Chance and Efficient Markets
/f the movement of stock prices is random or near-random, then the tools of technical analysis are nothing more than comforting blather giving one the illusion of control and the pleasure of a specialized jargon. They can prove especially attractive to those who tend to infuse random events with personal significance.
Even some social scientists don’t seem to realize that if you search for a correlation between any two randomly selected attributes in a very large population, you will likely find some small but statistically significant association. It doesn’t matter if the attributes are ethnicity and hip circumference, or (some measure of) anxiety and hair color, or perhaps the amount of sweet corn consumed annually and the number of mathematics courses taken. Despite the correlation’s statistical significance (its unlikelihood of occurring by chance), it is probably not practically significant because of the presence of so many confounding variables. Furthermore, it will not necessarily support the (often ad hoc) story that accompanies it, the one purporting to explain why people who eat a lot of corn take more math. Superficially plausible tales are always available: Corn-eaters are more likely to be from the upper Midwest, where dropout rates are low.
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Geniuses, Idiots, or Neither
Around stock market rises and declines, people are prone to devise just-so stories to satisfy various needs and concerns. During the bull markets of the ’90s investors tended to see themselves as “perspicacious geniuses.” During the more recent bear markets they’ve tended toward self-descriptions such as “benighted idiots.”
My own family is not immune to the temptation to make up pat after-the-fact stories explaining past financial gains and losses. When I was a child, my grandfather would regale me with anecdotes about topics as disparate as his childhood in Greece, odd people he’d known, and the exploits of the Chicago White Sox and their feisty second baseman “Fox Nelson” (whose real name was Nelson Fox). My grandfather was voluble, funny, and opinionated. Only rarely and succinctly, however, did he refer to the financial reversal that shaped his later life. As a young and uneducated immigrant, he worked in restaurants and candy stores. Over the years he managed to buy up eight of the latter and two of the former. His candy stores required sugar, which led him eventually to speculate in sugar markets and—he was always a bit vague about the details—to place a big bet on several train cars full of sugar. He apparently put everything he had into the deal a few weeks before the sugar market crashed. Another version attributed his loss to underinsurance of the sugar shipment. In any case, he lost it all and never really recovered financially. I remember him saying ruefully, “Johnny, I would have been a very, very rich man. I should have known.” The bare facts of the story registered with me then, but my recent less calamitous experience with WorldCom has made his pain more palpable.
This powerful natural proclivity to invest random events with meaning on many different levels makes us vulnerable to people who tell engaging stories about these events. In the
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Rorschach blot that chance provides us, we often see what we want to see or what is pointed out to us by business prognosticators, distinguishable from carnival psychics only by the size of their fees. Confidence, whether justified or not, is convincing, especially when there aren’t many “facts of the matter.” This may be why market pundits seem so much more certain than, say, sports commentators, who are comparatively frank in acknowledging the huge role of chance.
Efficiency and Random Walks
The Efficient Market Hypothesis formally dates from the 1964 dissertation of Eugene Fama, the work of Nobel prizewinning economist Paul Samuelson, and others in the 1960s. Its pedigree, however, goes back much earlier, to a dissertation in 1900 by Louis Bachelier, a student of the great French mathematician Henri Poincare. The hypothesis maintains that at any given time, stock prices reflect all relevant information about the stock. In Fama’s words: “In an efficient market, competition among the many intelligent participants leads to a situation where, at any point in time, actual prices of individual securities already reflect the effects of information based both on events that have already occurred and on events which, as of now, the market expects to take place in the future.”
There are various versions of the hypothesis, depending on what information is assumed to be reflected in the stock price. The weakest form maintains that all information about past market prices is already reflected in the stock price. A consequence of this is that all of the rules and patterns of technical analysis discussed in chapter 3 are useless. A stronger version maintains that all publicly available information about a company is already reflected in its stock price. A consequence
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of this version is that the earnings, interest, and other elements of fundamental analysis discussed in chapter 5 are useless. The strongest version maintains that all information of all sorts is already reflected in the stock price. A consequence of this is that even inside information is useless.
It was probably this last, rather ludicrous version of the hypothesis that prompted the joke about the two efficient market theorists walking down the street: They spot a hundred dollar bill on the sidewalk and pass by it, reasoning that if it were real, it would have been picked up already. And of course there is the obligatory light-bulb joke. Question: How many efficient market theorists does it take to change a light bulb? Answer: None. If the light bulb needed changing the market would have already done it. Efficient market theorists tend to believe in passive investments such as broad-gauged index funds, which attempt to track a given market index such as the S&P 500. John Bogle, the crusading founder of Vanguard and presumably a believer in efficient markets, was the first to offer such a fund to the general investing public. His Vanguard 500 fund is unmanaged, offers broad diversification and very low fees, and generally beats the more expensive, managed funds. Investing in it does have a cost, however: One must give up the fantasy of a perspicacious gunslinger/investor outwitting the market.
And why do such theorists believe the market to be efficient? They point to a legion of investors of all sorts all seeking to make money by employing all sorts of strategies. These investors sniff out and pounce upon any tidbit of information even remotely relevant to a company’s stock price, quickly driving it up or down. Through the actions of this investing horde the market rapidly responds to the new information, efficiently adjusting prices to reflect it. Opportunities to make an excess profit by utilizing technical rules or fundamental analyses, so the story continues, disappear before they can be
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fully exploited, and investors who pursue them will see their excess profits shrink to zero, especially after taking into account brokers’ fees and other transaction costs. Once again, it’s not that subscribers to technical or fundamental analysis won’t make money; they generally will. They just won’t make more than, say, the S&P 500.
(That exploitable opportunities tend to gradually disappear is a general phenomenon that occurs throughout economics and in a variety of fields. Consider an argument about baseball put forward by Steven Jay Gould in his book Full House: The Spread of Excellence front Plato to Darwin. The absence of .400 hitters in the years since Ted Williams hit .406 in 1941, he maintained, was not due to any decline in baseball ability but the reverse: a gradual increase in the athleticism of all players and a consequent decrease in the disparity between the worst and best players. When players are as physically gifted and well trained as they are now, the distribution of batting averages and earned run averages shows less variability. There are few “easy” pitchers for hitters and few “easy” hitters for pitchers. One result is that .400 averages are now very scarce. The athletic prowess of hitters and pitchers makes the “market” between them more efficient.)
There is, moreover, a close connection between the Efficient Market Hypothesis and the proposition that the movement of stock prices is random. If present stock prices already reflect all available information (that is, if the information is common knowledge in the sense of chapter 1), then future stock prices must be unpredictable. Any news that might be relevant in predicting a stock’s future price has already been weighed and responded to by investors whose buying and selling have adjusted the present price to reflect the news. Oddly enough, as markets become more efficient, they tend to become less predictable. What will move stock prices in the future are truly new developments (or new shadings of old
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developments), news that is, by definition, impossible to anticipate. The conclusion is that in an efficient market, stock prices move up and down randomly. Evincing no memory of their past, they take what is commonly called a random walk, each step of which is independent of past steps. There is over time, however, an upward trend, as if the coin being flipped were slightly biased.
There is a story I’ve always liked that is relevant to the impossibility of anticipating new developments. It concerns a college student who completed a speed-reading course. He noted this fact in a letter to his mother. His mother responded with a long, chatty letter of her own in the middle of which she wrote, “Now that you’ve taken that speed-reading course, you’ve probably finished reading this letter by now.”
Likewise, true scientific breakthroughs or applications, by definition, cannot be foreseen. It would be preposterous to have expected a newspaper headline in 1890 proclaiming “Only 15 Years Until Relativity.” It is similarly foolhardy, the efficient market theorist reiterates, to predict changes in a company’s business environment. To the extent these predictions reflect a consensus of opinion, they’re already accounted for. To the extent that they don’t, they’re tantamount to forecasting coin flips.
Whatever your views on the subject, the arguments for an efficient market spelled out in Burton Malkiel’s A Random Walk Down Wall Street and elsewhere can’t be grossly wrong. After all, most mutual fund managers continue to generate average gains less than those of, say, the Vanguard Index 500 fund. (This has always seemed to me a rather scandalous fact.) There is other evidence for a fairly efficient market as well. There are few opportunities for risk-free money-making or arbitrage, prices seem to adjust rapidly in response to news, and the autocorrelation of the stock prices from day to day, week to week, month to month, and year to year is small (albeit not
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zero). That is, if the market has done well (or poorly) over a given time period in the past, there is no strong tendency for it to do well (or poorly) during the next time period.
Nevertheless, in the last few years I have qualified my view of the Efficient Market Hypothesis and random-walk theory. One reason is the accounting scandals involving Enron, Adel- phia, Global Crossing, Qwest, Tyco, WorldCom, Andersen, and many others from corporate America’s Hall of Infamy, which make it hard to believe that available information about a stock always quickly becomes common knowledge.
Pennies and the Perception of Pattern
The Wall Street Journal has famously conducted a regular series of stock-picking contests between a rotating collection of stock analysts, whose selections are a result of their own studies, and dart-throwers, whose selections are determined randomly. Over many six-month trials, the pros’ selections have performed marginally better than the darts’ selections, but not overwhelmingly so, and there is some feeling that the pros’ picks may influence others to buy the same stocks and hence drive up their price. Mutual funds, although less volatile than individual stocks, also display a disregard for analysts’ pronouncements, often showing up in the top quarter of funds one year and in the bottom quarter the next.
Whether or not you believe in efficient markets and the random movement of stock prices, the huge element of chance present in the market cannot be denied. For this reason an examination of random behavior sheds light on many market phenomena. (So does study of a standard tome on probability such as that by Sheldon Ross.) Sources for such random behavior are penny stocks or, more accessible and more random, stocks of pennies, so let’s imagine flipping a
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penny repeatedly and keeping track of the sequence of heads and tails. We’ll assume the coin and the flip are fair (although, if we wish, the penny can be altered slightly to reflect the small upward bias of the market over time).
One odd and little-known fact about such a series of coin flips concerns the proportion of time that the number of heads exceeds the number of tails. It’s seldom close to 50 percent!
To illustrate, imagine two contestants, Henry and Tommy, who bet that heads and tails respectively will be the outcome of a daily coin flip, a ritual that goes on for years. (Let’s not ask why.) Henry is ahead on any given day if up to that day there have been more heads than tails, and Tommy is ahead if up to that day there have been more tails. The coin is fair, so they’re equally likely to be in the lead, but one of them will probably be in the lead during most of their rather stultifying contest.
Stated numerically, the claim is that if there have been
1,000 coin flips, then it’s considerably more probable that Henry (or Tommy) has been ahead more than, say, 96 percent of the time than that either one has been ahead between 48 percent and 52 percent of the time.
People find this result hard to believe. Many subscribe to the “gambler’s fallacy” and believe that the coin’s deviations from a 50-50 split between heads and tails are governed by a probabilistic rubber band: the greater the deviation, the greater the equalizing push toward an even split. But even if Henry were way ahead, with 525 heads to Tommy’s 475 tails, his lead would be as likely to grow as to shrink. Likewise, a stock that’s fallen on a truly random trajectory is as likely to fall further as it is to rise.
The rarity with which the lead switches sides in no way contradicts the fact that the proportion of heads approaches 1/2 as the number of flips increases. Nor does it contradict the phenomenon of regression to the mean. If Henry and Tommy
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were to start over and flip their penny another 1,000 times, it’s quite likely that the number of heads would be smaller than 525.
Given the relative rarity with which Henry and Tommy overtake one another in their penny-flipping contest, it wouldn’t be surprising if one of them came to be known as a “winner” and the other a “loser” despite their complete lack of control over the penny. If one professional stock picker outperformed another by a margin of 525 to 475, he might even be interviewed on Moneyline or profiled in Fortune magazine. Yet he might, like Henry or Tommy, owe his success to nothing more than getting “stuck” by chance on the up side of a 50-50 split.
But what about such stellar “value investors” as Warren Buffet? His phenomenal success, like that of Peter Lynch, John Neff, and others, is often cited as an argument against the market’s randomness. This assumes, however, that Buffett’s choices have no effect on the market. Originally no doubt they didn’t, but now his selections themselves and his ability to create synergies among them can influence others. His performance is therefore a bit less remarkable than it first appears.
A different argument points to the near certainty of some stocks, funds, or analysts doing well over an extended period merely by chance. Of 1,000 stocks (or funds or analysts), for example, roughly 500 might be expected to outperform the market next year simply by chance, say by the flipping of a coin. Of these 500, roughly 250 might be expected to do well for a second year. And of these 250, roughly 125 might be expected to continue the pattern, doing well three years in a row simply by chance. Iterating in this way, we might reasonably expect there to be a stock (or fund or analyst) among the thousand that does well for ten consecutive years by chance alone. Once again, some in the business media are likely to go gaga over the performance.
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The surprising length and frequency of consecutive runs of heads or tails is yet another lesson of penny flipping. If Henry and Tommy were to continue flipping pennies once a day, then there’s a better-than-even chance that within about two months Henry will have won at least five flips in a row, as will Tommy. If they continue flipping for six years, there’s a better-than-even chance that each will have won at least ten flips in a row.
When people are asked to write down a series of heads and tails that simulates a series of coin flips, they almost always fail to include enough runs of consecutive heads or consecutive tails. In particular, they fail to include any very long runs of heads or tails, and their series are thus easily distinguishable from a real series of coin flips.
But try telling people that long streaks are due to chance alone, whether the streak is a basketball player’s shots, a stock analyst’s picks, or a series of coin flips. The fact is that random events can frequently seem quite ordered.
To literally see this, take out a large piece of paper and partition it into little squares in a checkerboard pattern. Flip a coin repeatedly and color the squares white or black depending upon whether the coin lands heads or tails. After the checkerboard has been completely filled in, look it over and see if you can discern any patterns or clusters of similarly colored squares. Chances are you will, and if you felt the need to explain these patterns, you would invent a story that might sound superficially plausible or intriguing, but, given how the colors were determined, would necessarily be false.
The same illusion of pattern would result if you were to graph (with time on the horizontal axis) the results of the coin flips, up one unit for a head, down one for a tail. Some chartists and technicians would no doubt see “head and shoulders,” “triple tops,” or “ascending channels” patterns in these zigzag, up-and-down movements, and they would expatiate
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on their significance. (One difference between coin flips and models of random stock movements is that in the latter it is generally assumed that stocks move up or down not by a fixed amount per unit time, but by a fixed percentage.)
Leaving aside, once again, the question whether the market is perfectly efficient or whether stock movements follow a truly random walk, we can nevertheless say that phenomena that are truly random often appear almost indistinguishable from real-market behavior. This should, but probably won’t, give pause to commentators who provide a neat post hoc explanation for every rally, every sell-off, and everything in between. Such commentators generally don’t make remarks analogous to the observation that the penny happened by chance to land heads a few more times than it did tails. Instead they will refer to Tommy’s profit-taking, Henry’s increased confidence, labor problems in the copper mines, or countless other factors.
Because so much information is available—business pages, companies’ annual reports, earnings expectations, alleged scandals, on-line sites, and commentary—something insightful- sounding can always be said. All we need do is filter the sea of numbers until we catch a plausible nugget of speculation. Like flipping a penny, doing so is a snap.
A Stock-Newsletter Scam
The accounting scandals involving WorldCom, Enron, and others derived from the data being selected, spun, and filtered. A scam I first discussed in my book Innumeracy derives instead from the recipients of the data being selected, spun, and filtered. It goes like this. Someone claiming to be the publisher of a stock newsletter rents a mailbox in a fancy neighborhood, has expensive stationery made up, and sends out letters to
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potential subscribers boasting of his sophisticated stockpicking software, financial acumen, and Wall Street connections. He writes also of his amazing track record, but notes that the recipients of his letters needn’t take his word for it.
Assume you are one of these recipients and for the next six weeks you receive correct predictions about a certain common stock index. Would you subscribe to the newsletter? What if you received ten consecutive correct predictions?
Here’s the scam. The newsletter publisher sends out 64,000 letters to potential subscribers. (Using email would save postage, but might appear to be a “spam scam” and hence be less credible.) To 32,000 of the recipients, he predicts the index in question will rise the following week and to the other 32,000, he predicts it will decline. No matter what happens to the index the next week, he will have made a correct prediction to
32.000 people. To 16,000 of them he sends another letter predicting a rise in the index for the following week, and to the other 16,000 he predicts a decline. Again, no matter what happens to the index the next week, he will have made correct predictions for two consecutive weeks to 16,000 people. To
8.000 of them he sends a third letter predicting a rise for the third week and to the other 8,000 he predicts a decline.
Focusing at each stage on the people to whom he’s made only correct predictions and winnowing out the rest, he iterates this procedure a few more times until there are 1,000 people left to whom he’s made six straight correct “predictions.” To these he sends a different sort of follow-up letter, pointing out his successes and saying that they can continue to receive these oracular pronouncements if they pay the $1,000 subscription price to the newsletter. If they all pay, that’s a million dollars for someone who need know nothing about stock, indices, trends, or dividends. If this is done knowingly, it is illegal. But what if it’s done unknowingly by earnest, confident, and ignorant newsletter publishers? (Compare the faithhealer who takes credit for any accidental improvements.)
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There is so much complexity in the market, there are so many different measures of success and ways to spin a story, that most people can manage to convince themselves that they’ve been, or are about to be, inordinately successful. If people are desperate enough, they’ll manage to find some seeming order in random happenings.
Similar to the newsletter scam, but with a slightly different twist, is a story related to me by an acquaintance who described his father’s business and its sad demise. He claimed that his father, years before, had run a large college-preparation service in a South American country whose identity I’ve forgotten. My friend’s father advertised that he knew how to drastically improve applicants’ chances of getting into the elite national university. Hinting at inside contacts and claiming knowledge of the various forms, deadlines, and procedures, he charged an exorbitant fee for his service, which he justified by offering a money-back guarantee to students who were not accepted.
One day, the secret of his business model came to light. All the material that prospective students had sent him over the years was found unopened in a trash dump. Upon investigation it turned out that he had simply been collecting the students’ money (or rather their parents’ money) and doing nothing for it. The trick was that his fees were so high and his marketing so focused that only the children of affluent parents subscribed to his service, and almost all of them were admitted to the university anyway. He refunded the fees of those few who were not admitted. He was also sent to prison for his efforts.
Are stock brokers in the same business as my acquaintance’s father? Are stock analysts in the same business as the newsletter publisher? Not exactly, but there is scant evidence that they possess any unusual predictive powers. That’s why I thought news stories in November 2002 recounting New York Attorney General Eliot Spitzer’s criticism of Institutional Investor magazine’s analyst awards were a tad superfluous. Spitzer noted that the stock-picking performances of
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most of the winning analysts were, in fact, quite mediocre. Maybe Donald Trump will hold a press conference pointing out that the country’s top gamblers don’t do particularly well at roulette.
Decimals and Other Changes
Like analysts and brokers, market makers (who make their money on the spread between the bid and the ask price for a stock) have received more than their share of criticism in recent years. One result has been a quiet reform that makes the market a bit more efficient. Wall Street’s surrender to radical “decacrats” occurred a couple of years ago, courtesy of a Congressional mandate and a direct order from the Securities and Exchange Commission. Since then stock prices have been expressed in dollars and cents, and we no longer hear “profittaking drove XYZ down 2 and 1/8” or “news of the deal sent PQR up 4 and 5/16.”
Although there may be less romance associated with declines of 2.13 and rises of 4.31, decimalization makes sense for a number of reasons. The first is that price rises and declines are immediately comparable since we no longer must perform the tiresome arithmetic of, say, dividing 11 by 16. Mentally calculating the difference between two decimals generally requires less time than subtracting 3 5/8 from 5 3/16. Another benefit is global uniformity of pricing, as American securities are now denominated in the same decimal units as those in the rest of the world. Foreign securities no longer need to be rounded to the nearest multiple of 1/16, a perverse arithmetical act if there ever was one.
More importantly, the common spread between the bid and ask prices has shrunk. Once generally 1/16 (.0625, that is), the spread in many cases has become .01 and, by so shrivA
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eling, will save investors billions of dollars over the years. Market makers aside, most investors applaud this consequence of decimalization.
The last reason for cheering the change is more mathematical. There is a sense in which the old system of halves, quarters, eighths, and sixteenths is more natural than decimals. It is, after all, only a slightly disguised binary system, based on powers of 2 (2, 4, 8, 16) rather than powers of 10. It doesn’t inherit any of the prestige of the binary system, however, because it awkwardly combines the base 2 fractional part of a stock price with the base 10 whole-number part.
Thus it is that Ten extends its imperial reach to Wall Street. From the biblical Commandments to David Letterman’s lists, the number 10 is ubiquitous. Not unrelated to the perennial yearning for the simplicity of the metric system, 10 envy has also come to be associated with rationality and efficiency. It is thus fitting that all stocks are now expressed in decimals. Still, I suspect that many market veterans miss those pesky fractions and their role in stories of past killings and baths. Except for generation X-ers (Roman numeral ten-ers), many others will too. Anyway, that’s my two cents (.02, l/50th) worth on the subject.
The replacement of marks, francs, drachmas, and other European currencies by euros on stock exchanges and in stores is another progressive step that nevertheless rouses a touch of nostalgia. The coins and bills from my past travels that are scattered about in drawers are suddenly out of work and will never see the inside of a wallet again.
Yet another vast change in trading practices is the greater self-reliance among investors. Despite the faulty accounting that initially disguised their sickly returns, the ladies of Beardstown, Illinois, helped popularize investment clubs. Even more significant in this regard is the advent of effortless online trading, which has further hastened the decline of the
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traditional broker. The ease with which I clicked on simple icons to buy and sell (specifically sell reasonably performing funds and buy more WCOM shares) was always a little frightening, and I sometimes felt as if there were a loaded gun on my desk. Some studies have linked online trading and day trading to increased volatility in the late ’90s, although it’s not clear that they remain factors in the ’00s.
What’s undeniable is that buying and selling online remains easy, so easy that I think it might not be a bad idea were small pictures of real-world items to pop up before every stock purchase or sale as a reminder of the approximate value of what’s being traded. If your transaction were for $35,000, a luxury car might appear; if it were for $100,000, a small cottage; and if it were for a penny stock, a candy bar. Investors can now check stock quotations, the size and the number of the bids and the asks, and megabytes of other figures on so- called level-two screens available in (almost) real-time on their personal computers. Millions of little desktop brokerages! Unfortunately, librarian Jesse Sherra’s paraphrase of Coleridge often seems apt: Data, data everywhere, but not a thought to think.
Benford's Law and Looking Out for Number One
I mentioned that people find it very difficult to simulate a series of coin flips. Are there other human disabilities that might allow someone to look at a company’s books, say Enron’s or WorldCom’s, and determine whether or not they had been cooked? There may have been, and the mathematical principle involved is easily stated, but counterintuitive.
Benford’s Law states that in a wide variety of circumstances, numbers—as diverse as the drainage areas of rivers, physical properties of chemicals, populations of small towns,
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figures in a newspaper or magazine, and the half-lives of radioactive atoms—have “ 1 ” as their first non-zero digit disproportionately often. Specifically, they begin with “1” about 30 percent of the time, with “2” about 18 percent of the time, with “3” about 12.5 percent, and with larger digits progressively less often. Less than 5 percent of the numbers in these circumstances begin with the digit “9.” Note that this is in stark contrast to many other situations where each of the digits has an equal chance of appearing.
Benford’s Law goes back one hundred years to the astronomer Simon Newcomb (note the letters WCOM in his name), who noticed that books of logarithm tables were much dirtier near the front, indicating that people more frequently looked up numbers with a low first digit. This odd phenomenon remained a little-known curiosity until it was rediscovered in 1938 by the physicist Frank Benford. It wasn’t until 1996, however, that Ted Hill, a mathematician at Georgia Tech, established what sort of situations generate numbers in accord with Benford’s Law. Then a mathematically inclined accountant named Mark Nigrini generated considerable buzz when he noted that Benford’s Law could be used to catch fraud in income tax returns and other accounting documents.
The following example suggests why collections of numbers governed by Benford’s Law arise so frequently:
Imagine that you deposit $1,000 in a bank at 10 percent compound interest per year. Next year you’ll have $1,100, the year after that $1,210, then $1,331, and so on. (Compounding is discussed further in chapter 5.) The first digit of your account balance remains a “1” for a long time. When your account grows to over $2,000, the first digit will remain a “2” for a shorter period. And when your deposit finally grows to over $9,000, the 10 percent growth will result in more than $10,000 in your account the following year and a long return to “1” as the first digit. If you record your account balance
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each year for many years, these numbers will thus obey Ben- ford’s Law.
The law is also “scale-invariant” in that the dimensions of the numbers don’t matter. If you expressed your $1,000 in euros or pounds (or the now defunct francs or marks) and watched it grow at 10 percent per year, about 30 percent of the yearly values would begin with a “1,” about 18 percent with a “2,” and so on.
More generally, Hill showed that such collections of numbers arise whenever we have what he calls a “distribution of distributions,” a random collection of random samples of data. Big, motley collections of numbers will follow Benford’s Law.
This brings us back to Enron, WorldCom, accounting, and Mark Nigrini, who reasoned that the numbers on accounting forms, which often come from a variety of company operations and a variety of sources, should be governed by Benford’s Law. That is, these numbers should begin disproportionately with the digit “1,” and progressively less often with bigger digits, and if they don’t, that is a sign that the books have been cooked. When people fake plausible-seeming numbers, they generally use more “5s” and “6s” as initial digits, for example, than Benford’s Law would predict.
Nigrini’s work has been well publicized and has surely been noted by accountants and by prosecutors. Whether the Enron, WorldCom, and Anderson people have heard of it is unknown, but investigators might want to check if the distribution of leading digits in the Enron documents accords with Benford’s Law. Such checks are not foolproof and sometimes lead to false-positive results, but they provide an extra tool that might be useful in certain situations.
It would be amusing if, in looking out for number one, the culprits forgot to look out for their “Is.” Imagine the Anderson accountants muttering anxiously that there weren’t enough leading “Is” on the documents they were feeding into the shredders. A 1-derful fantasy!
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The Numbers Man—A Screen Treatment
An astonishing amount of attention has been paid recently to fictional and narrative treatments of mathematical topics. The movies Good Will Hunting, Pi, and The Croupier come to mind; so do plays such as Copenhagen, Arcadia, and The Proof, the two biographies of Paul Erdos, A Beautiful Mind, the biography of John Nash (with its accompanying Academy Award-winning movie), TV specials on Fermat’s Last Theorem, and other mathematical topics, as well as countless books on popular mathematics and mathematicians. The plays and movies, in particular, prompted me to expand the idea in the stock-newsletter scam discussed above (I changed the focus, however, from stocks to sports) into a sort of abbreviated screen treatment that highlights the relevant mathematics a bit more than has been the case in the productions just cited. Yet another instance of what columnist Charles Krauthammer has dubbed “Disturbed Nerd Chic,” the treatment might even be developed into an intriguing and amusing film. In fact, I rate it a “strong buy” for any studio executive or independent filmmaker.
Rough Idea: Math nerd runs a clever sports-betting scam and accidentally nets an innumerate mobster.
Act One
Louis is a short, lecherous, somewhat nerdy man who dropped out of math graduate school about ten years ago (in the late ’80s) and now works at home as a technical consultant. He looks and acts a bit like the young Woody Allen. He’s playing cards with his pre-teenage kids and has just finished telling them a funny story. His kids are smart and they ask him how it is that he always knows the right story to tell. His wife, Marie, is uninterested. True to form, he begins telling them the Leo Rosten story about the famous rabbi who was asked by an
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admiring student how it was that the rabbi always had a perfect parable for any subject. Louis pauses to make sure they see the relevance.
When they smile and his wife rolls her eyes again, he continues. He tells them that the rabbi replied to his students with a parable. It was about a recruiter in the Tsar’s army who was riding through a small town and noticed dozens of chalked circular targets on the side of a barn, each with a bullet hole through the bull’s-eye. The recruiter was impressed and asked a neighbor who this perfect shooter might be. The neighbor responded, “Oh that’s Shepsel, the shoemaker’s son. He’s a little peculiar.” The enthusiastic recruiter was undeterred until the neighbor added, “You see, first Shepsel shoots and then he draws the chalk circles around the bullet hole.” The rabbi grinned. “That’s the way it is with me. I don’t look for a parable to fit the subject. I introduce only subjects for which I have parables.”
Louis and his kids laugh until a distracted, stricken look crosses his face. Closing the book, Louis hurries his kids off to bed, interrupts Marie’s prattling about her new pearl necklace and her Main Line parents’ nasty neighbors, distractedly bids her good night, and retreats to his study where he starts scribbling, making calls, and performing calculations. The next day he stops by the bank and the post office and a stationery store, does some research online, and then has a long discussion with his friend, a sportswriter on the local suburban New Jersey newspaper. The conversation revolves around the names, addresses, and intelligence of big sports bettors around the country.
The idea for a lucrative con game has taken shape in his mind. For the next several days he sends letters and emails to many thousands of known sports bettors “predicting” the outcome of a certain sporting event. His wife is uncomprehending when Louis mumbles that, Shepsel-like, he can’t lose
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since whatever happens in the sporting event, his prediction is bound to be right for half the bettors. The reason, it will turn out, is that to half of these people he predicts a certain team will win, and to the other half he predicts that it will lose.
Tall, blond, plain, and dim-witted, Marie is left wondering what exactly her sneaky husband is up to now. She finds the new postage meter behind the computer, notes the increasingly frequent secret telephone calls, and nags him about their worsening financial and marital situation. He replies that she doesn’t really need three closets full of clothes and a small fortune of jewelry when she spends all her time watching soaps and puts her off with some mathematical mumbo-jumbo about demographic research and new statistical techniques. She still doesn’t follow, but she is mollified by his promise that his mysterious endeavor will end up being lucrative.
They go out to eat to celebrate and Louis, intense and cad- like as always, talks up genetically modified food and tells the cute waitress that he wants to order whatever item on the menu has the most artificial ingredients. Much to Marie’s chagrin, he then involves the waitress in a classic mathematical trick by asking her to examine his three cards, one black on both sides, one red on both sides, and one black on one side and red on the other. He asks her for her cap, drops the cards into it, and tells her to pick a card, but only to look at one side of it. The side is red, and Louis notes that the card she picked couldn’t possibly be the card that was black on both sides, and therefore it must be one of the other two cards—the red-red card or the red-black card. He guesses that it’s the red-red card and offers to double her 15 percent tip if it’s the red-black card and stiff her if it’s the red-red card. He looks at Marie for approbation that is not forthcoming. The waitress accepts and loses.
Tone-deaf to Marie’s discomfort, Louis thinks he’s making amends with her by explaining the trick. She is less than
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enthralled. He tells her that it’s not an even bet even though at first glance it appears to be one. There are, after all, two cards it could be, and he bet on one, and the waitress bet on the other. The rub is, he gleefully runs on with his mouth full, there are two ways he can win and only one way the waitress can win. The visible side of the card the waitress picked could be the red side of the red-black card, in which case she wins, or it could be one side of the red-red card, in which case he wins, or it could be the other side of the red-red card, in which case he also wins. His chances of winning are thus 2/3, he concludes exultantly, and the average tip he gives is reduced by a third. Marie yawns and checks her Rolex. He breaks to go to the men’s room where he calls his girlfriend May Lee to apologize for some vague indiscretion.
The next week he explains the sports-betting con to May Lee, who looks a bit like Lucy Liu and is considerably smarter than Marie and even more materialistic. They’re in her apartment. She is interested in the con and asks clarifying questions. He enthuses to her that he needs her secretarial help. He’s sending out more letters and making a second prediction in them, but this time just to the half of the people to whom he sent a correct first prediction; the other half he plans to ignore. To half of this smaller group, he will predict a win in a second sporting event, to the other half a loss. Again for half of this group his prediction is going to be right, and so for one-fourth of the original group he’s going to be right two times in a row. “And to this one-fourth of the bettors?” she asks knowingly and excitedly. A mathematico-sexual tension develops.
He smiles rakishly and continues. To half of this one-fourth he will predict a win the following week, to the other half a loss; he again will ignore those to whom he’s made an incorrect prediction. Once again he will be right—this time for the third straight time—although for only one-eighth of the original population. May Lee helps with the mailings as he continues
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this process, focusing only on those to whom he’s made correct predictions and winnowing out those to whom he’s made incorrect ones. There is a sex scene amid all the letters, and they joke about winning whether the teams in question do or not, whether the predictions are right or wrong. Whether up or down, they’ll be happy.
As the mailings go on, so does his other life as a bored consultant, cyber-surfer, and ardent sports fan. He continues to extend his string of successful predictions to a smaller and smaller group of people until finally with great anticipation he sends a letter to the small group of people who are left. In it he points to his impressive string of successes and requests a substantial payment to keep these valuable and seemingly oracular “predictions” coming.
Act Two
He receives many payments and makes a further prediction. Again he’s right for half of the remaining people and drops the half for which he’s wrong. He asks the former group for even more money for another prediction, receives it, and continues. Things improve with Marie and with May Lee as the money rolls in and Louis realizes his plan is working even better than he expected. He takes his kids and, in turn, each woman to sports events or to Atlantic City, where he comments smugly on the losers who, unlike him, bet on iffy propositions. When Marie worries aloud about shark attacks off the beach, Louis tells her that more Americans die from falling airplane parts each year than from shark attacks. He makes similar pronouncements throughout the trip.
He plays a little blackjack and counts cards while doing so. He complains that it requires too much low-level concentration and that, unless one has a lot of money already, the rate at which one makes money is so slow and uneven that one might as well get a job. Still, he goes on, it’s the only game
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where a strategy exists for winning. All the other games are for mentally flabby losers. He goes to one of the casino restaurants where he shows his kids the waitress tip-cheating game. They think it’s great.
Back home in suburban New Jersey again, the sports- betting con resumes. Now there are only a few people left among the original thousands of sports-bettors. One of them, a rough underworld type named Otto, tracks him down, follows him to the parking lot of the basketball arena, and, politely at first and then more and more insistently, demands a prediction on an upcoming game on which he plans to bet a lot of money. Louis dismisses him and Otto, who looks a little like Stephen Segal, promptly orders him into his car at gunpoint and threatens to harm his family. He knows where they live.
Not understanding how he could be the recipient of so many consecutive correct predictions, Otto doesn’t believe Louis’s protestations that this is a con game. Louis makes some mathematical points in an effort to convince Otto of the possible falsity of any particular prediction. But no matter how he tries, he can’t quite convince Otto of the fact that there will always be some people who receive many consecutive correct predictions by chance alone.
Marooned in Otto’s basement, the math-nerd scam artist and the bald muscled extortionist are a study in contrasts. They speak different languages and have different frames of reference. Otto claims, for example, that every bet is more or less a 50-50 proposition because you either win or lose. Louis talks of his basketball buddies Lewis Carroll and Bertrand Russell and the names go over Otto’s head, of course. Oddly, they have similar attitudes toward women and money and also share an interest in cards, which they play to while away the time. Otto proudly shows off his riffle shuffle that he claims completely mixes the cards, while Louis prefers soliA
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taire and silently scoffs at Otto’s lottery expenditures and gambling misconceptions. When they forget why they’re there they get along well enough, although now and then Otto renews his threats and Louis renews his disavowal of any special sports knowledge and his plea to go home.
Finally granting that he might receive an incorrect prediction occasionally, Otto still insists that Louis give him his take on who’s going to win an upcoming football game. In addition to not being too bright, Otto, it appears, is in serious debt. Under extreme duress (with a gun to his head), Louis makes a prediction that happens to be right, and Otto, desperate and still convinced that he is in control of a money tree, now wants to bet funds borrowed from his gambling associates on Louis’s next prediction.
Act Three
Louis at last convinces Otto to let him go home and do research for his next big sports prediction. He and May Lee, whose need for money, baubles, and clothes has all along provided the impetus for the scam, discuss his predicament and realize they must exploit Otto’s only weaknesses, his stupidity and gullibility, and his only intellectual interests, money and playing cards.
Both go over to Otto’s apartment. Otto is charmed by May Lee, who flirts with him and offers him a deal. She wordlessly takes two decks of cards from her purse and asks Otto to shuffle each of them. Otto is pleased to show off to a more appreciative audience. She then gives him one of the decks and asks him to turn over one card at a time as she, keeping pace with him, does the same thing with the other deck. May Lee asks, what does he think is the likelihood that the cards they turn over will ever match, denomination and suit exactly the same? He scoffs but is entranced by May Lee and is amazed when after a tense minute or so that is exactly what
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happens. She explains that it will happen more often than not and perhaps he can use this fact to make some money. After all, Louis is a mathematical genius and he’s proved that it will. Louis smiles proudly.
Otto is puzzled. May Lee tells Otto again that the sports- betting was a scam and that he’s more likely to make money with the card tricks that Louis can teach him. Louis steps forward with the same two decks, which he’s now arranged so that the cards in each deck alternate colors. In one, it’s red- black, red-black, red-black .... In the other it’s black-red, black-red, black-red .... He gives the two decks to Otto and challenges him to do one of his great riffle shuffles of one deck into the other so that the cards will be mixed. Otto does and arrogantly announces that the cards are completely mixed now, whereupon Louis takes the combined two decks, puts them behind his back, pretends to be manipulating them, and brings forward two cards, one black and one red. So, Otto asks? Louis brings forth two more cards, one of each color, and then he does this again and again. I really shuffled them, Otto observes. How’d you do that? Louis explains that it involves no skill; the cards no longer alternate color in the combined deck, but any two from the top on down are always of different color.
There is a collage scene in which Louis explains various card tricks to Otto and the ways in which they can be exploited to make money. There’s always some order, some deviation from randomness, that a card man like you can use to get rich, Louis says to Otto. He even explains to him how he avoids paying waitresses tips. The deal, of course, is that Otto releases them, understanding, vaguely at least, how the betting scheme works and, more precisely, how the new card tricks do. Louis promises a one-day crash course on how to exploit the tricks for money.
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In the last scene Louis is seen working the same scam but this time with predictions about the movements of a stock market index. Since he doesn’t want any more Ottos, but a higher class clientele, he’s redefined himself as the publisher of a stock newsletter. The house he’s in is more sumptuous and May Lee, to whom he’s now married, bustles about in an expensive suit as Louis plays cards with his slightly older children, occasionally doodling little bull’s-eyes and targets on an envelope. He excuses himself and goes to his study to make a secret telephone call to an apartment on Central Park West that he’s just purchased for his new mistress.
Even some social scientists don’t seem to realize that if you search for a correlation between any two randomly selected attributes in a very large population, you will likely find some small but statistically significant association. It doesn’t matter if the attributes are ethnicity and hip circumference, or (some measure of) anxiety and hair color, or perhaps the amount of sweet corn consumed annually and the number of mathematics courses taken. Despite the correlation’s statistical significance (its unlikelihood of occurring by chance), it is probably not practically significant because of the presence of so many confounding variables. Furthermore, it will not necessarily support the (often ad hoc) story that accompanies it, the one purporting to explain why people who eat a lot of corn take more math. Superficially plausible tales are always available: Corn-eaters are more likely to be from the upper Midwest, where dropout rates are low.
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Geniuses, Idiots, or Neither
Around stock market rises and declines, people are prone to devise just-so stories to satisfy various needs and concerns. During the bull markets of the ’90s investors tended to see themselves as “perspicacious geniuses.” During the more recent bear markets they’ve tended toward self-descriptions such as “benighted idiots.”
My own family is not immune to the temptation to make up pat after-the-fact stories explaining past financial gains and losses. When I was a child, my grandfather would regale me with anecdotes about topics as disparate as his childhood in Greece, odd people he’d known, and the exploits of the Chicago White Sox and their feisty second baseman “Fox Nelson” (whose real name was Nelson Fox). My grandfather was voluble, funny, and opinionated. Only rarely and succinctly, however, did he refer to the financial reversal that shaped his later life. As a young and uneducated immigrant, he worked in restaurants and candy stores. Over the years he managed to buy up eight of the latter and two of the former. His candy stores required sugar, which led him eventually to speculate in sugar markets and—he was always a bit vague about the details—to place a big bet on several train cars full of sugar. He apparently put everything he had into the deal a few weeks before the sugar market crashed. Another version attributed his loss to underinsurance of the sugar shipment. In any case, he lost it all and never really recovered financially. I remember him saying ruefully, “Johnny, I would have been a very, very rich man. I should have known.” The bare facts of the story registered with me then, but my recent less calamitous experience with WorldCom has made his pain more palpable.
This powerful natural proclivity to invest random events with meaning on many different levels makes us vulnerable to people who tell engaging stories about these events. In the
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Rorschach blot that chance provides us, we often see what we want to see or what is pointed out to us by business prognosticators, distinguishable from carnival psychics only by the size of their fees. Confidence, whether justified or not, is convincing, especially when there aren’t many “facts of the matter.” This may be why market pundits seem so much more certain than, say, sports commentators, who are comparatively frank in acknowledging the huge role of chance.
Efficiency and Random Walks
The Efficient Market Hypothesis formally dates from the 1964 dissertation of Eugene Fama, the work of Nobel prizewinning economist Paul Samuelson, and others in the 1960s. Its pedigree, however, goes back much earlier, to a dissertation in 1900 by Louis Bachelier, a student of the great French mathematician Henri Poincare. The hypothesis maintains that at any given time, stock prices reflect all relevant information about the stock. In Fama’s words: “In an efficient market, competition among the many intelligent participants leads to a situation where, at any point in time, actual prices of individual securities already reflect the effects of information based both on events that have already occurred and on events which, as of now, the market expects to take place in the future.”
There are various versions of the hypothesis, depending on what information is assumed to be reflected in the stock price. The weakest form maintains that all information about past market prices is already reflected in the stock price. A consequence of this is that all of the rules and patterns of technical analysis discussed in chapter 3 are useless. A stronger version maintains that all publicly available information about a company is already reflected in its stock price. A consequence
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of this version is that the earnings, interest, and other elements of fundamental analysis discussed in chapter 5 are useless. The strongest version maintains that all information of all sorts is already reflected in the stock price. A consequence of this is that even inside information is useless.
It was probably this last, rather ludicrous version of the hypothesis that prompted the joke about the two efficient market theorists walking down the street: They spot a hundred dollar bill on the sidewalk and pass by it, reasoning that if it were real, it would have been picked up already. And of course there is the obligatory light-bulb joke. Question: How many efficient market theorists does it take to change a light bulb? Answer: None. If the light bulb needed changing the market would have already done it. Efficient market theorists tend to believe in passive investments such as broad-gauged index funds, which attempt to track a given market index such as the S&P 500. John Bogle, the crusading founder of Vanguard and presumably a believer in efficient markets, was the first to offer such a fund to the general investing public. His Vanguard 500 fund is unmanaged, offers broad diversification and very low fees, and generally beats the more expensive, managed funds. Investing in it does have a cost, however: One must give up the fantasy of a perspicacious gunslinger/investor outwitting the market.
And why do such theorists believe the market to be efficient? They point to a legion of investors of all sorts all seeking to make money by employing all sorts of strategies. These investors sniff out and pounce upon any tidbit of information even remotely relevant to a company’s stock price, quickly driving it up or down. Through the actions of this investing horde the market rapidly responds to the new information, efficiently adjusting prices to reflect it. Opportunities to make an excess profit by utilizing technical rules or fundamental analyses, so the story continues, disappear before they can be
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fully exploited, and investors who pursue them will see their excess profits shrink to zero, especially after taking into account brokers’ fees and other transaction costs. Once again, it’s not that subscribers to technical or fundamental analysis won’t make money; they generally will. They just won’t make more than, say, the S&P 500.
(That exploitable opportunities tend to gradually disappear is a general phenomenon that occurs throughout economics and in a variety of fields. Consider an argument about baseball put forward by Steven Jay Gould in his book Full House: The Spread of Excellence front Plato to Darwin. The absence of .400 hitters in the years since Ted Williams hit .406 in 1941, he maintained, was not due to any decline in baseball ability but the reverse: a gradual increase in the athleticism of all players and a consequent decrease in the disparity between the worst and best players. When players are as physically gifted and well trained as they are now, the distribution of batting averages and earned run averages shows less variability. There are few “easy” pitchers for hitters and few “easy” hitters for pitchers. One result is that .400 averages are now very scarce. The athletic prowess of hitters and pitchers makes the “market” between them more efficient.)
There is, moreover, a close connection between the Efficient Market Hypothesis and the proposition that the movement of stock prices is random. If present stock prices already reflect all available information (that is, if the information is common knowledge in the sense of chapter 1), then future stock prices must be unpredictable. Any news that might be relevant in predicting a stock’s future price has already been weighed and responded to by investors whose buying and selling have adjusted the present price to reflect the news. Oddly enough, as markets become more efficient, they tend to become less predictable. What will move stock prices in the future are truly new developments (or new shadings of old
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developments), news that is, by definition, impossible to anticipate. The conclusion is that in an efficient market, stock prices move up and down randomly. Evincing no memory of their past, they take what is commonly called a random walk, each step of which is independent of past steps. There is over time, however, an upward trend, as if the coin being flipped were slightly biased.
There is a story I’ve always liked that is relevant to the impossibility of anticipating new developments. It concerns a college student who completed a speed-reading course. He noted this fact in a letter to his mother. His mother responded with a long, chatty letter of her own in the middle of which she wrote, “Now that you’ve taken that speed-reading course, you’ve probably finished reading this letter by now.”
Likewise, true scientific breakthroughs or applications, by definition, cannot be foreseen. It would be preposterous to have expected a newspaper headline in 1890 proclaiming “Only 15 Years Until Relativity.” It is similarly foolhardy, the efficient market theorist reiterates, to predict changes in a company’s business environment. To the extent these predictions reflect a consensus of opinion, they’re already accounted for. To the extent that they don’t, they’re tantamount to forecasting coin flips.
Whatever your views on the subject, the arguments for an efficient market spelled out in Burton Malkiel’s A Random Walk Down Wall Street and elsewhere can’t be grossly wrong. After all, most mutual fund managers continue to generate average gains less than those of, say, the Vanguard Index 500 fund. (This has always seemed to me a rather scandalous fact.) There is other evidence for a fairly efficient market as well. There are few opportunities for risk-free money-making or arbitrage, prices seem to adjust rapidly in response to news, and the autocorrelation of the stock prices from day to day, week to week, month to month, and year to year is small (albeit not
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zero). That is, if the market has done well (or poorly) over a given time period in the past, there is no strong tendency for it to do well (or poorly) during the next time period.
Nevertheless, in the last few years I have qualified my view of the Efficient Market Hypothesis and random-walk theory. One reason is the accounting scandals involving Enron, Adel- phia, Global Crossing, Qwest, Tyco, WorldCom, Andersen, and many others from corporate America’s Hall of Infamy, which make it hard to believe that available information about a stock always quickly becomes common knowledge.
Pennies and the Perception of Pattern
The Wall Street Journal has famously conducted a regular series of stock-picking contests between a rotating collection of stock analysts, whose selections are a result of their own studies, and dart-throwers, whose selections are determined randomly. Over many six-month trials, the pros’ selections have performed marginally better than the darts’ selections, but not overwhelmingly so, and there is some feeling that the pros’ picks may influence others to buy the same stocks and hence drive up their price. Mutual funds, although less volatile than individual stocks, also display a disregard for analysts’ pronouncements, often showing up in the top quarter of funds one year and in the bottom quarter the next.
Whether or not you believe in efficient markets and the random movement of stock prices, the huge element of chance present in the market cannot be denied. For this reason an examination of random behavior sheds light on many market phenomena. (So does study of a standard tome on probability such as that by Sheldon Ross.) Sources for such random behavior are penny stocks or, more accessible and more random, stocks of pennies, so let’s imagine flipping a
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penny repeatedly and keeping track of the sequence of heads and tails. We’ll assume the coin and the flip are fair (although, if we wish, the penny can be altered slightly to reflect the small upward bias of the market over time).
One odd and little-known fact about such a series of coin flips concerns the proportion of time that the number of heads exceeds the number of tails. It’s seldom close to 50 percent!
To illustrate, imagine two contestants, Henry and Tommy, who bet that heads and tails respectively will be the outcome of a daily coin flip, a ritual that goes on for years. (Let’s not ask why.) Henry is ahead on any given day if up to that day there have been more heads than tails, and Tommy is ahead if up to that day there have been more tails. The coin is fair, so they’re equally likely to be in the lead, but one of them will probably be in the lead during most of their rather stultifying contest.
Stated numerically, the claim is that if there have been
1,000 coin flips, then it’s considerably more probable that Henry (or Tommy) has been ahead more than, say, 96 percent of the time than that either one has been ahead between 48 percent and 52 percent of the time.
People find this result hard to believe. Many subscribe to the “gambler’s fallacy” and believe that the coin’s deviations from a 50-50 split between heads and tails are governed by a probabilistic rubber band: the greater the deviation, the greater the equalizing push toward an even split. But even if Henry were way ahead, with 525 heads to Tommy’s 475 tails, his lead would be as likely to grow as to shrink. Likewise, a stock that’s fallen on a truly random trajectory is as likely to fall further as it is to rise.
The rarity with which the lead switches sides in no way contradicts the fact that the proportion of heads approaches 1/2 as the number of flips increases. Nor does it contradict the phenomenon of regression to the mean. If Henry and Tommy
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were to start over and flip their penny another 1,000 times, it’s quite likely that the number of heads would be smaller than 525.
Given the relative rarity with which Henry and Tommy overtake one another in their penny-flipping contest, it wouldn’t be surprising if one of them came to be known as a “winner” and the other a “loser” despite their complete lack of control over the penny. If one professional stock picker outperformed another by a margin of 525 to 475, he might even be interviewed on Moneyline or profiled in Fortune magazine. Yet he might, like Henry or Tommy, owe his success to nothing more than getting “stuck” by chance on the up side of a 50-50 split.
But what about such stellar “value investors” as Warren Buffet? His phenomenal success, like that of Peter Lynch, John Neff, and others, is often cited as an argument against the market’s randomness. This assumes, however, that Buffett’s choices have no effect on the market. Originally no doubt they didn’t, but now his selections themselves and his ability to create synergies among them can influence others. His performance is therefore a bit less remarkable than it first appears.
A different argument points to the near certainty of some stocks, funds, or analysts doing well over an extended period merely by chance. Of 1,000 stocks (or funds or analysts), for example, roughly 500 might be expected to outperform the market next year simply by chance, say by the flipping of a coin. Of these 500, roughly 250 might be expected to do well for a second year. And of these 250, roughly 125 might be expected to continue the pattern, doing well three years in a row simply by chance. Iterating in this way, we might reasonably expect there to be a stock (or fund or analyst) among the thousand that does well for ten consecutive years by chance alone. Once again, some in the business media are likely to go gaga over the performance.
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The surprising length and frequency of consecutive runs of heads or tails is yet another lesson of penny flipping. If Henry and Tommy were to continue flipping pennies once a day, then there’s a better-than-even chance that within about two months Henry will have won at least five flips in a row, as will Tommy. If they continue flipping for six years, there’s a better-than-even chance that each will have won at least ten flips in a row.
When people are asked to write down a series of heads and tails that simulates a series of coin flips, they almost always fail to include enough runs of consecutive heads or consecutive tails. In particular, they fail to include any very long runs of heads or tails, and their series are thus easily distinguishable from a real series of coin flips.
But try telling people that long streaks are due to chance alone, whether the streak is a basketball player’s shots, a stock analyst’s picks, or a series of coin flips. The fact is that random events can frequently seem quite ordered.
To literally see this, take out a large piece of paper and partition it into little squares in a checkerboard pattern. Flip a coin repeatedly and color the squares white or black depending upon whether the coin lands heads or tails. After the checkerboard has been completely filled in, look it over and see if you can discern any patterns or clusters of similarly colored squares. Chances are you will, and if you felt the need to explain these patterns, you would invent a story that might sound superficially plausible or intriguing, but, given how the colors were determined, would necessarily be false.
The same illusion of pattern would result if you were to graph (with time on the horizontal axis) the results of the coin flips, up one unit for a head, down one for a tail. Some chartists and technicians would no doubt see “head and shoulders,” “triple tops,” or “ascending channels” patterns in these zigzag, up-and-down movements, and they would expatiate
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on their significance. (One difference between coin flips and models of random stock movements is that in the latter it is generally assumed that stocks move up or down not by a fixed amount per unit time, but by a fixed percentage.)
Leaving aside, once again, the question whether the market is perfectly efficient or whether stock movements follow a truly random walk, we can nevertheless say that phenomena that are truly random often appear almost indistinguishable from real-market behavior. This should, but probably won’t, give pause to commentators who provide a neat post hoc explanation for every rally, every sell-off, and everything in between. Such commentators generally don’t make remarks analogous to the observation that the penny happened by chance to land heads a few more times than it did tails. Instead they will refer to Tommy’s profit-taking, Henry’s increased confidence, labor problems in the copper mines, or countless other factors.
Because so much information is available—business pages, companies’ annual reports, earnings expectations, alleged scandals, on-line sites, and commentary—something insightful- sounding can always be said. All we need do is filter the sea of numbers until we catch a plausible nugget of speculation. Like flipping a penny, doing so is a snap.
A Stock-Newsletter Scam
The accounting scandals involving WorldCom, Enron, and others derived from the data being selected, spun, and filtered. A scam I first discussed in my book Innumeracy derives instead from the recipients of the data being selected, spun, and filtered. It goes like this. Someone claiming to be the publisher of a stock newsletter rents a mailbox in a fancy neighborhood, has expensive stationery made up, and sends out letters to
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potential subscribers boasting of his sophisticated stockpicking software, financial acumen, and Wall Street connections. He writes also of his amazing track record, but notes that the recipients of his letters needn’t take his word for it.
Assume you are one of these recipients and for the next six weeks you receive correct predictions about a certain common stock index. Would you subscribe to the newsletter? What if you received ten consecutive correct predictions?
Here’s the scam. The newsletter publisher sends out 64,000 letters to potential subscribers. (Using email would save postage, but might appear to be a “spam scam” and hence be less credible.) To 32,000 of the recipients, he predicts the index in question will rise the following week and to the other 32,000, he predicts it will decline. No matter what happens to the index the next week, he will have made a correct prediction to
32.000 people. To 16,000 of them he sends another letter predicting a rise in the index for the following week, and to the other 16,000 he predicts a decline. Again, no matter what happens to the index the next week, he will have made correct predictions for two consecutive weeks to 16,000 people. To
8.000 of them he sends a third letter predicting a rise for the third week and to the other 8,000 he predicts a decline.
Focusing at each stage on the people to whom he’s made only correct predictions and winnowing out the rest, he iterates this procedure a few more times until there are 1,000 people left to whom he’s made six straight correct “predictions.” To these he sends a different sort of follow-up letter, pointing out his successes and saying that they can continue to receive these oracular pronouncements if they pay the $1,000 subscription price to the newsletter. If they all pay, that’s a million dollars for someone who need know nothing about stock, indices, trends, or dividends. If this is done knowingly, it is illegal. But what if it’s done unknowingly by earnest, confident, and ignorant newsletter publishers? (Compare the faithhealer who takes credit for any accidental improvements.)
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There is so much complexity in the market, there are so many different measures of success and ways to spin a story, that most people can manage to convince themselves that they’ve been, or are about to be, inordinately successful. If people are desperate enough, they’ll manage to find some seeming order in random happenings.
Similar to the newsletter scam, but with a slightly different twist, is a story related to me by an acquaintance who described his father’s business and its sad demise. He claimed that his father, years before, had run a large college-preparation service in a South American country whose identity I’ve forgotten. My friend’s father advertised that he knew how to drastically improve applicants’ chances of getting into the elite national university. Hinting at inside contacts and claiming knowledge of the various forms, deadlines, and procedures, he charged an exorbitant fee for his service, which he justified by offering a money-back guarantee to students who were not accepted.
One day, the secret of his business model came to light. All the material that prospective students had sent him over the years was found unopened in a trash dump. Upon investigation it turned out that he had simply been collecting the students’ money (or rather their parents’ money) and doing nothing for it. The trick was that his fees were so high and his marketing so focused that only the children of affluent parents subscribed to his service, and almost all of them were admitted to the university anyway. He refunded the fees of those few who were not admitted. He was also sent to prison for his efforts.
Are stock brokers in the same business as my acquaintance’s father? Are stock analysts in the same business as the newsletter publisher? Not exactly, but there is scant evidence that they possess any unusual predictive powers. That’s why I thought news stories in November 2002 recounting New York Attorney General Eliot Spitzer’s criticism of Institutional Investor magazine’s analyst awards were a tad superfluous. Spitzer noted that the stock-picking performances of
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most of the winning analysts were, in fact, quite mediocre. Maybe Donald Trump will hold a press conference pointing out that the country’s top gamblers don’t do particularly well at roulette.
Decimals and Other Changes
Like analysts and brokers, market makers (who make their money on the spread between the bid and the ask price for a stock) have received more than their share of criticism in recent years. One result has been a quiet reform that makes the market a bit more efficient. Wall Street’s surrender to radical “decacrats” occurred a couple of years ago, courtesy of a Congressional mandate and a direct order from the Securities and Exchange Commission. Since then stock prices have been expressed in dollars and cents, and we no longer hear “profittaking drove XYZ down 2 and 1/8” or “news of the deal sent PQR up 4 and 5/16.”
Although there may be less romance associated with declines of 2.13 and rises of 4.31, decimalization makes sense for a number of reasons. The first is that price rises and declines are immediately comparable since we no longer must perform the tiresome arithmetic of, say, dividing 11 by 16. Mentally calculating the difference between two decimals generally requires less time than subtracting 3 5/8 from 5 3/16. Another benefit is global uniformity of pricing, as American securities are now denominated in the same decimal units as those in the rest of the world. Foreign securities no longer need to be rounded to the nearest multiple of 1/16, a perverse arithmetical act if there ever was one.
More importantly, the common spread between the bid and ask prices has shrunk. Once generally 1/16 (.0625, that is), the spread in many cases has become .01 and, by so shrivA
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eling, will save investors billions of dollars over the years. Market makers aside, most investors applaud this consequence of decimalization.
The last reason for cheering the change is more mathematical. There is a sense in which the old system of halves, quarters, eighths, and sixteenths is more natural than decimals. It is, after all, only a slightly disguised binary system, based on powers of 2 (2, 4, 8, 16) rather than powers of 10. It doesn’t inherit any of the prestige of the binary system, however, because it awkwardly combines the base 2 fractional part of a stock price with the base 10 whole-number part.
Thus it is that Ten extends its imperial reach to Wall Street. From the biblical Commandments to David Letterman’s lists, the number 10 is ubiquitous. Not unrelated to the perennial yearning for the simplicity of the metric system, 10 envy has also come to be associated with rationality and efficiency. It is thus fitting that all stocks are now expressed in decimals. Still, I suspect that many market veterans miss those pesky fractions and their role in stories of past killings and baths. Except for generation X-ers (Roman numeral ten-ers), many others will too. Anyway, that’s my two cents (.02, l/50th) worth on the subject.
The replacement of marks, francs, drachmas, and other European currencies by euros on stock exchanges and in stores is another progressive step that nevertheless rouses a touch of nostalgia. The coins and bills from my past travels that are scattered about in drawers are suddenly out of work and will never see the inside of a wallet again.
Yet another vast change in trading practices is the greater self-reliance among investors. Despite the faulty accounting that initially disguised their sickly returns, the ladies of Beardstown, Illinois, helped popularize investment clubs. Even more significant in this regard is the advent of effortless online trading, which has further hastened the decline of the
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traditional broker. The ease with which I clicked on simple icons to buy and sell (specifically sell reasonably performing funds and buy more WCOM shares) was always a little frightening, and I sometimes felt as if there were a loaded gun on my desk. Some studies have linked online trading and day trading to increased volatility in the late ’90s, although it’s not clear that they remain factors in the ’00s.
What’s undeniable is that buying and selling online remains easy, so easy that I think it might not be a bad idea were small pictures of real-world items to pop up before every stock purchase or sale as a reminder of the approximate value of what’s being traded. If your transaction were for $35,000, a luxury car might appear; if it were for $100,000, a small cottage; and if it were for a penny stock, a candy bar. Investors can now check stock quotations, the size and the number of the bids and the asks, and megabytes of other figures on so- called level-two screens available in (almost) real-time on their personal computers. Millions of little desktop brokerages! Unfortunately, librarian Jesse Sherra’s paraphrase of Coleridge often seems apt: Data, data everywhere, but not a thought to think.
Benford's Law and Looking Out for Number One
I mentioned that people find it very difficult to simulate a series of coin flips. Are there other human disabilities that might allow someone to look at a company’s books, say Enron’s or WorldCom’s, and determine whether or not they had been cooked? There may have been, and the mathematical principle involved is easily stated, but counterintuitive.
Benford’s Law states that in a wide variety of circumstances, numbers—as diverse as the drainage areas of rivers, physical properties of chemicals, populations of small towns,
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figures in a newspaper or magazine, and the half-lives of radioactive atoms—have “ 1 ” as their first non-zero digit disproportionately often. Specifically, they begin with “1” about 30 percent of the time, with “2” about 18 percent of the time, with “3” about 12.5 percent, and with larger digits progressively less often. Less than 5 percent of the numbers in these circumstances begin with the digit “9.” Note that this is in stark contrast to many other situations where each of the digits has an equal chance of appearing.
Benford’s Law goes back one hundred years to the astronomer Simon Newcomb (note the letters WCOM in his name), who noticed that books of logarithm tables were much dirtier near the front, indicating that people more frequently looked up numbers with a low first digit. This odd phenomenon remained a little-known curiosity until it was rediscovered in 1938 by the physicist Frank Benford. It wasn’t until 1996, however, that Ted Hill, a mathematician at Georgia Tech, established what sort of situations generate numbers in accord with Benford’s Law. Then a mathematically inclined accountant named Mark Nigrini generated considerable buzz when he noted that Benford’s Law could be used to catch fraud in income tax returns and other accounting documents.
The following example suggests why collections of numbers governed by Benford’s Law arise so frequently:
Imagine that you deposit $1,000 in a bank at 10 percent compound interest per year. Next year you’ll have $1,100, the year after that $1,210, then $1,331, and so on. (Compounding is discussed further in chapter 5.) The first digit of your account balance remains a “1” for a long time. When your account grows to over $2,000, the first digit will remain a “2” for a shorter period. And when your deposit finally grows to over $9,000, the 10 percent growth will result in more than $10,000 in your account the following year and a long return to “1” as the first digit. If you record your account balance
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each year for many years, these numbers will thus obey Ben- ford’s Law.
The law is also “scale-invariant” in that the dimensions of the numbers don’t matter. If you expressed your $1,000 in euros or pounds (or the now defunct francs or marks) and watched it grow at 10 percent per year, about 30 percent of the yearly values would begin with a “1,” about 18 percent with a “2,” and so on.
More generally, Hill showed that such collections of numbers arise whenever we have what he calls a “distribution of distributions,” a random collection of random samples of data. Big, motley collections of numbers will follow Benford’s Law.
This brings us back to Enron, WorldCom, accounting, and Mark Nigrini, who reasoned that the numbers on accounting forms, which often come from a variety of company operations and a variety of sources, should be governed by Benford’s Law. That is, these numbers should begin disproportionately with the digit “1,” and progressively less often with bigger digits, and if they don’t, that is a sign that the books have been cooked. When people fake plausible-seeming numbers, they generally use more “5s” and “6s” as initial digits, for example, than Benford’s Law would predict.
Nigrini’s work has been well publicized and has surely been noted by accountants and by prosecutors. Whether the Enron, WorldCom, and Anderson people have heard of it is unknown, but investigators might want to check if the distribution of leading digits in the Enron documents accords with Benford’s Law. Such checks are not foolproof and sometimes lead to false-positive results, but they provide an extra tool that might be useful in certain situations.
It would be amusing if, in looking out for number one, the culprits forgot to look out for their “Is.” Imagine the Anderson accountants muttering anxiously that there weren’t enough leading “Is” on the documents they were feeding into the shredders. A 1-derful fantasy!
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The Numbers Man—A Screen Treatment
An astonishing amount of attention has been paid recently to fictional and narrative treatments of mathematical topics. The movies Good Will Hunting, Pi, and The Croupier come to mind; so do plays such as Copenhagen, Arcadia, and The Proof, the two biographies of Paul Erdos, A Beautiful Mind, the biography of John Nash (with its accompanying Academy Award-winning movie), TV specials on Fermat’s Last Theorem, and other mathematical topics, as well as countless books on popular mathematics and mathematicians. The plays and movies, in particular, prompted me to expand the idea in the stock-newsletter scam discussed above (I changed the focus, however, from stocks to sports) into a sort of abbreviated screen treatment that highlights the relevant mathematics a bit more than has been the case in the productions just cited. Yet another instance of what columnist Charles Krauthammer has dubbed “Disturbed Nerd Chic,” the treatment might even be developed into an intriguing and amusing film. In fact, I rate it a “strong buy” for any studio executive or independent filmmaker.
Rough Idea: Math nerd runs a clever sports-betting scam and accidentally nets an innumerate mobster.
Act One
Louis is a short, lecherous, somewhat nerdy man who dropped out of math graduate school about ten years ago (in the late ’80s) and now works at home as a technical consultant. He looks and acts a bit like the young Woody Allen. He’s playing cards with his pre-teenage kids and has just finished telling them a funny story. His kids are smart and they ask him how it is that he always knows the right story to tell. His wife, Marie, is uninterested. True to form, he begins telling them the Leo Rosten story about the famous rabbi who was asked by an
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admiring student how it was that the rabbi always had a perfect parable for any subject. Louis pauses to make sure they see the relevance.
When they smile and his wife rolls her eyes again, he continues. He tells them that the rabbi replied to his students with a parable. It was about a recruiter in the Tsar’s army who was riding through a small town and noticed dozens of chalked circular targets on the side of a barn, each with a bullet hole through the bull’s-eye. The recruiter was impressed and asked a neighbor who this perfect shooter might be. The neighbor responded, “Oh that’s Shepsel, the shoemaker’s son. He’s a little peculiar.” The enthusiastic recruiter was undeterred until the neighbor added, “You see, first Shepsel shoots and then he draws the chalk circles around the bullet hole.” The rabbi grinned. “That’s the way it is with me. I don’t look for a parable to fit the subject. I introduce only subjects for which I have parables.”
Louis and his kids laugh until a distracted, stricken look crosses his face. Closing the book, Louis hurries his kids off to bed, interrupts Marie’s prattling about her new pearl necklace and her Main Line parents’ nasty neighbors, distractedly bids her good night, and retreats to his study where he starts scribbling, making calls, and performing calculations. The next day he stops by the bank and the post office and a stationery store, does some research online, and then has a long discussion with his friend, a sportswriter on the local suburban New Jersey newspaper. The conversation revolves around the names, addresses, and intelligence of big sports bettors around the country.
The idea for a lucrative con game has taken shape in his mind. For the next several days he sends letters and emails to many thousands of known sports bettors “predicting” the outcome of a certain sporting event. His wife is uncomprehending when Louis mumbles that, Shepsel-like, he can’t lose
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since whatever happens in the sporting event, his prediction is bound to be right for half the bettors. The reason, it will turn out, is that to half of these people he predicts a certain team will win, and to the other half he predicts that it will lose.
Tall, blond, plain, and dim-witted, Marie is left wondering what exactly her sneaky husband is up to now. She finds the new postage meter behind the computer, notes the increasingly frequent secret telephone calls, and nags him about their worsening financial and marital situation. He replies that she doesn’t really need three closets full of clothes and a small fortune of jewelry when she spends all her time watching soaps and puts her off with some mathematical mumbo-jumbo about demographic research and new statistical techniques. She still doesn’t follow, but she is mollified by his promise that his mysterious endeavor will end up being lucrative.
They go out to eat to celebrate and Louis, intense and cad- like as always, talks up genetically modified food and tells the cute waitress that he wants to order whatever item on the menu has the most artificial ingredients. Much to Marie’s chagrin, he then involves the waitress in a classic mathematical trick by asking her to examine his three cards, one black on both sides, one red on both sides, and one black on one side and red on the other. He asks her for her cap, drops the cards into it, and tells her to pick a card, but only to look at one side of it. The side is red, and Louis notes that the card she picked couldn’t possibly be the card that was black on both sides, and therefore it must be one of the other two cards—the red-red card or the red-black card. He guesses that it’s the red-red card and offers to double her 15 percent tip if it’s the red-black card and stiff her if it’s the red-red card. He looks at Marie for approbation that is not forthcoming. The waitress accepts and loses.
Tone-deaf to Marie’s discomfort, Louis thinks he’s making amends with her by explaining the trick. She is less than
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enthralled. He tells her that it’s not an even bet even though at first glance it appears to be one. There are, after all, two cards it could be, and he bet on one, and the waitress bet on the other. The rub is, he gleefully runs on with his mouth full, there are two ways he can win and only one way the waitress can win. The visible side of the card the waitress picked could be the red side of the red-black card, in which case she wins, or it could be one side of the red-red card, in which case he wins, or it could be the other side of the red-red card, in which case he also wins. His chances of winning are thus 2/3, he concludes exultantly, and the average tip he gives is reduced by a third. Marie yawns and checks her Rolex. He breaks to go to the men’s room where he calls his girlfriend May Lee to apologize for some vague indiscretion.
The next week he explains the sports-betting con to May Lee, who looks a bit like Lucy Liu and is considerably smarter than Marie and even more materialistic. They’re in her apartment. She is interested in the con and asks clarifying questions. He enthuses to her that he needs her secretarial help. He’s sending out more letters and making a second prediction in them, but this time just to the half of the people to whom he sent a correct first prediction; the other half he plans to ignore. To half of this smaller group, he will predict a win in a second sporting event, to the other half a loss. Again for half of this group his prediction is going to be right, and so for one-fourth of the original group he’s going to be right two times in a row. “And to this one-fourth of the bettors?” she asks knowingly and excitedly. A mathematico-sexual tension develops.
He smiles rakishly and continues. To half of this one-fourth he will predict a win the following week, to the other half a loss; he again will ignore those to whom he’s made an incorrect prediction. Once again he will be right—this time for the third straight time—although for only one-eighth of the original population. May Lee helps with the mailings as he continues
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this process, focusing only on those to whom he’s made correct predictions and winnowing out those to whom he’s made incorrect ones. There is a sex scene amid all the letters, and they joke about winning whether the teams in question do or not, whether the predictions are right or wrong. Whether up or down, they’ll be happy.
As the mailings go on, so does his other life as a bored consultant, cyber-surfer, and ardent sports fan. He continues to extend his string of successful predictions to a smaller and smaller group of people until finally with great anticipation he sends a letter to the small group of people who are left. In it he points to his impressive string of successes and requests a substantial payment to keep these valuable and seemingly oracular “predictions” coming.
Act Two
He receives many payments and makes a further prediction. Again he’s right for half of the remaining people and drops the half for which he’s wrong. He asks the former group for even more money for another prediction, receives it, and continues. Things improve with Marie and with May Lee as the money rolls in and Louis realizes his plan is working even better than he expected. He takes his kids and, in turn, each woman to sports events or to Atlantic City, where he comments smugly on the losers who, unlike him, bet on iffy propositions. When Marie worries aloud about shark attacks off the beach, Louis tells her that more Americans die from falling airplane parts each year than from shark attacks. He makes similar pronouncements throughout the trip.
He plays a little blackjack and counts cards while doing so. He complains that it requires too much low-level concentration and that, unless one has a lot of money already, the rate at which one makes money is so slow and uneven that one might as well get a job. Still, he goes on, it’s the only game
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where a strategy exists for winning. All the other games are for mentally flabby losers. He goes to one of the casino restaurants where he shows his kids the waitress tip-cheating game. They think it’s great.
Back home in suburban New Jersey again, the sports- betting con resumes. Now there are only a few people left among the original thousands of sports-bettors. One of them, a rough underworld type named Otto, tracks him down, follows him to the parking lot of the basketball arena, and, politely at first and then more and more insistently, demands a prediction on an upcoming game on which he plans to bet a lot of money. Louis dismisses him and Otto, who looks a little like Stephen Segal, promptly orders him into his car at gunpoint and threatens to harm his family. He knows where they live.
Not understanding how he could be the recipient of so many consecutive correct predictions, Otto doesn’t believe Louis’s protestations that this is a con game. Louis makes some mathematical points in an effort to convince Otto of the possible falsity of any particular prediction. But no matter how he tries, he can’t quite convince Otto of the fact that there will always be some people who receive many consecutive correct predictions by chance alone.
Marooned in Otto’s basement, the math-nerd scam artist and the bald muscled extortionist are a study in contrasts. They speak different languages and have different frames of reference. Otto claims, for example, that every bet is more or less a 50-50 proposition because you either win or lose. Louis talks of his basketball buddies Lewis Carroll and Bertrand Russell and the names go over Otto’s head, of course. Oddly, they have similar attitudes toward women and money and also share an interest in cards, which they play to while away the time. Otto proudly shows off his riffle shuffle that he claims completely mixes the cards, while Louis prefers soliA
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taire and silently scoffs at Otto’s lottery expenditures and gambling misconceptions. When they forget why they’re there they get along well enough, although now and then Otto renews his threats and Louis renews his disavowal of any special sports knowledge and his plea to go home.
Finally granting that he might receive an incorrect prediction occasionally, Otto still insists that Louis give him his take on who’s going to win an upcoming football game. In addition to not being too bright, Otto, it appears, is in serious debt. Under extreme duress (with a gun to his head), Louis makes a prediction that happens to be right, and Otto, desperate and still convinced that he is in control of a money tree, now wants to bet funds borrowed from his gambling associates on Louis’s next prediction.
Act Three
Louis at last convinces Otto to let him go home and do research for his next big sports prediction. He and May Lee, whose need for money, baubles, and clothes has all along provided the impetus for the scam, discuss his predicament and realize they must exploit Otto’s only weaknesses, his stupidity and gullibility, and his only intellectual interests, money and playing cards.
Both go over to Otto’s apartment. Otto is charmed by May Lee, who flirts with him and offers him a deal. She wordlessly takes two decks of cards from her purse and asks Otto to shuffle each of them. Otto is pleased to show off to a more appreciative audience. She then gives him one of the decks and asks him to turn over one card at a time as she, keeping pace with him, does the same thing with the other deck. May Lee asks, what does he think is the likelihood that the cards they turn over will ever match, denomination and suit exactly the same? He scoffs but is entranced by May Lee and is amazed when after a tense minute or so that is exactly what
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happens. She explains that it will happen more often than not and perhaps he can use this fact to make some money. After all, Louis is a mathematical genius and he’s proved that it will. Louis smiles proudly.
Otto is puzzled. May Lee tells Otto again that the sports- betting was a scam and that he’s more likely to make money with the card tricks that Louis can teach him. Louis steps forward with the same two decks, which he’s now arranged so that the cards in each deck alternate colors. In one, it’s red- black, red-black, red-black .... In the other it’s black-red, black-red, black-red .... He gives the two decks to Otto and challenges him to do one of his great riffle shuffles of one deck into the other so that the cards will be mixed. Otto does and arrogantly announces that the cards are completely mixed now, whereupon Louis takes the combined two decks, puts them behind his back, pretends to be manipulating them, and brings forward two cards, one black and one red. So, Otto asks? Louis brings forth two more cards, one of each color, and then he does this again and again. I really shuffled them, Otto observes. How’d you do that? Louis explains that it involves no skill; the cards no longer alternate color in the combined deck, but any two from the top on down are always of different color.
There is a collage scene in which Louis explains various card tricks to Otto and the ways in which they can be exploited to make money. There’s always some order, some deviation from randomness, that a card man like you can use to get rich, Louis says to Otto. He even explains to him how he avoids paying waitresses tips. The deal, of course, is that Otto releases them, understanding, vaguely at least, how the betting scheme works and, more precisely, how the new card tricks do. Louis promises a one-day crash course on how to exploit the tricks for money.
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In the last scene Louis is seen working the same scam but this time with predictions about the movements of a stock market index. Since he doesn’t want any more Ottos, but a higher class clientele, he’s redefined himself as the publisher of a stock newsletter. The house he’s in is more sumptuous and May Lee, to whom he’s now married, bustles about in an expensive suit as Louis plays cards with his slightly older children, occasionally doodling little bull’s-eyes and targets on an envelope. He excuses himself and goes to his study to make a secret telephone call to an apartment on Central Park West that he’s just purchased for his new mistress.
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