Devlin's Angle

October 2008

The Big Mortgage Surprise

The ramifications of the mortgage meltdown continue to reverberate through the economy, and many homeowners still face foreclosure. Should we really have been surprised when all those highly risky real estate loans started to turn sour? One group who will surely make money from the episode are the academics now busily writing books analyzing what led to the crash.

To me, the really surprising thing about mortgages, however, is that they exist at all. When you stop and think about it, the idea that millions of people of very ordinary means can actually own their own homes seems to defy common sense. The amount of money required to purchase a house is so large that it typically takes 20 to 30 years to pay off a mortgage loan, if indeed it ever gets paid off. During that long period, all sorts of things can and surely will happen, ranging from fluctuations in the interest rates to natural disasters, which may result in total destruction of the one piece of collateral for the loan, namely the house itself. How can a mortgage lender possibly stay in business, let alone make an attractive profit, in a competitive finance industry, selling loans that people will be willing to buy and which stretch over many decades?

The answer is the enormous power of a mathematical crystal ball that traces its origins to a letter sent from one Frenchman to another on Monday August 24, 1654. In fewer than three-thousand words, mathematician Blaise Pascal summarized and analyzed a solution that had just been worked out by the letter's recipient, Pierre de Fermat, to a problem that had puzzled gamblers and mathematicians alike for decades. Known as the problem of the Unfinished Game, or sometimes the Problem of the Points, the puzzle asked how the pot should be divided when a game of dice has to be abandoned before it has been completed, based on how many rounds each has won at that point.

Today, the solution to the problem can be explained to high school students in a few minutes, but that is because we have all grown up in a world where we are inundated with probabilistic predictions about the future - the 30% chance of rain tomorrow, the 49% to 42% advantage Obama has over McCain in the upcoming election, the likelihood that Google stock will increase in value, etc. Modern life, including house purchase, is built upon our ability to manage risk. We might not be able to predict the future, but we can calculate the chances that various things will happen, often with extraordinary precision.

Until Pascal wrote his letter to Fermat, no one thought it was possible to use mathematics to predict the future. What will be will be, people believed - it's all a matter of fate. Fortunately for those of us who enjoy modern life, and (still) own our homes, those two mathematicians persevered in the face of thinking that they were attempting the impossible, and came up with an ingenious way to calculate numerical likelihoods of how the future will turn out.

The idea is to list all the different possible ways the future might turn out, and then just count the various kinds of outcomes. It's not obvious that this can be done at all, let alone clear how you do it, and much of Pascal's letter focuses on how to do the counting. In fact, one of the more fascinating aspects of the letter is the insight it provides into how new mathematics is developed. The letter was never intended for publication, and as a result it lays bear the messy process of mathematical discovery, showing how two of the greatest mathematicians the world has ever known struggle to find the solution, stumbling, making "elementary" errors, arguing about what it the "right" way to proceed, and being unsure that what they have done is correct. Sound familiar? I suggest that all mathematics students should be encouraged read this letter.

So powerful was the breakthrough described in the letter that, within a few years, all of the trappings of quantitative risk management, actuarial science, and modern statistics had come into being. Ordinary people could buy cars, take expensive vacations, purchase life insurance, and buy homes.

On that summer day in 1654, the world learned how to predict the future and manage risk, and the modern world began. Unfortunately, in recent years, some lenders forgot the lessons learned 350 years ago.

The entire letter, together with the story of how it came to be written and how it changed the way people view the future, is described in my most recent book, The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published this month by Basic Books.

Marcia Sward

Finally, I can't send this month's column off to the MAA without expressing my great sadness at the passing of Marcia Sward, MAA Executive Director from 1989 to 1999, who died on September 21. This column is a consequence of her initiative. In 1989, at the height of Prime Minister Margaret Thatcher's assault on the British university system, I finally bowed to the pressure from my UK university to resign and relocate to the United States. (It was a money thing.) Having been highly active in both the UK mathematics world and in trying to promote mathematics awareness among the general population (since 1984 I had written a twice-monthly column in The Guardian newspaper, among other activities), I was eager to contribute in a similar fashion in my newly adopted country, but had little by way of a network through which to do it. It turned out I did not need one to get me started. In the summer of 1991, Marcia phoned me up to ask me if I was interested in becoming the next editor of Focus. I said yes immediately. What I did not know at the time was that Focus was Marcia's very own creation, and one she cared deeply about. It was only later that I realized just how much trust she was putting in someone she had never met. Of course, the appointment was made by a committee, after considering the issue at length, doubtless including reading many of my Guardian columns, but I have always assumed - though without any concrete evidence - that I got the position primarily because Marcia wanted me to have it.

I took Focus in the new direction that I said I would before being appointed editor, including a "British-style" editorial that presented my views (or often a position that I did not actually agree with but felt needed an airing), not necessarily the official position of the MAA. Not everyone at the MAA agreed with everything I did, particularly the editorial, always a tricky matter in a news magazine of a professional organization. (It would have been surprising if everyone had agreed.) I'm not sure Marcia fully agreed either, but from the start she made it clear that she wanted me to take "her Focus" and develop it as I (and my editorial committee) thought fit, and she stood behind me throughout. No editor can ask for more from his boss. Thank you, Marcia. You were a class act. I miss you. We all do.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition.