Trends, Crowds, and Waves
/Isa predictor of stock prices, psychology goes only so far. /lMany investors subscribe to “technical analysis,” an approach generally intent on discerning the short-term direction of the market via charts and patterns and then devising rules for pursuing it. Adherents of technical analysis, which is not all that technical and would more accurately be termed “trend analysis,” believe that “the trend is their friend,” that “momentum investing” makes sense, that crowds should be followed. Whatever the validity of these beliefs and of technical analysis in general (and I’ll get to this shortly), I must admit to an a priori distaste for the herdish behavior it often seems to counsel: Figure out where the pack is going and follow it. It was this distaste, perhaps, that prevented me from selling WCOM and that caused me to sputter continually to myself that the company was the victim of bad public relations, investor misunderstanding, media bashing, anger at the CEO, a poisonous business climate, unfortunate timing, or panic selling. In short, I thought the crowd was wrong and hated the idea that it must be obeyed. As I slowly learned, however, disdaining the crowd is sometimes simply hubris.
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Technical Analysis: Following the Followers
My own prejudices aside, the justification for technical analysis is murky at best. To the extent there is one, it most likely derives from psychology, perhaps in part from the Keynesian idea of conventionally anticipating the conventional response, or perhaps from some as yet unarticulated systemic interactions. “Unarticulated” is the key word here: The quasi-mathematical jargon of technical analysis seldom hangs together as a coherent theory. I’ll begin my discussion of it with one of its less plausible manifestations, the so-called Elliott wave theory.
Ralph Nelson Elliott famously believed that the market moved in waves that enabled investors to predict the behavior of stocks. Outlining his theory in 1939, Elliott wrote that stock prices move in cycles based upon the Fibonacci numbers (1, 2, 3, 5, 8,13, 21, 34, 59, 93,..., each successive number in the sequence being the sum of the two previous ones). Most commonly the market rises in five distinct waves and declines in three distinct waves for obscure psychological or systemic reasons. Elliott believed as well that these patterns exist at many levels and that any given wave or cycle is part of a larger one and contains within it smaller waves and cycles. (To give Elliott his due, this idea of small waves within larger ones having the same structure does seem to presage mathematician Benoit Mandelbrot’s more sophisticated notion of a fractal, to which I’ll return later.) Using Fibonacci-inspired rules, the investor buys on rising waves and sells on falling ones.
The problem arises when these investors try to identify where on a wave they find themselves. They must also decide whether the larger or smaller cycle of which the wave is inevitably a part may temporarily be overriding the signal to buy or sell. To save the day, complications are introduced into the theory, so many, in fact, that the theory soon becomes
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incapable of being falsified. Such complications and unfalsifi- ability are reminiscent of the theory of biorhythmns and many other pseudosciences. (Biorhythm theory is the idea that various aspects of one’s life follow rigid periodic cycles that begin at birth and are often connected to the numbers 23 and 28, the periods of some alleged male and female principles, respectively.) It also brings to mind the ancient Ptolemaic system of describing the planets’ movements, in which more and more corrections and ad hoc exceptions had to be created to make the system jibe with observation. Like most other such schemes, Elliott wave theory founders on the simple question: Why should anyone expect it to work?
For some, of course, what the theory has going for it is the mathematical mysticism associated with the Fibonacci numbers, any two adjacent ones of which are alleged to stand in an aesthetically appealing relation. Natural examples of Fibonacci series include whorls on pine cones and pineapples; the number of leaves, petals, and stems on plants; the numbers of left and right spirals in a sunflower; the number of rabbits in succeeding generations; and, insist Elliott enthusiasts, the waves and cycles in stock prices.
It’s always pleasant to align the nitty-gritty activities of the market with the ethereal purity of mathematics.
The Euro and the Golden Ratio
Before moving on to less barren financial theories, I invite you to consider a brand new instance of financial numerology. An email from a British correspondent apprised me of an interesting connection between the euro-pound and pound-euro exchange rates on March 19, 2002.
To appreciate it, one needs to know the definition of the golden ratio from classical Greek mathematics. (Those for
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whom the confluence of Greek, mathematics, and finance is a bit much may want to skip to the next section.) If a point on a straight line divides the line so that the ratio of the longer part to the shorter is equal to the ratio of the whole to the longer part, the point is said to divide the line in a golden ratio. Rectangles whose length and width stand in a golden ratio are also said to be golden, and many claim that rectangles of this shape, for example, the facade of the Parthenon, are particularly pleasing to the eye. Note that a 3-by-5 card is almost a golden rectangle since 5/3 (or 1.666 . . . ) is approximately equal to (5 + 3)/5 (or 1.6).
The value of the golden ratio, symbolized by the Greek letter phi, is 1.618 ... (the number is irrational and so its decimal representation never repeats). It is not difficult to prove that phi has the striking property that it is exactly equal to 1 plus its reciprocal (the reciprocal of a number is simply 1 divided by the number). Thus 1.618 ... is equal to 1 + 1/1.618
This odd fact returns us to the euro and the pound. An announcer on the BBC on the day in question, March 19, 2002, observed that the exchange rate for 1 pound sterling was 1 euro and 61.8 cents (1.618 euros) and that, lo and behold, this meant that the reciprocal exchange rate for 1 euro was 61.8 pence (.618 pounds). This constituted, the announcer went on, “a kind of symmetry.” The announcer probably didn’t realize how profound this symmetry was.
In addition to the aptness of “golden” in this financial context, there is the following well-known relation between the golden ratio and the Fibonacci numbers. The ratio of any Fibonacci number to its predecessor is close to the golden ratio of 1.618 ..., and the bigger the numbers involved, the closer the two ratios become. Consider again, the Fibonacci numbers, 1, 2, 3, 5, 8, 13, 21, 34, 59, .... The ratios, 5/3, 8/5,
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13/8, 21/13, . . ., of successive Fibonacci numbers approach the golden ratio of 1.618 ... !
There’s no telling how an Elliott wave theorist dabbling in currencies at the time of the above exchange rate coincidence would have reacted to this beautiful harmony between money and mathematics. An unscrupulous, but numerate hoaxer might have even cooked up some flapdoodle sufficiently plausible to make money from such a “cosmic” connection.
The story could conceivably form the basis of a movie like Pi, since there are countless odd facts about phi that could be used to give various investing schemes a superficial plausibility. (The protagonist of Pi was a numerologically obsessed mathematician who thought he’d found the secret to just about everything in the decimal expansion of pi. He was pursued by religious zealots, greedy financiers, and others. The only sane character, his mentor, had a stroke, and the syncopated black-and-white cinematography was anxiety-inducing. Appealing as it was, the movie was mathematically nonsensical.) Unfortunately for investors and mathematicians alike, the lesson again is that more than beautiful harmonies are needed to make money on Wall Street. And Phi can’t match the cachet of Pi as a movie title either.
Moving Averages, Big Picture
People, myself included, sometimes ridicule technical analysis and the charts associated with it in one breath and then in the next reveal how much in (perhaps unconscious) thrall to these ideas they really are. They bring to mind the old joke about the man who complains to his doctor that his wife has for several years believed she’s a chicken. He would have sought help sooner, he says, “but we needed the eggs.” Without reading
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too much into this story except that we do sometimes seem to need the notions of technical analysis, let me finally proceed to examine some of these notions.
Investors naturally want to get a broad picture of the movement of the market and of particular stocks, and for this the simple technical notion of a moving average is helpful. When a quantity varies over time (such as the stock price of a company, the noontime temperature in Milwaukee, or the cost of cabbage in Kiev), one can, each day, average its values over, say, the previous 200 days. The averages in this sequence vary and hence the sequence is called a moving average, but the value of such a moving average is that it doesn’t move nearly as much as the stock price itself; it might be termed the phlegmatic average.
For illustration, consider the three-day moving average of a company whose stock is very volatile, its closing prices on successive days being: 8, 9, 10, 5, 6, 9. On the day the stock closed at 10, its three-day moving average was (8 + 9 + 10)/3 or 9. On the next day, when the stock closed at 5, its three- day moving average was (9 + 10 + 5)/3 or 8. When the stock closed at 6, its three-day moving average was (10 + 5 + 6)/3 or 7. And the next day, when it closed at 9, its three-day moving average was (5 + 6 + 9)/3 or 6.67.
If the stock oscillates in a very regular way and you are careful about the length of time you pick, the moving average may barely move at all. Consider an extreme case, the twenty- day moving average of a company whose closing stock prices oscillate with metronomic regularity. On successive days they are: 51, 52, 53, 54, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 46, 47, 48, 49, 50, 51, 52, 53, and so on, moving up and down around a price of 50. The twenty-day moving average on the day marked in bold is 50 (obtained by averaging the 20 numbers up to and including it). Likewise, the twenty-day moving average on the next day, when the stock is at 51, is
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also 50. It’s the same for the next day. In fact, if the stock price oscillates in this regular way and repeats itself every twenty days, the twenty-day moving average is always 50.
There are variations in the definition of moving averages (some weight recent days more heavily, others take account of the varying volatility of the stock), but they are all designed to smooth out the day-to-day fluctuations in a stock’s price in order to give the investor a look at broader trends. Software and online sites allow easy comparison of the stock’s daily movements with the slower-moving averages.
Technical analysts use the moving average to generate buy-sell rules. The most common such rule directs you to buy a stock when it exceeds its X-day moving average. Context determines the value of X, which is usually 10, 50, or 200 days. Conversely, the rule directs you to sell when the stock falls below its X-day moving average. With the regularly oscillating stock above, the rule would not lead to any gains or losses. It would call for you to buy the stock when it moves from 50, its moving average, to 51, and for you to sell it when it moves from 50 to 49. In the previous example of the three-day moving average, the rule would require that you buy the stock at the end of the third day and sell it at the end of the fourth, leading in this particular case to a loss.
The rule can work well when a stock fluctuates about a long-term upward- or downward-sloping course. The rationale for it is that trends should be followed, and that when a stock moves above its X-day moving average, this movement signals that a bullish trend has begun. Conversely, when a stock moves below its X-day moving average, the movement signals a bearish trend. I reiterate that mere upward (downward) movement of the stock is not enough to signal a buy (sell) order; a stock must move above (below) its moving average.
Alas, had I followed any sort of moving average rule, I would have been out of WCOM, which moved more or less
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steadily downhill for almost three years, long before I lost most of my investment in it. In fact, I never would have bought it in the first place. The security guard mentioned in chapter 1 did, in effect, use such a rule to justify the sale of the stocks in his pension plan.
There are a few studies, which I’ll get to later, suggesting that a moving average rule is sometimes moderately effective. Even so, however, there are several problems. One is that it can cost you a lot in commissions if the stock price hovers around the moving average and moves through it many times in both directions. Thus you have to modify the rule so that the price must move above or below its moving average by a non-trivial amount. You must also decide whether to buy at the end of the day the price exceeds the moving average or at the beginning of the next day or later still.
You can mine the voluminous time-series data on stock prices to find the X that has given the best returns for adhering to the X-day moving average buy-sell rule. Or you can complicate the rule by comparing moving averages over different intervals and buying or selling when these averages cross each other. You can even adapt the idea to day trading by using X-minute moving averages defined in terms of the mathematical notion of an integral. Optimal strategies can always be found after the fact. The trick is getting something that will work in the future; everyone’s very good at predicting the past. This brings us to the most trenchant criticism of the moving-average strategy. If the stock market is efficient, that is, if information about a stock is almost instantaneously incorporated into its price, then any stock’s future moves will be determined by random external events. Its past behavior, in particular its moving average, is irrelevant, and its future movement is unpredictable.
Of course, the market may not be all that efficient. There’ll be much more on this question in later chapters.
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Resistance and Support and All That
Two other important ideas from technical analysis are resistance and support levels. The argument for them assumes that people usually remember when they’ve been burned, insulted, or left out; in particular, they remember what they paid, or wish they had paid, for a stock. Assume a stock has been selling for $40 for a while and then drops to $32 before slowly rising again. The large number of people who bought it around $40 are upset and anxious to recoup their losses, so if the stock moves back up to $40, they’re likely to sell it, thereby driving the price down again. The $40 price is termed a resistance level and is considered an obstacle to further upward movement of the stock price.
Likewise, investors who considered buying at $32 but did not are envious of those who did buy at that price and reaped the 25 percent returns. They are eager to get these gains, so if the stock falls back to $32, they’re likely to buy it, driving the price up again. The $32 price is termed a support level and is considered an obstacle to further downward movement.
Since stocks often seem to meander between their support and resistance levels, one rule followed by technical analysts is to buy the stock when it “bounces” off its support level and sell it when it “bumps” up against its resistance level. The rule can, of course, be applied to the market as a whole, inducing investors to wait for the Dow or the S&P to definitively turn up (or down) before buying (or selling).
Since chartists tend to view support levels as shaky, often temporary, floors and resistance levels as slightly stronger, but still temporary, ceilings, there is a more compelling rule involving these notions. It instructs you to buy the stock if the rising price breaks through the resistance level and to sell it if the falling price breaks through the support level. In both these cases breaking through indicates that the stock has
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moved out of its customary channel and the rule counsels investors to follow the new trend.
As with the moving-average rules, there are a few studies that indicate that resistance-support rules sometimes lead to moderate increases in returns. Against this there remains the perhaps dispiriting efficient-market hypothesis, which maintains that past prices, trends, and resistance and support levels provide no evidence about future movements.
Innumerable variants of these rules exist and they can be combined in ever more complicated ways. The resistance and support levels can change and trend up or down in a channel or with the moving average, for example, rather than remain fixed. The rules can also be made to take account of variations in a stock’s volatility as well.
These variants depend on price patterns that often come equipped with amusing names. The “head and shoulders” pattern, for example, develops after an extended upward trend. It is comprised of three peaks, the middle and highest one being the head, and the smaller left and right ones (earlier and later ones, that is) being the shoulders. After falling below the right shoulder and breaking through the support line connecting the lows on either side of the head, the stock price has, technical chartists aver, reversed direction and a downward trend has begun, so sell.
Similar metaphors describe the double-bottom trend reversal. It develops after an extended downward trend and is comprised of two successive troughs or bottoms with a small peak between them. After bouncing off the second bottom, the stock has, technical chartists again aver, reversed direction and an upward trend has begun, so buy.
These are nice stories, and technical analysts tell them with great earnestness and conviction. Even if everyone told the same stories (and they don’t), why should they be true? Presumably the rationale is ultimately psychological or perhaps
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sociological or systemic, but exactly what principles justify these beliefs? Why not triple or quadruple bottoms? Or two heads and shoulders? Or any of innumerable other equally plausible, equally risible patterns? What combination of psychological, financial, or other principles has sufficient specificity to generate effective investment rules?
As with Elliott waves, scale is an issue. If we go to the level of ticks, we can find small double bottoms and little heads and tiny shoulders all over. We find them also in the movement of broad market indices. And do these patterns mean for the market as whole what they are purported to mean for individual stocks? Is the “double-dip” recession discussed in early 2002 simply a double bottom?
Predictability and Trends
I often hear people swear that they make money using the rules of technical analysis. Do they really? The answer, of course, is that they do. People make money using all sorts of strategies, including some involving tea leaves and sunspots. The real question is: Do they make more money than they would investing in a blind index fund that mimics the performance of the market as a whole? Do they achieve excess returns? Most financial theorists doubt this, but there is some tantalizing evidence for the effectiveness of momentum strategies or short-term trend-following. Economists Nara- simhan Jegadeesh and Sheridan Titman, for example, have written several papers arguing that momentum strategies result in moderate excess returns and that, having done so over the years, their success is not the result of data mining. Whether this alleged profitability—many dispute it—is due to overreactions among investors or to the short-term persistence of the impact of companies’ earnings reports, they don’t
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say. They do seem to point to behavioral models and psychological factors as relevant.
William Brock, Josef Lakonishok, and Blake LeBaron have also found some evidence that rules based on moving averages and the notions of resistance and support are moderately effective. They focus on the simplest rules, but many argue that their results have not been replicated on new stock data.
More support for the existence of technical exploitability comes from Andrew Lo, who teaches at M.I.T., and Craig MacKinlay, from the Wharton School. They argue in their book, A Non-Random Walk Down Wall Street, that in the short run overall market returns are, indeed, slightly positively correlated, much like the local weather. A hot, sunny day is a bit more likely to be followed by another one, just as a good week in the market is a bit more likely to be followed by another one. Likewise for rainy days and bad markets. Employing state-of-the-art tools, Lo and MacKinlay also claim that in the long term the prognosis changes: Individual stock prices display a slight negative correlation. Winners are a bit more likely to be losers three to five years hence and vice versa.
They also bring up an interesting theoretical possibility. Weeding out some of the details, let’s assume for the sake of the argument (although Lo and MacKinlay don’t) that the thesis of Burton Malkiel’s classic book, A Random Walk Down Wall Street, is true and that the movement of the market as a whole is entirely random. Let’s also assume that each stock, when its fluctuations are examined in isolation, moves randomly. Given these assumptions it would nevertheless still be possible that the price movements of, say, 5 percent of stocks accurately predict the price movements of a different 5 percent of stocks one week later.
The predictability comes from cross-correlations over time between stocks. (These associations needn’t be causal, but might merely be brute facts.) More concretely, let’s say stock
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X, when looked at in isolation, fluctuates randomly from week to week, as does stock Y. Yet if X’s price this week often predicts Y’s next week, this would be an exploitable opportunity and the strict random-walk hypothesis would be wrong. Unless we delved deeply into such possible cross-correlations among stocks, all we would see would be a randomly fluctuating market populated by randomly fluctuating stocks. Of course, I’ve employed the typical mathematical gambit of considering an extreme case, but the example does suggest that there may be relatively simple elements of order in a market that appears to fluctuate randomly.
There are other sorts of stock price anomalies that can lead to exploitable opportunities. Among the most well-known are so-called calendar effects whereby the prices of stocks, primarily small-firm stocks, rise disproportionately in January, especially during the first week of January. (The price of WCOM rose significantly in January 2001, and I was hoping this rise would repeat itself in January 2002. It didn’t.) There has been some effort to explain this by citing tax law concerns that end with the close of the year, but the effect also seems to hold in countries with different tax laws. Moreover, unusual returns (good or bad) occur not only at the turn of the year, but, as Richard Thaler and others have observed, at the turn of the month, week, and day as well as before holidays. Again, poorly understood behavioral factors seem to be involved.
Technical Strategies and Blackjack
Most academic financial experts believe in some form of the random-walk theory and consider technical analysis almost indistinguishable from a pseudoscience whose predictions are either worthless or, at best, so barely discernibly better than chance as to be unexploitable because of transaction costs.
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I’ve always leaned toward this view, but I’ll reserve my more nuanced judgment for later in the book. In the meantime, I’d like to point out a parallel between market strategies such as technical analysis in one of its many forms and blackjack strategies. (There are, of course, great differences too.)
Blackjack is the only casino game of chance whose outcomes depend on past outcomes. In roulette, the previous spins of the wheel have no effect on future spins. The probability of red on the next spin is 18/38, even if red has come up on the five previous spins. The same is true with dice, which are totally lacking in memory. The probability of rolling a 7 with a pair of dice is 1/6, even if the four previous rolls have not resulted in a single 7. The probability of six reds in a row is (18/38)6; the probability of five 7s in a row is (1/6)5. Each spin and each roll are independent of the past.
A game of blackjack, on the other hand, is sensitive to its past. The probability of drawing two aces in a row from a deck of cards is not (4/52 x 4/52) but rather (4/52 x 3/51). The second factor, 3/51, is the probability of choosing another ace given that the first card chosen was an ace. In the same way the probability that a card drawn from a deck will be a face card (jack, queen, or king) given that only three of the thirty cards drawn so far have been face cards is not 12/52, but a much higher 9/22.
This fact—that (conditional) probabilities change according to the composition of the remaining portion of the deck—is the basis for various counting strategies in blackjack that involve keeping track of how many cards of each type have already been drawn and increasing one’s bet size when the odds are (occasionally and slightly) in one’s favor. Some of these strategies, followed carefully, do work. This is evidenced by the fact that some casinos supply burly guards free of charge to abruptly escort successful counting practitioners from the premises.
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The vast majority of people who try these strategies (or, worse, others of their own devising) lose money. It would make no sense, however, to point to the unrelenting average losses of blackjack players and maintain that this proves that there is no effective betting strategy for playing the game.
Blackjack is much simpler than the stock market, of course, which depends on vastly more factors as well as on the actions and beliefs of other investors. But the absence of conclusive evidence for the effectiveness of various investing rules, technical or otherwise, does not imply that no effective rules exist. If the market’s movements are not completely random, then it has a kind of memory within it, and investing rules depending on this memory might be effective. Whether they would remain so if widely known is very dubious, but that is another matter.
Interestingly, if there were an effective technical trading strategy, it wouldn’t need any convincing rationale. Most investors would be quite pleased to use it, as most blackjack players use the standard counting strategy, without understanding why it works. With blackjack, however, there is a compelling mathematical explanation for those who care to study it. By contrast an effective technical trading strategy might be found that was beyond the comprehension not only of the people using it but of everyone. It might simply work, at least temporarily. In Plato’s allegory of the cave the benighted see only the shadows on the wall of the cave and not the real objects behind them that are causing the shadows. If they were really predictive, investors would be quite content with the shadows alone and would simply take the cave to be a bargain basement.
The next segment is a bit of a lark. It offers a suggestive hint for developing a novel and counterintuitive investment strategy that has a bit of the feel of technical analysis.
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Winning Through Losing?
The old joke about the store owner losing money on every sale but making it up in volume may have a kernel of truth to it. An interesting new paradox by Juan Parrondo, a Spanish physicist, brings the joke to mind. It deals with two games, each of which results in steady losses over time. When these games are played in succession in random order, however, the result is a steady gain. Bad bets strung together to produce big winnings—very strange indeed!
To understand Parrondo’s paradox, let’s switch from a financial to a spatial metaphor. Imagine you are standing on stair 0, in the middle of a very long staircase with 1,001 stairs numbered from -500 to 500 (-500, -499, -498, . . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, ..., 498, 499, 500). You want to go up rather than down the staircase and which direction you move depends on the outcome of coin flips. The first game—let’s call it game S—is very Simple. You flip a coin and move up a stair whenever it comes up heads and down a stair whenever it comes up tails. The coin is slightly biased, however, and comes up heads 49.5 percent of the time and tails 50.5 percent. It’s clear that this is not only a boring game but a losing one. If you played it long enough, you would move up and down for a while, but almost certainly you would eventually reach the bottom of the staircase.
The second game—let’s continue to wax poetic and call it game C—is more Complicated, so bear with me. It involves two coins, one of which, the bad one, comes up heads only 9.5 percent of the time, tails 90.5 percent. The other coin, the good one, comes up heads 74.5 percent of the time, tails 25.5 percent. As in game S, you move up a stair if the coin you flip comes up heads and you move down one if it comes up tails.
But which coin do you flip? If the number of the stair you’re on is a multiple of 3 (that is, ..., -9, -6, -3, 0, 3, 6, 9,
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12, . . . ), you flip the bad coin. If the number of the stair you’re on is not a multiple of 3, you flip the good coin. (Note: Changing these odd percentages and constraints may affect the game’s outcome.)
Let’s go through game C’s dance steps. If you were on stair number 5, you would flip the good coin to determine your direction, whereas if you were on stair number 6, you would flip the bad coin. The same holds for the negatively numbered stairs. If you were on stair number -2 and playing game C, you would flip the good coin, whereas if you were on stair number -9, you would flip the bad coin.
Though less obviously so than in game S, game C is also a losing game. If you played it long enough, you would almost certainly reach the bottom of the staircase eventually. Game C is a losing game because the number of the stair you’re on is a multiple of 3 more often than a third of the time and thus you must flip the bad coin more often than a third of the time. Take my word for this or read the next paragraph to get a better feel for why it is.
(Assume that you’ve just started playing game C. Since you’re on stair number 0, and 0 is a multiple of 3, you would flip the bad coin, which lands heads with probability less than 10 percent, and you would very likely move down to stair number -1. Then, since -1 is not a multiple of 3, you would flip the good coin, which lands heads with probability almost 75 percent, and would probably move back up to stair 0. You may move up and down like this for a while. Occasionally, however, after the bad coin lands tails, the good coin, which lands tails almost 25 percent of the time, will land tails twice in succession, and you would move down to stair number -3, where the pattern will likely begin again. This latter downward pattern happens slightly more frequently (with probability .905 x .255 x .255) than does a rare head on the bad coin being followed by two heads on the good one (with
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probability .095 x .745 x .745) and your moving up three stairs as a consequence. So-called Markov chains are needed for a fuller analysis.)
So far, so what? Game S is simple and results in steady movement down the staircase to the bottom, and game C is complicated and also results in steady movement down the staircase to the bottom. Parrondo’s fascinating discovery is that if you play these two games in succession in random order (keeping your place on the staircase as you switch between games), you will steadily ascend to the top of the staircase. Alternatively, if you play two games of S followed by two games of C followed by two games of S and so on, all the while keeping your place on the staircase as you switch between games, you will also steadily rise to the top of the staircase. (You might want to look up M. C. Escher’s paradoxical drawing, “Ascending and Descending” for a nice visual analog to Parrondo’s paradox.)
Standard stock-market investments cannot be modeled by games of this type, but variations of these games might conceivably give rise to counterintuitive investment strategies. The probabilities might be achieved, for example, by complicated combinations of various financial instruments (options, derivatives, and so on), but the decision which coin (which investment, that is) to flip (to make) in game C above would, it seems, have to depend upon something other than whether one’s holdings were worth a multiple of $3.00 (or a multiple of $3,000.00). Perhaps the decision could depend in some way on the cross-correlation between a pair of stocks or turn on the value of some index being a multiple of 3.
If strategies like this could be made to work, they would yield what one day might be referred to as Parrondo profits.
Finally, let’s consider a companion paradox of sorts that might be called “losing through winning” and that may help explain why companies often overpaid for small companies
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they were purchasing during the bubble in the late ’90s. Professor Martin Shubik has regularly auctioned off $1 to students in his classes at Yale. The bidding takes place at fifty-four intervals, and the highest bidder gets the dollar, of course, but the second highest bidder is required to pay his bid as well. Thus, if the highest bid is 504 and you are second highest at 454, the leader stands to make 504 on the deal and you stand to lose 454 on it if bidding stops. You have an incentive to up your bid to at least 554, but after you’ve done so the other bidder has an even bigger incentive to raise his bid as well. In this way a one dollar bill can be successfully auctioned off for two, three, four, or more dollars.
If several companies are bidding on a small company and the cost of the preliminary legal, financial, and psychological efforts required to purchase the company are a reasonable fraction of the cost of the company, the situation is formally similar to Shubik’s auction. One or more of the bidding companies might feel compelled to make an exorbitant preemptive offer to avoid the fate of the losing bidder on the $1. WorldCom’s purchase of the web-hosting company Digex in 2000 for $6 billion was, I suspect, such an offer. John Sidg- more, the CEO who succeeded Bernie Ebbers, says that Digex was worth no more than $50 million, but that Ebbers was obsessed with beating out Global Crossing for the company.
The purchase is much more bizarre than Parrondo’s paradox
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Technical Analysis: Following the Followers
My own prejudices aside, the justification for technical analysis is murky at best. To the extent there is one, it most likely derives from psychology, perhaps in part from the Keynesian idea of conventionally anticipating the conventional response, or perhaps from some as yet unarticulated systemic interactions. “Unarticulated” is the key word here: The quasi-mathematical jargon of technical analysis seldom hangs together as a coherent theory. I’ll begin my discussion of it with one of its less plausible manifestations, the so-called Elliott wave theory.
Ralph Nelson Elliott famously believed that the market moved in waves that enabled investors to predict the behavior of stocks. Outlining his theory in 1939, Elliott wrote that stock prices move in cycles based upon the Fibonacci numbers (1, 2, 3, 5, 8,13, 21, 34, 59, 93,..., each successive number in the sequence being the sum of the two previous ones). Most commonly the market rises in five distinct waves and declines in three distinct waves for obscure psychological or systemic reasons. Elliott believed as well that these patterns exist at many levels and that any given wave or cycle is part of a larger one and contains within it smaller waves and cycles. (To give Elliott his due, this idea of small waves within larger ones having the same structure does seem to presage mathematician Benoit Mandelbrot’s more sophisticated notion of a fractal, to which I’ll return later.) Using Fibonacci-inspired rules, the investor buys on rising waves and sells on falling ones.
The problem arises when these investors try to identify where on a wave they find themselves. They must also decide whether the larger or smaller cycle of which the wave is inevitably a part may temporarily be overriding the signal to buy or sell. To save the day, complications are introduced into the theory, so many, in fact, that the theory soon becomes
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incapable of being falsified. Such complications and unfalsifi- ability are reminiscent of the theory of biorhythmns and many other pseudosciences. (Biorhythm theory is the idea that various aspects of one’s life follow rigid periodic cycles that begin at birth and are often connected to the numbers 23 and 28, the periods of some alleged male and female principles, respectively.) It also brings to mind the ancient Ptolemaic system of describing the planets’ movements, in which more and more corrections and ad hoc exceptions had to be created to make the system jibe with observation. Like most other such schemes, Elliott wave theory founders on the simple question: Why should anyone expect it to work?
For some, of course, what the theory has going for it is the mathematical mysticism associated with the Fibonacci numbers, any two adjacent ones of which are alleged to stand in an aesthetically appealing relation. Natural examples of Fibonacci series include whorls on pine cones and pineapples; the number of leaves, petals, and stems on plants; the numbers of left and right spirals in a sunflower; the number of rabbits in succeeding generations; and, insist Elliott enthusiasts, the waves and cycles in stock prices.
It’s always pleasant to align the nitty-gritty activities of the market with the ethereal purity of mathematics.
The Euro and the Golden Ratio
Before moving on to less barren financial theories, I invite you to consider a brand new instance of financial numerology. An email from a British correspondent apprised me of an interesting connection between the euro-pound and pound-euro exchange rates on March 19, 2002.
To appreciate it, one needs to know the definition of the golden ratio from classical Greek mathematics. (Those for
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whom the confluence of Greek, mathematics, and finance is a bit much may want to skip to the next section.) If a point on a straight line divides the line so that the ratio of the longer part to the shorter is equal to the ratio of the whole to the longer part, the point is said to divide the line in a golden ratio. Rectangles whose length and width stand in a golden ratio are also said to be golden, and many claim that rectangles of this shape, for example, the facade of the Parthenon, are particularly pleasing to the eye. Note that a 3-by-5 card is almost a golden rectangle since 5/3 (or 1.666 . . . ) is approximately equal to (5 + 3)/5 (or 1.6).
The value of the golden ratio, symbolized by the Greek letter phi, is 1.618 ... (the number is irrational and so its decimal representation never repeats). It is not difficult to prove that phi has the striking property that it is exactly equal to 1 plus its reciprocal (the reciprocal of a number is simply 1 divided by the number). Thus 1.618 ... is equal to 1 + 1/1.618
This odd fact returns us to the euro and the pound. An announcer on the BBC on the day in question, March 19, 2002, observed that the exchange rate for 1 pound sterling was 1 euro and 61.8 cents (1.618 euros) and that, lo and behold, this meant that the reciprocal exchange rate for 1 euro was 61.8 pence (.618 pounds). This constituted, the announcer went on, “a kind of symmetry.” The announcer probably didn’t realize how profound this symmetry was.
In addition to the aptness of “golden” in this financial context, there is the following well-known relation between the golden ratio and the Fibonacci numbers. The ratio of any Fibonacci number to its predecessor is close to the golden ratio of 1.618 ..., and the bigger the numbers involved, the closer the two ratios become. Consider again, the Fibonacci numbers, 1, 2, 3, 5, 8, 13, 21, 34, 59, .... The ratios, 5/3, 8/5,
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13/8, 21/13, . . ., of successive Fibonacci numbers approach the golden ratio of 1.618 ... !
There’s no telling how an Elliott wave theorist dabbling in currencies at the time of the above exchange rate coincidence would have reacted to this beautiful harmony between money and mathematics. An unscrupulous, but numerate hoaxer might have even cooked up some flapdoodle sufficiently plausible to make money from such a “cosmic” connection.
The story could conceivably form the basis of a movie like Pi, since there are countless odd facts about phi that could be used to give various investing schemes a superficial plausibility. (The protagonist of Pi was a numerologically obsessed mathematician who thought he’d found the secret to just about everything in the decimal expansion of pi. He was pursued by religious zealots, greedy financiers, and others. The only sane character, his mentor, had a stroke, and the syncopated black-and-white cinematography was anxiety-inducing. Appealing as it was, the movie was mathematically nonsensical.) Unfortunately for investors and mathematicians alike, the lesson again is that more than beautiful harmonies are needed to make money on Wall Street. And Phi can’t match the cachet of Pi as a movie title either.
Moving Averages, Big Picture
People, myself included, sometimes ridicule technical analysis and the charts associated with it in one breath and then in the next reveal how much in (perhaps unconscious) thrall to these ideas they really are. They bring to mind the old joke about the man who complains to his doctor that his wife has for several years believed she’s a chicken. He would have sought help sooner, he says, “but we needed the eggs.” Without reading
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too much into this story except that we do sometimes seem to need the notions of technical analysis, let me finally proceed to examine some of these notions.
Investors naturally want to get a broad picture of the movement of the market and of particular stocks, and for this the simple technical notion of a moving average is helpful. When a quantity varies over time (such as the stock price of a company, the noontime temperature in Milwaukee, or the cost of cabbage in Kiev), one can, each day, average its values over, say, the previous 200 days. The averages in this sequence vary and hence the sequence is called a moving average, but the value of such a moving average is that it doesn’t move nearly as much as the stock price itself; it might be termed the phlegmatic average.
For illustration, consider the three-day moving average of a company whose stock is very volatile, its closing prices on successive days being: 8, 9, 10, 5, 6, 9. On the day the stock closed at 10, its three-day moving average was (8 + 9 + 10)/3 or 9. On the next day, when the stock closed at 5, its three- day moving average was (9 + 10 + 5)/3 or 8. When the stock closed at 6, its three-day moving average was (10 + 5 + 6)/3 or 7. And the next day, when it closed at 9, its three-day moving average was (5 + 6 + 9)/3 or 6.67.
If the stock oscillates in a very regular way and you are careful about the length of time you pick, the moving average may barely move at all. Consider an extreme case, the twenty- day moving average of a company whose closing stock prices oscillate with metronomic regularity. On successive days they are: 51, 52, 53, 54, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 46, 47, 48, 49, 50, 51, 52, 53, and so on, moving up and down around a price of 50. The twenty-day moving average on the day marked in bold is 50 (obtained by averaging the 20 numbers up to and including it). Likewise, the twenty-day moving average on the next day, when the stock is at 51, is
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also 50. It’s the same for the next day. In fact, if the stock price oscillates in this regular way and repeats itself every twenty days, the twenty-day moving average is always 50.
There are variations in the definition of moving averages (some weight recent days more heavily, others take account of the varying volatility of the stock), but they are all designed to smooth out the day-to-day fluctuations in a stock’s price in order to give the investor a look at broader trends. Software and online sites allow easy comparison of the stock’s daily movements with the slower-moving averages.
Technical analysts use the moving average to generate buy-sell rules. The most common such rule directs you to buy a stock when it exceeds its X-day moving average. Context determines the value of X, which is usually 10, 50, or 200 days. Conversely, the rule directs you to sell when the stock falls below its X-day moving average. With the regularly oscillating stock above, the rule would not lead to any gains or losses. It would call for you to buy the stock when it moves from 50, its moving average, to 51, and for you to sell it when it moves from 50 to 49. In the previous example of the three-day moving average, the rule would require that you buy the stock at the end of the third day and sell it at the end of the fourth, leading in this particular case to a loss.
The rule can work well when a stock fluctuates about a long-term upward- or downward-sloping course. The rationale for it is that trends should be followed, and that when a stock moves above its X-day moving average, this movement signals that a bullish trend has begun. Conversely, when a stock moves below its X-day moving average, the movement signals a bearish trend. I reiterate that mere upward (downward) movement of the stock is not enough to signal a buy (sell) order; a stock must move above (below) its moving average.
Alas, had I followed any sort of moving average rule, I would have been out of WCOM, which moved more or less
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steadily downhill for almost three years, long before I lost most of my investment in it. In fact, I never would have bought it in the first place. The security guard mentioned in chapter 1 did, in effect, use such a rule to justify the sale of the stocks in his pension plan.
There are a few studies, which I’ll get to later, suggesting that a moving average rule is sometimes moderately effective. Even so, however, there are several problems. One is that it can cost you a lot in commissions if the stock price hovers around the moving average and moves through it many times in both directions. Thus you have to modify the rule so that the price must move above or below its moving average by a non-trivial amount. You must also decide whether to buy at the end of the day the price exceeds the moving average or at the beginning of the next day or later still.
You can mine the voluminous time-series data on stock prices to find the X that has given the best returns for adhering to the X-day moving average buy-sell rule. Or you can complicate the rule by comparing moving averages over different intervals and buying or selling when these averages cross each other. You can even adapt the idea to day trading by using X-minute moving averages defined in terms of the mathematical notion of an integral. Optimal strategies can always be found after the fact. The trick is getting something that will work in the future; everyone’s very good at predicting the past. This brings us to the most trenchant criticism of the moving-average strategy. If the stock market is efficient, that is, if information about a stock is almost instantaneously incorporated into its price, then any stock’s future moves will be determined by random external events. Its past behavior, in particular its moving average, is irrelevant, and its future movement is unpredictable.
Of course, the market may not be all that efficient. There’ll be much more on this question in later chapters.
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Resistance and Support and All That
Two other important ideas from technical analysis are resistance and support levels. The argument for them assumes that people usually remember when they’ve been burned, insulted, or left out; in particular, they remember what they paid, or wish they had paid, for a stock. Assume a stock has been selling for $40 for a while and then drops to $32 before slowly rising again. The large number of people who bought it around $40 are upset and anxious to recoup their losses, so if the stock moves back up to $40, they’re likely to sell it, thereby driving the price down again. The $40 price is termed a resistance level and is considered an obstacle to further upward movement of the stock price.
Likewise, investors who considered buying at $32 but did not are envious of those who did buy at that price and reaped the 25 percent returns. They are eager to get these gains, so if the stock falls back to $32, they’re likely to buy it, driving the price up again. The $32 price is termed a support level and is considered an obstacle to further downward movement.
Since stocks often seem to meander between their support and resistance levels, one rule followed by technical analysts is to buy the stock when it “bounces” off its support level and sell it when it “bumps” up against its resistance level. The rule can, of course, be applied to the market as a whole, inducing investors to wait for the Dow or the S&P to definitively turn up (or down) before buying (or selling).
Since chartists tend to view support levels as shaky, often temporary, floors and resistance levels as slightly stronger, but still temporary, ceilings, there is a more compelling rule involving these notions. It instructs you to buy the stock if the rising price breaks through the resistance level and to sell it if the falling price breaks through the support level. In both these cases breaking through indicates that the stock has
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moved out of its customary channel and the rule counsels investors to follow the new trend.
As with the moving-average rules, there are a few studies that indicate that resistance-support rules sometimes lead to moderate increases in returns. Against this there remains the perhaps dispiriting efficient-market hypothesis, which maintains that past prices, trends, and resistance and support levels provide no evidence about future movements.
Innumerable variants of these rules exist and they can be combined in ever more complicated ways. The resistance and support levels can change and trend up or down in a channel or with the moving average, for example, rather than remain fixed. The rules can also be made to take account of variations in a stock’s volatility as well.
These variants depend on price patterns that often come equipped with amusing names. The “head and shoulders” pattern, for example, develops after an extended upward trend. It is comprised of three peaks, the middle and highest one being the head, and the smaller left and right ones (earlier and later ones, that is) being the shoulders. After falling below the right shoulder and breaking through the support line connecting the lows on either side of the head, the stock price has, technical chartists aver, reversed direction and a downward trend has begun, so sell.
Similar metaphors describe the double-bottom trend reversal. It develops after an extended downward trend and is comprised of two successive troughs or bottoms with a small peak between them. After bouncing off the second bottom, the stock has, technical chartists again aver, reversed direction and an upward trend has begun, so buy.
These are nice stories, and technical analysts tell them with great earnestness and conviction. Even if everyone told the same stories (and they don’t), why should they be true? Presumably the rationale is ultimately psychological or perhaps
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sociological or systemic, but exactly what principles justify these beliefs? Why not triple or quadruple bottoms? Or two heads and shoulders? Or any of innumerable other equally plausible, equally risible patterns? What combination of psychological, financial, or other principles has sufficient specificity to generate effective investment rules?
As with Elliott waves, scale is an issue. If we go to the level of ticks, we can find small double bottoms and little heads and tiny shoulders all over. We find them also in the movement of broad market indices. And do these patterns mean for the market as whole what they are purported to mean for individual stocks? Is the “double-dip” recession discussed in early 2002 simply a double bottom?
Predictability and Trends
I often hear people swear that they make money using the rules of technical analysis. Do they really? The answer, of course, is that they do. People make money using all sorts of strategies, including some involving tea leaves and sunspots. The real question is: Do they make more money than they would investing in a blind index fund that mimics the performance of the market as a whole? Do they achieve excess returns? Most financial theorists doubt this, but there is some tantalizing evidence for the effectiveness of momentum strategies or short-term trend-following. Economists Nara- simhan Jegadeesh and Sheridan Titman, for example, have written several papers arguing that momentum strategies result in moderate excess returns and that, having done so over the years, their success is not the result of data mining. Whether this alleged profitability—many dispute it—is due to overreactions among investors or to the short-term persistence of the impact of companies’ earnings reports, they don’t
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say. They do seem to point to behavioral models and psychological factors as relevant.
William Brock, Josef Lakonishok, and Blake LeBaron have also found some evidence that rules based on moving averages and the notions of resistance and support are moderately effective. They focus on the simplest rules, but many argue that their results have not been replicated on new stock data.
More support for the existence of technical exploitability comes from Andrew Lo, who teaches at M.I.T., and Craig MacKinlay, from the Wharton School. They argue in their book, A Non-Random Walk Down Wall Street, that in the short run overall market returns are, indeed, slightly positively correlated, much like the local weather. A hot, sunny day is a bit more likely to be followed by another one, just as a good week in the market is a bit more likely to be followed by another one. Likewise for rainy days and bad markets. Employing state-of-the-art tools, Lo and MacKinlay also claim that in the long term the prognosis changes: Individual stock prices display a slight negative correlation. Winners are a bit more likely to be losers three to five years hence and vice versa.
They also bring up an interesting theoretical possibility. Weeding out some of the details, let’s assume for the sake of the argument (although Lo and MacKinlay don’t) that the thesis of Burton Malkiel’s classic book, A Random Walk Down Wall Street, is true and that the movement of the market as a whole is entirely random. Let’s also assume that each stock, when its fluctuations are examined in isolation, moves randomly. Given these assumptions it would nevertheless still be possible that the price movements of, say, 5 percent of stocks accurately predict the price movements of a different 5 percent of stocks one week later.
The predictability comes from cross-correlations over time between stocks. (These associations needn’t be causal, but might merely be brute facts.) More concretely, let’s say stock
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X, when looked at in isolation, fluctuates randomly from week to week, as does stock Y. Yet if X’s price this week often predicts Y’s next week, this would be an exploitable opportunity and the strict random-walk hypothesis would be wrong. Unless we delved deeply into such possible cross-correlations among stocks, all we would see would be a randomly fluctuating market populated by randomly fluctuating stocks. Of course, I’ve employed the typical mathematical gambit of considering an extreme case, but the example does suggest that there may be relatively simple elements of order in a market that appears to fluctuate randomly.
There are other sorts of stock price anomalies that can lead to exploitable opportunities. Among the most well-known are so-called calendar effects whereby the prices of stocks, primarily small-firm stocks, rise disproportionately in January, especially during the first week of January. (The price of WCOM rose significantly in January 2001, and I was hoping this rise would repeat itself in January 2002. It didn’t.) There has been some effort to explain this by citing tax law concerns that end with the close of the year, but the effect also seems to hold in countries with different tax laws. Moreover, unusual returns (good or bad) occur not only at the turn of the year, but, as Richard Thaler and others have observed, at the turn of the month, week, and day as well as before holidays. Again, poorly understood behavioral factors seem to be involved.
Technical Strategies and Blackjack
Most academic financial experts believe in some form of the random-walk theory and consider technical analysis almost indistinguishable from a pseudoscience whose predictions are either worthless or, at best, so barely discernibly better than chance as to be unexploitable because of transaction costs.
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I’ve always leaned toward this view, but I’ll reserve my more nuanced judgment for later in the book. In the meantime, I’d like to point out a parallel between market strategies such as technical analysis in one of its many forms and blackjack strategies. (There are, of course, great differences too.)
Blackjack is the only casino game of chance whose outcomes depend on past outcomes. In roulette, the previous spins of the wheel have no effect on future spins. The probability of red on the next spin is 18/38, even if red has come up on the five previous spins. The same is true with dice, which are totally lacking in memory. The probability of rolling a 7 with a pair of dice is 1/6, even if the four previous rolls have not resulted in a single 7. The probability of six reds in a row is (18/38)6; the probability of five 7s in a row is (1/6)5. Each spin and each roll are independent of the past.
A game of blackjack, on the other hand, is sensitive to its past. The probability of drawing two aces in a row from a deck of cards is not (4/52 x 4/52) but rather (4/52 x 3/51). The second factor, 3/51, is the probability of choosing another ace given that the first card chosen was an ace. In the same way the probability that a card drawn from a deck will be a face card (jack, queen, or king) given that only three of the thirty cards drawn so far have been face cards is not 12/52, but a much higher 9/22.
This fact—that (conditional) probabilities change according to the composition of the remaining portion of the deck—is the basis for various counting strategies in blackjack that involve keeping track of how many cards of each type have already been drawn and increasing one’s bet size when the odds are (occasionally and slightly) in one’s favor. Some of these strategies, followed carefully, do work. This is evidenced by the fact that some casinos supply burly guards free of charge to abruptly escort successful counting practitioners from the premises.
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The vast majority of people who try these strategies (or, worse, others of their own devising) lose money. It would make no sense, however, to point to the unrelenting average losses of blackjack players and maintain that this proves that there is no effective betting strategy for playing the game.
Blackjack is much simpler than the stock market, of course, which depends on vastly more factors as well as on the actions and beliefs of other investors. But the absence of conclusive evidence for the effectiveness of various investing rules, technical or otherwise, does not imply that no effective rules exist. If the market’s movements are not completely random, then it has a kind of memory within it, and investing rules depending on this memory might be effective. Whether they would remain so if widely known is very dubious, but that is another matter.
Interestingly, if there were an effective technical trading strategy, it wouldn’t need any convincing rationale. Most investors would be quite pleased to use it, as most blackjack players use the standard counting strategy, without understanding why it works. With blackjack, however, there is a compelling mathematical explanation for those who care to study it. By contrast an effective technical trading strategy might be found that was beyond the comprehension not only of the people using it but of everyone. It might simply work, at least temporarily. In Plato’s allegory of the cave the benighted see only the shadows on the wall of the cave and not the real objects behind them that are causing the shadows. If they were really predictive, investors would be quite content with the shadows alone and would simply take the cave to be a bargain basement.
The next segment is a bit of a lark. It offers a suggestive hint for developing a novel and counterintuitive investment strategy that has a bit of the feel of technical analysis.
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Winning Through Losing?
The old joke about the store owner losing money on every sale but making it up in volume may have a kernel of truth to it. An interesting new paradox by Juan Parrondo, a Spanish physicist, brings the joke to mind. It deals with two games, each of which results in steady losses over time. When these games are played in succession in random order, however, the result is a steady gain. Bad bets strung together to produce big winnings—very strange indeed!
To understand Parrondo’s paradox, let’s switch from a financial to a spatial metaphor. Imagine you are standing on stair 0, in the middle of a very long staircase with 1,001 stairs numbered from -500 to 500 (-500, -499, -498, . . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, ..., 498, 499, 500). You want to go up rather than down the staircase and which direction you move depends on the outcome of coin flips. The first game—let’s call it game S—is very Simple. You flip a coin and move up a stair whenever it comes up heads and down a stair whenever it comes up tails. The coin is slightly biased, however, and comes up heads 49.5 percent of the time and tails 50.5 percent. It’s clear that this is not only a boring game but a losing one. If you played it long enough, you would move up and down for a while, but almost certainly you would eventually reach the bottom of the staircase.
The second game—let’s continue to wax poetic and call it game C—is more Complicated, so bear with me. It involves two coins, one of which, the bad one, comes up heads only 9.5 percent of the time, tails 90.5 percent. The other coin, the good one, comes up heads 74.5 percent of the time, tails 25.5 percent. As in game S, you move up a stair if the coin you flip comes up heads and you move down one if it comes up tails.
But which coin do you flip? If the number of the stair you’re on is a multiple of 3 (that is, ..., -9, -6, -3, 0, 3, 6, 9,
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12, . . . ), you flip the bad coin. If the number of the stair you’re on is not a multiple of 3, you flip the good coin. (Note: Changing these odd percentages and constraints may affect the game’s outcome.)
Let’s go through game C’s dance steps. If you were on stair number 5, you would flip the good coin to determine your direction, whereas if you were on stair number 6, you would flip the bad coin. The same holds for the negatively numbered stairs. If you were on stair number -2 and playing game C, you would flip the good coin, whereas if you were on stair number -9, you would flip the bad coin.
Though less obviously so than in game S, game C is also a losing game. If you played it long enough, you would almost certainly reach the bottom of the staircase eventually. Game C is a losing game because the number of the stair you’re on is a multiple of 3 more often than a third of the time and thus you must flip the bad coin more often than a third of the time. Take my word for this or read the next paragraph to get a better feel for why it is.
(Assume that you’ve just started playing game C. Since you’re on stair number 0, and 0 is a multiple of 3, you would flip the bad coin, which lands heads with probability less than 10 percent, and you would very likely move down to stair number -1. Then, since -1 is not a multiple of 3, you would flip the good coin, which lands heads with probability almost 75 percent, and would probably move back up to stair 0. You may move up and down like this for a while. Occasionally, however, after the bad coin lands tails, the good coin, which lands tails almost 25 percent of the time, will land tails twice in succession, and you would move down to stair number -3, where the pattern will likely begin again. This latter downward pattern happens slightly more frequently (with probability .905 x .255 x .255) than does a rare head on the bad coin being followed by two heads on the good one (with
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probability .095 x .745 x .745) and your moving up three stairs as a consequence. So-called Markov chains are needed for a fuller analysis.)
So far, so what? Game S is simple and results in steady movement down the staircase to the bottom, and game C is complicated and also results in steady movement down the staircase to the bottom. Parrondo’s fascinating discovery is that if you play these two games in succession in random order (keeping your place on the staircase as you switch between games), you will steadily ascend to the top of the staircase. Alternatively, if you play two games of S followed by two games of C followed by two games of S and so on, all the while keeping your place on the staircase as you switch between games, you will also steadily rise to the top of the staircase. (You might want to look up M. C. Escher’s paradoxical drawing, “Ascending and Descending” for a nice visual analog to Parrondo’s paradox.)
Standard stock-market investments cannot be modeled by games of this type, but variations of these games might conceivably give rise to counterintuitive investment strategies. The probabilities might be achieved, for example, by complicated combinations of various financial instruments (options, derivatives, and so on), but the decision which coin (which investment, that is) to flip (to make) in game C above would, it seems, have to depend upon something other than whether one’s holdings were worth a multiple of $3.00 (or a multiple of $3,000.00). Perhaps the decision could depend in some way on the cross-correlation between a pair of stocks or turn on the value of some index being a multiple of 3.
If strategies like this could be made to work, they would yield what one day might be referred to as Parrondo profits.
Finally, let’s consider a companion paradox of sorts that might be called “losing through winning” and that may help explain why companies often overpaid for small companies
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they were purchasing during the bubble in the late ’90s. Professor Martin Shubik has regularly auctioned off $1 to students in his classes at Yale. The bidding takes place at fifty-four intervals, and the highest bidder gets the dollar, of course, but the second highest bidder is required to pay his bid as well. Thus, if the highest bid is 504 and you are second highest at 454, the leader stands to make 504 on the deal and you stand to lose 454 on it if bidding stops. You have an incentive to up your bid to at least 554, but after you’ve done so the other bidder has an even bigger incentive to raise his bid as well. In this way a one dollar bill can be successfully auctioned off for two, three, four, or more dollars.
If several companies are bidding on a small company and the cost of the preliminary legal, financial, and psychological efforts required to purchase the company are a reasonable fraction of the cost of the company, the situation is formally similar to Shubik’s auction. One or more of the bidding companies might feel compelled to make an exorbitant preemptive offer to avoid the fate of the losing bidder on the $1. WorldCom’s purchase of the web-hosting company Digex in 2000 for $6 billion was, I suspect, such an offer. John Sidg- more, the CEO who succeeded Bernie Ebbers, says that Digex was worth no more than $50 million, but that Ebbers was obsessed with beating out Global Crossing for the company.
The purchase is much more bizarre than Parrondo’s paradox
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