Sunday, August 19, 2007

Where Credit is Due

I remember the night Visa found me. It was 1993 and I was living in England. I was working late at the Mathematical Institute in Oxford. The phone rang. I was used to getting calls from my friends at all hours of the night - for one thing because of the time difference with the United States, and for another thing because my friends are the kind of people who are awake at all hours of the night no matter what time zone they're in. But there was no friend on the other end of this line. Visa had found me.

I left college in 1991 with more debt than I could handle and a bad habit of letting myself forget about it for months at a time. During my senior year of college I had fought a pretty good bout against American Express. I won on points: they wrote off the debt, and I gave back the card. But with Visa I got on my bicycle. I thought if I rode it all the way to England, I'd be safe. I didn't count on my mother turning me in.


My grad school years were lean, but they were also fat, because in Berkeley we lived like paupers but ate like kings. If one of us got a fellowship check, all of us ate Roquefort. I liked to say back then that my friends and I would be the first graduate students in the history of graduate school to come down with gout.

None of this lent itself particularly well to paying bills. I remember when I was first beginning to date a particular young woman, and the two of us went back to my apartment for the first time. We walked in, and I flicked the light switch. Nothing. PG&E had shut me down. Gamely, she lit some candles, then said she'd be just a minute in the bathroom. Soon, word came through the closed door: no toilet paper. Gamely, she accepted the paper coffee filters I passed through to her. Gamely, she paid my electric bill the next day. Surprisingly after such a beginning, our relationship lasted another two and a half years.

But to return to the issue of credit. In my fourth year at Berkeley, Capital One, an unknown but clearly hungry new company, offered me my first credit card in years, with a thousand-dollar limit. I took them up on the offer. On the very same day the card arrived, I maxed it out with a plane ticket to Amsterdam. Somebody over at Capital One lost their job that day, I'm pretty sure. But it was legitimate! I was scheduled to give a paper at a conference on quantum mechanics. But I didn't have funding for the airfare, so the card arrived just in time. My relationship with Capital One has lasted to this day, and it's had all the ups and downs of any relationship, though in this case mostly to do with APRs.


People ask me mathematical questions all the time. A friend once sent me the following cryptic email:

situation: test.
pool of possible questions: 5
possible amount on test to choose from: 2 or 3.
number of questions to answer: 1.

If I just choose two questions to study, what are the odds of having one of those show up on the test as a posed question to answer?

More recently, a friend asked how best to pay off a high-interest credit card. She'd been paying $600 per month on a $10,000 balance at 21.9% interest, and she was now considering cashing in a 403(b) account to eliminate the debt. After finding out some more about the person's situation, I advised taking out a home equity line of credit to pay off the debt. Assuming she stops using the card completely, then in three years, she can have the home equity loan paid off, and by that time she can also have $10,000 cash reserves in the bank - all for the same $600 per month she's paying now on the credit card. (The caveat of course is that you really do have to stop using the credit card, otherwise in three years you'll just find yourself maxed out again.)

In my own life, I have often wondered, if I owe X amount on a credit card at Y% interest, and I pay Z per month, then how long will it take me to pay off the card, assuming I don't charge anything else on it? The mathematics required to answer this is not trivial. There are online calculators that will do the computation for you - see this one for example. But the downside of using a calculator is that you don't get any insight into the problem you're facing. So I sat down one day several years ago and derived the answer once and for all. Here it is:

In this formula, b_0 is the initial balance, p is what you have decided to pay every month, and i is the daily interest rate, that is, your APR divided by 365. Note that the payoff time N turns out to be a function not of b_0 and p separately, but rather a function of the ratio p/b_0. Physicists will have seen this coming, thanks to their penchant for dimensional analysis. (The answer to the problem - a number of months - has no dollar signs on it; so the dollar signs on p and b_0 have to be got rid of. The only way to do that is to work with the ratio p/b_0.)

The derivation is here.

Getting into the details of this problem gives you a gut-level feel for an important financial fact: payments to the credit card company are mathematically the same as investments that grow with a guaranteed rate of return. Millionaires pay ludicrous fees to hedge funds in exchange for a guaranteed rate of return. Schmucks like us can do the same thing just by overpaying our credit card bill.

For convenience, I have put together the following table, suitable for printing out and stashing away in the utility drawer (click to enlarge):

For example, suppose your starting balance is $10,000, your interest rate is 12.9%, and you figure you can afford to pay $400 per month. Your monthly payment equals 4% of the starting balance. So look down the 4% column until you get to the row for an APR of 13%. You see that it will take you 29 months to pay off the card, or about two and a half years. (After a few months of making payments, call the company every so often to request a lower APR.)

Another example. Your starting balance is $10,000, your interest rate is 12.9%, and you really want to have this card paid off in a year, because you're planning to refinance your house in 18 months, and you want your credit score to be as high as possible. [Can somebody tell me when I suddenly became this middle-aged??] Looking at the table in the 13% row, you see that in order to have a 12-month payoff time, you're going to have to pay 9% of the starting balance, or $900, every month.


I liked Jean Chatzky's advice on her financial webpage. But the trouble with Chatzky's program is that it has nine steps. Who can remember nine steps? Who can carry them out? The people who can follow nine steps are not the people with maxed-out credit cards. So, as a recovered credit-card delinquent, I thought I would share my own program, which is so simple it has only one step:

Tithe 10 percent.

Tithing 10 percent means that any time any money whatsoever comes into your household - be it a paycheck, tax refund, honorarium, royalty, or even a good night's poker winnings - sit down that very night and send a check to your credit card company for 10 percent of whatever amount you brought in that day. Don't even wait for your account statement; just keep those checks moving out the door.

(If you have more than one credit card bill, divide your 10% tithe among the various cards in a ratio that makes the most sense to you.)

If your annual net income is, say, $30,000 per year, then tithing will divert $3,000 a year to your credit card, and what's more that'll be on a continual basis, hammering away at the debt before interest can pile up. They say interest never sleeps; well, don't let your payments sleep either. Think of your steady stream of checks as a flurry of jabs that will keep the credit card companies constantly on their heels.

If your tithe isn't bringing the balances down, then you can increase the percentage, make an extra payment when the account statement comes in the mail, and, most important of all, STOP USING THE CARD.

The system works, because in all honesty you can probably spare 10% of whatever you make, especially if you part with it immediately. After all, with the money gone, you can't very well blow it on Roquefort, can you?


Joe Holt said...

I've survived 20 years without a credit card, and have only borrowed money to buy the truly big things: houses. But I'm not writing this to brag (maybe just a little), but to air a nagging concern I have about credit: My "credit" rating according to a recent report is near the top. Is there meat to the cultural meme about the necessity of gaming one's credit rating? I pay my bills on time and I've avoided debt. That seems to account for something. What do you think about manipulating your credit payments in order to "improve" your rating?

JasonZimba said...

There is some truth in the general concept that you can & occasionally should game the system. For example, at one point a few years ago, I was going to cancel my faithful Capital One card, because I never use it. But it turns out that a significant factor in the credit score is the average length of time you've had your cards; getting rid of the Capital One would have lowered my average a lot, and brought my score down.

Not all of the FICO methodology makes sense to me, but on the other hand I realize their correlation coefficients are based on hundreds of millions of data points. In any case, given that they do have a formula, it only makes sense for us on the other end to reverse-engineer it.

danimal said...

I like the recommendation of "tithe 10 percent". I came out of graduate school in 2003 with about $3000 credit card debt, and managed to pay it off in a little over a year by prioritizing it as my first payment, and living off the remainder. Since I was single in rural Mississippi at the time, I made out fine, although it can be tougher with more expenses I am sure.

Found out a couple of interesting things looking into credit card debt in Europe; I had thought that legislation in Europe might protect consumers more by, e.g., forcing credit companies to clearly show the number of months to payoff if the "minimum monthly payment" below the this minimum payment amount (people have tried a couple times unsucessfully to pass this in the U.S.). The debt in Europe is somewhat lower, but increasing fairly fast, especially in England:

Also, this article points out that we should avoid thinking of the distribution of credit card debt in the US as a Gaussian centered around the oft-quoted figure of $9000 avg. debt/household. It has a large peak (55%) with little or no debt, and then a distribution of the remaining 45% which must average about $20000.
Note: I don't think the tone of this article is too helpful to people working to pay off credit-card debt, but this "bi-modal" distribution fits better with my experience of what I know of credit card debt among family/friends (and myself).

Also, if you like this blog by Jason, I recommend "The Motley Fool's Guide to Personal Finance"! It is ** very ** entertainingly written, and has (what seemed to me to be) sensible advice, based on calculations of interest and basic financial math much like this post. Now, having read and enjoyed the book, I just need to start applying their advice more...

danimal said...

The link to the second article was cut off in the previous post. Here it is in three lines (you'll have to cut and paste, sorry!):

Joe Holt said...

Unrelated to finance, I find sites like useful for posting otherwise long and unwieldy urls.

Joe Holt said...

Jason, here are some thoughts on credit scores vs. risk scores, in light of the current "crisis". From the livejournal of Bram Cohen:

Anonymous said...

um, Mr. Zimba? what's the answer to your friend's question about situation: test?

Is it gonna be on the test?

JasonZimba said...

hee hee

For the sake of interest, here is the response I gave at the time:

This is basically an investment strategy question. You have a finite resource (time) and you want to know how to invest it in studying, to achieve a certain return (test score). As with any investment question, the probabilities can only give you the raw data. The final decision you make always depends on how willing you are to tolerate risk.

For now, I assume that the test will have two questions. Later I'll rewrite it for the case of a three-question test.

At one extreme, suppose you study only one question. This strategy has the best upside, because if that question is on the test, then you knock it out of the park. But this strategy also has the most severe downside risk, because there is a 60% chance that your question will not even be on the test. Most of us would not seek to maximize our possible upside if it meant exposing us to a 60% risk of complete failure.

At the other extreme, suppose you study four questions. This strategy has the least downside risk, because it is guaranteed that one of your questions will be on the test. But you pay this security by losing upside: your score on the question will not be as high, because you won't have studied it as thoroughly. This tradeoff may make sense if failing the test is a disaster to be avoided at all costs.

The middle-of-the-road strategies are to study 2 or 3 questions. Here the downside risk is not as high as when you study 1 question, and the score attained if a question is on the test is higher than if you study 4 questions.

The story is much the same if the test has three questions. Except now, you only have to study 3 questions to be completely certain that one of your questions will be on the test.

To summarize the basic data:

Assuming there are two questions on the test, chosen from a pool of five, and you have to answer one:

If you study... Best poss. score... Strategy value... Chance that none of your question(s) will show up on the test is...

1 question 100 40 60%
2 questions 50 35 30%
3 questions 33 30 10%
4 questions 25 25 0%

Assuming there are three questions on the test, chosen from a pool of five, and you have to answer one:

1 question 100 60 40%
2 questions 50 45 10%
3 questions 33 33 0%
4 questions 25 25 0%

The best possible score assumes a simplistic picture according to which, if you spend all the available time on one question, you get a perfect score on it, and if you share the available time among N problems, then your score on any one of them would be 100/N. This is a severe dropoff, but it's just so one can assign some numbers.

The "Strategy value" is the weighted average of the best possible score (100/N) and the worst possible score (zero), with weights given by the appropriate probabilities.

Now the basic investment game is, as always, picking a strategy that compromises between high value and low risk. The math won't tell you that...

Personally, I am willing to tolerate risk when it comes to money, but not when it comes to test score. :) I'd probably end up studying 3 or 4 questions.