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You Can't Become Rich In Your Pocket Until You Become Rich In Your Mind | ||||
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Someone claiming to be the publisher of a stock newsletterGiven the relative rarity with which Henry and Tommy overtake one another in their penny-flipping contest, it wouldnt 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 markets randomness. This assumes, however, that Buffetts choices have no effect on the market. Originally no doubt they didnt, 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. 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 theres 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, theres a better-than-even chance that each will have won at least ten f l ips 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 players shots, a stock analysts 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 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 wont, give pause to commentators who provide a neat post hoc explanation for every rally, every sell-off, and everything in between. Such commentators generally dont 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 Tommys profit-taking, Henrys 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 insightfulsounding 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 f i ltered. 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 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 neednt 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? Heres 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 hes 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 hes 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, thats 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 its done unknowingly by earnest, confident, and ignorant newsletter publishers? (Compare the faithhealer who takes credit for any accidental improvements.) 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 theyve been, or are about to be, inordinately successful. If people are desperate enough, theyll 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 fathers 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 Ive forgotten. My friends 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 acquaintances 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. Thats why Ithought news stories in November 2002 recounting New York Attorney General Eliot Spitzers criticism of Institutional Investor magazines analyst awards were a tad superfluous. Spitzer noted that the stock-picking performances of most of the winning analysts were, in fact, quite mediocre. Maybe Donald Trump will hold a press conference pointing out that the countrys top gamblers dont 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 Streets 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 shriveling, 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 doesnt 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 Lettermans 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, thats my two cents (.02, 1/50th) 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 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 its not clear that they remain factors in the 00s. Whats 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 whats 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 socalled level-two screens available in (almost) real-time on their personal computers. Millions of little desktop brokerages! Unfortunately, librarian Jesse Sherras paraphrase of Coleridge often seems apt: Data, data everywhere, but not a thought to think. Benfords 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 companys books, say Enrons or WorldComs, and determine whether or not they had been cooked? There may have been, and the mathematical principle involved is easily stated, but counterintuitive. Benfords 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, 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. Benfords 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 wasnt until 1996, however, that Ted Hill, a mathematician at Georgia Tech, established what sort of situations generate numbers in accord with Benfords Law. Then a mathematically inclined accountant named Mark Nigrini generated considerable buzz when he noted that Benfords 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 Benfords Law arise so frequently: Imagine that you deposit $1,000 in a bank at 10 percent compound interest per year. Next year youll 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 each year for many years, these numbers will thus obey Benfords Law. The law is also scale-invariant in that the dimensions of the numbers dont 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 Benfords 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 Benfords Law. That is, these numbers should begin disproportionately with the digit 1, and progressively less often with bigger digits, and if they dont, 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 Benfords Law would predict. Nigrinis 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 Benfords 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 1s. Imagine the Anderson accountants muttering anxiously that there werent enough leading 1s on the documents they were feeding into the shredders. A 1-derful fantasy! 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 Fermats 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. |
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