minimum payment, so I just pay the minimum and never pay it off or whatever goes on. As the balance gets smaller, 10% of that balance gets smaller, too. The great thing about the guideline is it cuts your highest balances down to manageable sizes quickly, but that quality turns against you when the balances get lower. So I thought there was a mythical "perfect algorithm" that'll tell you to pay large amounts (\$500 for a \$5000 balance) but not face the steep linear decline that a simple divide-by-ten will obviously give you when the amount starts to get smaller. In practice, this is easily solved by putting a minimum cap (say, \$75/mo or divide the balance by the number of months I'd like to finish it off and pay that) but... I was doing MATH, dammit! That's too easy.

The other big thing was that I didn't want to use the fixed-payment calculator. I know that a \$1000 loan at 1.6% compounded monthly can be paid off in 12 ~\$93 payments. It is the most efficient, but when you're dealing with month-to-month finances with all kinds of unpredictable expenses, you would have to constantly re-calc the formula which would mean going to every creditor ahead of time, finding out the current balance (hey, I might have bought something), putting it into the spreadsheet, adjusting which month it'll be paid-off by if necessary, and then paying whatever it spit out... the gung-ho approach I've already tried and gave up on before. Too time-consuming. Whatever I did, I needed to be able to do the calculations in my head, or at least using only the information I had in front of me-- how much do I owe, how much money do I have, how aggressively do I want this particular account to be paid in full. It's a lot of work when a "\$100/mo" guideline would probably similarly effective.

I pretty much gave up on the endeavor when I realized that my methods were always going to result in infinite series. Divide-by-ten, for example, will obviously never actually reach zero. No multiplicative system would-- it simply isn't possible. It dawned on me quickly that I was really just looking for an exponential/logarithmic equation, and all I'd have to do is play with the constants/factors involved and I'd get my "perfect" formula. Except most logarithmic/exponential calculations can't really be done in your head.

BUT. If you've read all of this, I'm curious what a mathematician thinks of it all. In the various branches of math, is there anywhere some kind of in-your-head estimation technique that would be appropriate for this? Or is this a case of good old-fashioned common-sense being the best bet: just treat the lower balanced accounts differently. The low balance means that the monthly interest fee is just a few dollars at most, so it really isn't worth computing at all. You could simply decide that you pay a different \$X/mo for every \$100 change in balance. Then you can tune it exactly.

Anyway, here's the summary of the (I believe impossible) math I was trying to find:

  • A simple technique to figure out how much more than the minimum I should pay that is much easier than the fixed-payment calculation formula.
  • A range of values would be preferable to a specific one, much like how I can choose between 5-10% of the \$1000+ balances depending on my means that month.
  • Preferably, the value our range that results from this technique should also indicate how long it will take to pay the debt. If the number of months is a goal input into the technique, then this is satisfied.
  • The amount it would spit out would be high for high balances and lower for low balances, but would taper off so the balance would get to zero.
  • No historical knowledge necessary. I shouldn't need to know that we calculated a 12 month payoff and it is month 8 when I do the calculation. Perhaps the technique automatically assumes that I want to pay off a \$200 debt faster than a \$2000 one.
  • At most, it should only require the balance, the interest rate, and (maybe) a goal for how many months I want to pay it off in. Adjusting the last value could be how you get the "range" mentioned above. Depends on how difficult the technique is.
  • I probably won't use a technique that can't be performed in my head. However, if it's clever or cool, I'll upvote/choose answers that can be done by putting the values in a spreadsheet or better, one or two easy operations in Windows' Calculator.

If this is a stupid "question," I will accept answers that adequately explain why this is impossible, especially if there's a cool proof involved, or at least a reference to some article (Wikipedia or otherwise) that I can learn more from. Also be nice if you mention topics related to my inquiry, prior art, anything that I can learn from. I'll understand if there's no answer for this, but I would like to come out of it knowing more than I do at this moment.

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I don't have the patience for all of that, but here's the best way to take care of it. You should be looking for another credit card (or 2 or 3) with a 0% APR on balance transfers with little to no fees and moving the highest interest cards' balances to it.

If that is not an option, pay the minimums on all your cards but the one(s) with the highest interest. Put whatever cash you have free towards the highest interest card of the lot to minimize final debt payments. If all the interest rates are the same, it won't matter how you pay them off as long as you pay the minimums and give the maximum amount of money you can afford each month.

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  • $\begingroup$ I don't want to be rude, but I want to comment for future passers-by. When I wrote the first paragraph asking for no derisive comments about my personal affairs, I wrote it because I didn't want off-topic comments (or answers) like this one. I'm well-aware about my options concerning personal finance and I've made an informed decision. I don't need help there. But if you have anything related to the (admittedly) elementary math I mentioned, or even something interesting that's related, please do tell. But if no one thinks the math here is worth talking about, please let my question just die. $\endgroup$ – David Schwartz Sep 5 '14 at 4:34
  • $\begingroup$ I saw that you did ask for the best algorithm, and so I gave it to you (assuming 'best' means you pay the least possible). Just trying to help. To be honest all you need is excel and a compound interest formula to figure these things out on the fly. To be accurate you need to know precise dates of any payments and how the interest is compounded for each account you have. $\endgroup$ – David Peterson Sep 5 '14 at 4:47
  • $\begingroup$ I'm glad your intentions weren't derisive, then! Yeah, I did say "algorithm" which is a misnomer. I'm a programmer and we use that word very loosely. I actually am not really looking for the "best" algorithm, because I knew right away that you're right, a spreadsheet and the PMT macro is the "pay least" way. I was more hoping for a shortcut or simplification I might not know how to find by wandering the web that would be simple enough for me to do in my head rather than keep a spreadsheet up-to-date. It probably doesn't exist, but I wanted to ask, just in case it or something like it does. $\endgroup$ – David Schwartz Sep 5 '14 at 5:10

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