In A Mathematician Plays the Stock Market best-selling author John Allen Paulos demonstrates what the tools of mathematics can tell us about the vagaries of the stock market. Employing his trademark stories, vignettes, paradoxes, and puzzles (and even a film treatment), Paulos addresses every thinking reader's curiosity about the market: Is it efficient? Is it rational? Is there anything to technical analysis, fundamental analysis, and other supposedly time-tested methods of picking stocks? How can one quantify risk? What are the most common scams? What light do fractals, network theory, and common psychological foibles shed on investor behavior? Are there any approaches to investing that truly outperform the major indexes? Can a deeper knowledge of mathematics help beat the odds? All of these questions are explored with the engaging erudition that made Paulos's A Mathematician Reads the Newspaper and Innumeracy favorites with both armchair mathematicians and readers who want to think like them. Paulos also shares the cautionary tale of his own long and disastrous love affair with WorldCom. In the tradition of Burton Malkiel's A Random Walk Down Wall Street and Jeremy Siegel's Stocks for the Long Run, this wry and illuminating book is for anyone, investor or not, who follows the markets-or knows someone who does.
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About the Author
John Allen Paulos received his Ph.D. in mathematics from the University of Wisconsin and is professor of mathematics at Temple University. Dr. Paulos has written a number of scholarly papers on mathematical logic, probability, and the philosophy of science. He is also the author of Innumeracy: Mathematical Illiteracy and Its Consequences, Beyond Numeracy: Ruminations of a Numbers Man, Mathematics and Humor, and A Mathematician Reads the Newspaper. He lives in Philadelphia with his wife and two children.
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A Mathematician Plays the Stock Market
By John Allen Paulos
Basic BooksCopyright © 2004 John Allen Paulos
All right reserved.
Chapter OneAnticipating Others Anticipations
It was early 2000, the market was booming, and my investments in various index funds were doing well but not generating much excitement. Why investments should generate excitement is another issue, but it seemed that many people were genuinely enjoying the active management of their portfolios. So when I received a small and totally unexpected chunk of money, I placed it into what Richard Thaler, a behavioral economist I'll return to later, calls a separate mental account. I considered it, in effect, "mad money."
Nothing distinguished the money from other assets of mine except this private designation, but being so classified made my modest windfall more vulnerable to whim. In this case it entrained a series of ill-fated investment decisions that, even now, are excruciating to recall. The psychological ease with which such funds tend to be spent was no doubt a factor in my using the unexpected money to buy some shares of WorldCom (abbreviated WCOM), "the pre-eminent global communications company for the digital generation," as its ads boasted, at $47 per share. (Hereafter I'll generally use WCOM to refer to the stock and WorldCom to refer to the company.) Today, of course, WorldCom is synonymous with business fraud, but in the halcyon late 1990s it seemed an irrepressibly successful devourer of high-tech telecommunications companies. Bernie Ebbers, the founder and former CEO, is now viewed by many as a pirate, but then he was seen as a swashbuckler. I had read about the company, knew that high-tech guru George Gilder had been long and fervently singing its praises, and was aware that among its holdings were MCI, the huge long-distance telephone company, and UUNet, the "backbone" of the Internet. I spend a lot of time on the net (home is where you hang your @) so I found Gilder's lyrical writings on the "telecosm" and the glories of unlimited bandwidth particularly seductive.
I also knew that, unlike most dot-com companies with no money coming in and few customers, WorldCom had more than $25 billion in revenues and almost 25 million customers, and so when several people I knew told me that WorldCom was a "strong buy," I was receptive to their suggestion. Although the stock had recently fallen a little in price, it was, I was assured, likely to soon surpass its previous high of $64.
If this was all there was to it, there would have been no important financial consequences for me, and I wouldn't be writing about the investment now. Alas, there was something else, or rather a whole series of "something elses." After buying the shares, I found myself idly wondering, why not buy more? I don't think of myself as a gambler, but I willed myself not to think, willed myself simply to act, willed myself to buy more shares of WCOM, shares that cost considerably more than the few I'd already bought. Nor were these the last shares I would buy. Usually a hardheaded fellow, I was nevertheless falling disastrously in love.
Although my particular heartthrob was WCOM, almost all of what I will say about my experience is unfortunately applicable to many other stocks and many other investors. Wherever WCOM appears, you may wish to substitute the symbols for Lucent, Tyco, Intel, Yahoo, AOL-Time Warner, Global Crossing, Enron, Adelphia, or, perhaps, the generic symbols WOE or BANE. The time flame of the book-in the midst of a market collapse after a heady, nearly decade-long surge-may also appear rather more specific and constraining than it is. Almost all the points made herein are rather general or can be generalized with a little common sense.
Falling in Love with WorldCom
John Maynard Keynes, arguably the greatest economist of the twentieth century, likened the position of short-term investors in a stock market to that of readers in a newspaper beauty contest (popular in his day). The ostensible task of the readers is to pick the five prettiest out of, say, one hundred contestants, but their real job is more complicated. The reason is that the newspaper rewards them with small prizes only if they pick the five contestants who receive the most votes from readers. That is, they must pick the contestants that they think are most likely to be picked by the other readers, and the other readers must try to do the same. They're not to become enamored of any of the contestants or otherwise give undue weight to their own taste. Rather they must, in Keynes' words, anticipate "what average opinion expects the average opinion to be" (or, worse, anticipate what the average opinion expects the average opinion expects the average opinion to be).
Thus it may be that, as in politics, the golden touch derives oddly from being in tune with the brass masses. People might dismiss rumors, for example, about "Enronitis" or "WorldComism" affecting the companies in which they've invested, but if they believe others will believe the rumors, they can't afford to ignore them.
BWC (before WorldCom) such social calculations never interested me much. I didn't find the market particularly inspiring or exalted and viewed it simply as a way to trade shares in businesses. Studying the market wasn't nearly as engaging as doing mathematics or philosophy or watching the Comedy Network. Thus, taking Keynes literally and not having much confidence in my judgment of popular taste, I refrained from investing in individual stocks. In addition, I believed that stock movements were entirely random and that trying to outsmart dice was a fool's errand. The bulk of my money therefore went into broad-gauge stock index funds.
AWC, however, I deviated from this generally wise course. Fathoming the market, to the extent possible, and predicting it, if at all possible, suddenly became live issues. Instead of snidely dismissing the business talk shows' vapid talk, sportscaster-ish attitudes, and empty prognostication, I began to search for what of substance might underlie all the commentary about the market and slowly changed my mind about some matters. I also sought to account for my own sometimes foolish behavior, instances of which will appear throughout the book, and tried to reconcile it with my understanding of the mathematics underlying the market.
Lest you dread a cloyingly personal account of how I lost my shirt (or at least had my sleeves shortened), I should stress that my primary purpose here is to lay out, elucidate, and explore the basic conceptual mathematics of the market. I'll examine-largely via vignettes and stories rather than formulas and equations-various approaches to investing as well as a number of problems, paradoxes, and puzzles, some old, some new, that encapsulate issues associated with the market. Is it efficient? Random? Is there anything to technical analysis, fundamental analysis? How can one quantify risk? What is the role of cognitive illusion? Of common knowledge? What are the most common scams? What are options, portfolio theory, short-selling, the efficient market hypothesis? Does the normal bell-shaped curve explain the market's occasional extreme volatility? What about fractals, chaos, and other non-standard tools? There will be no explicit investment advice and certainly no segments devoted to the ten best stocks for the new millennium, the five smartest ways to jump-start your 401(k), or the three savviest steps you can take right now. In short, there'll be no financial pornography.
Often inseparable from these mathematical issues, however, is psychology, and so I'll begin with a discussion of the no-man's land between this discipline and mathematics.
Being Right Versus Being Right About the Market
There's something very reductive about the stock market. You can be right for the wrong reasons or wrong for the right reasons, but to the market you're just plain right or wrong. Compare this to the story of the teacher who asks if anyone in the class can name two pronouns. When no one volunteers, the teacher calls on Tommy who responds, "Who, me?" To the market, Tommy is right and therefore, despite being unlikely to get an A in English, he's rich.
Guessing right about the market usually leads to chortling. While waiting to give a radio interview at a studio in Philadelphia in June 2002, I mentioned to the security guard that I was writing this book. This set him off on a long disquisition on the market and how a couple of years before he had received two consecutive statements from his 401(k) administrator indicating that his retirement funds had declined. (He took this to be what in chapter 3 is called a technical sell signal.) "The first one I might think was an accident, but two in a row, no. Do you know I had to argue with that pension person there about getting out of stocks and into those treasury bills? She told me not to worry because I wasn't going to retire for years, but I insisted 'No, I want out now.' And I'm sure glad I did get out." He went on to tell me about "all the big shots at the station who cry like babies every day about how much money they lost. I warned them that two down statements and you get out, but they didn't listen to me."
I didn't tell the guard about my ill-starred WorldCom experience, but later I did say to the producer and sound man that the guard had told me about his financial foresight in response to my mentioning my book on the stock market. They both assured me that he would have told me no matter what. "He tells everyone," they said, with the glum humor of big shots who didn't take his advice and now cry like babies.
Such anecdotes bring up the question: "If you're so smart, why ain't you rich?" Anyone with a modicum of intelligence and an unpaid bill or two is asked this question repeatedly. But just as there is a distinction between being smart and being rich, there is a parallel distinction between being right and being right about the market.
Consider a situation in which the individuals in a group must simultaneously choose a number between 0 and 100. They are further directed to pick the number that they think will be closest to 80 percent of the average number chosen by the group. The one who comes closest will receive $100 for his efforts. Stop for a bit and think what number you would pick.
Some in the group might reason that the average number chosen is likely to be 50 and so these people would guess 40, which is 80 percent of this. Others might anticipate that people will guess 40 for this reason and so they would guess 32, which is 80 percent of 40. Still others might anticipate that people will guess 32 for this reason and so they would guess 25.6, which is 80 percent of 32.
If the group continues to play this game, they will gradually learn to engage in ever more iterations of this meta-reasoning about others' reasoning until they all reach the optimal response, which is 0. Since they all want to choose a number equal to 80 percent of the average, the only way they can all do this is by choosing 0, the only number equal to 80 percent of itself. (Choosing 0 leads to what is called the Nash equilibrium of this game. It results when individuals modify their actions until they can no longer benefit from changing them given what the others' actions are.)
The problem of guessing 80 percent of the average guess is a bit like Keynes's description of the investors' task. What makes it tricky is that anyone bright enough to cut to the heart of the problem and guess 0 right away is almost certain to be wrong, since different individuals will engage in different degrees of meta-reasoning about others' reasoning. Some, to increase their chances, will choose numbers a little above or a little below the natural guesses of 40 or 32 or 25.6 or 20.48. There will be some random guesses as well and some guesses of 50 or more. Unless the group is very unusual, few will guess 0 initially.
If a group plays this game only once or twice, guessing the average of all the guesses is as much a matter of reading the others' intelligence and psychology as it is of following an idea to its logical conclusion. By the same token, gauging investors is often as important as gauging investments. And it's likely to be more difficult.
Other situations, as well, require anticipating others' actions and adapting yours to theirs. Recall, for example, the television show on which contestants had to guess how their spouses would guess they would answer a particular question. There was also a show on which opposing teams had to guess the most common associations the studio audience had made with a collection of words. Or consider the game in which you have to pick the location in New York City (or simply the local shopping mall) that others would most likely look for you first. You win if the location you pick is chosen by most of the others. Instances of Keynes's beauty contest metaphor are widespread.
As I've related elsewhere, a number of years ago I taught a summer probability course at Temple University. It met every day and the pace was rapid, so to induce my students to keep up with the material I gave a short quiz every day. Applying a perverse idea I'd experimented with in other classes, I placed a little box at the bottom of each exam sheet and a notation next to it stating that students who crossed the box (placed an X in it) would have ten extra points added to their exam scores. A further notation stated that the points would be added only if less than half the class crossed the box. If more than half crossed the box, those crossing it would lose ten points on their exam scores. This practice, I admit, bordered on pedagogical cruelty.
A few brave souls crossed the box on the first quiz and received ten extra points. As the summer wore on, more and more students did so. One day I announced that more than half the students had crossed the box and that those who did had therefore been penalized ten points. Very few students crossed the box on the next exam. Gradually, however, the number crossing it edged up to around 40 percent of the class and stayed there. But it was always a different 40 percent, and it struck me that the calculation a student had to perform to decide whether to cross the box was quite difficult. It was especially so since the class was composed largely of foreign students who, despite my best efforts (which included this little game), seemed to have developed little camaraderie. Without any collusion that I could discern, the students had to anticipate other students' anticipations of their anticipations in a convoluted and very skittish self-referential tangle. Dizzying.
I've since learned that W.
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