From Paradox to Complexity
Groucho Marx vowed that he’d never join a club that would be willing to accept him as a member. Epimenides the Cretan exclaimed (almost) inconsistently, “All Cretans are liars.” The prosecutor booms, “You must answer Yes or No. Will your next word be ‘No’?” The talk show guest laments that her brother is an only child. The author of an investment book suggests that we follow the tens of thousands of his readers who have gone against the crowd.
Warped perhaps by my study of mathematical logic and its emphasis on paradoxes and self-reference, I’m naturally interested in the paradoxical and self-referential aspects of the market, particularly of the Efficient Market Hypothesis. Can it be proved? Can it be disproved? These questions beg a deeper question. The Efficient Market Hypothesis is, I think, neither necessarily true nor necessarily false.
The Paradoxical Efficient Market Hypothesis
If a large majority of investors believe in the hypothesis, they would all assume that new information about a stock would quickly be reflected in its price. Specifically, they would affirm that since news almost immediately moves the price up or
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down, and since news can’t be predicted, neither can changes in stock prices. Thus investors who subscribe to the Efficient Market Hypothesis would further believe that looking for trends and analyzing companies’ fundamentals is a waste of time. Believing this, they won’t pay much attention to new developments. But if relatively few investors are looking for an edge, the market will not respond quickly to new information. In this way an overwhelming belief in the hypothesis ensures its falsity.
To continue with this cerebral somersault, recall now a rule of logic: Sentences of the form “H implies I” are equivalent to those of the form “not I implies not H.” For example, the sentence “heavy rain implies that the ground will be wet” is logically equivalent to “dry ground implies the absence of heavy rain.” Using this equivalence, we can restate the claim that overwhelming belief in the Efficient Market Hypothesis leads to (or implies) its falsity. Alternatively phrased, the claim is that if the Efficient Market Hypothesis is true, then it’s not the case that most investors believe it to be true. That is, if it’s true, most investors believe it to be false (assuming almost all investors have an opinion and each either believes it or disbelieves it).
Consider now the inelegantly named Sluggish Market Hypothesis, the belief that the market is quite slow in responding to new information. If the vast majority of investors believe the Sluggish Market Hypothesis, then they all would believe that looking for trends and analyzing companies is well worth their time and, by so exercising themselves, they would bring about an efficient market. Thus, if most investors believe the Sluggish Market Hypothesis is true, they will by their actions make the Efficient Market Hypothesis true. We conclude that if the Efficient Market Hypothesis is false, then it’s not the case that most investors believe the
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Sluggish Market Hypothesis to be true. That is, if the Efficient Market Hypothesis is false, then most investors believe it (the EMH) to be true. (You may want to read over the last few sentences in a quiet corner.)
In summary, if the Efficient Market Hypothesis is true, most investors won’t believe it, and if it’s false, most investors will believe it. Alternatively stated, the Efficient Market Hypothesis is true if and only if a majority believes it to be false. (Note that the same holds for the Sluggish Market Hypothesis.) These are strange hypotheses indeed!
Of course, I’ve made some big assumptions that may not hold. One is that if an investor believes in one of the two hypotheses, then he disbelieves in the other, and almost all believe in one or the other. I’ve also assumed that it’s clear what “large majority” means, and I’ve ignored the fact that it sometimes requires very few investors to move the market. (The whole argument could be relativized to the set of knowledgeable traders only.)
Another gap in the argument is that any suspected deviations from the Efficient Market Hypothesis can always be attributed to mistakes in asset pricing models, and thus the hypothesis can’t be conclusively rejected for this reason either. Maybe some stocks or kinds of stock are riskier than our pricing models allow for and that’s why their returns are higher. Nevertheless, I think the point remains: The truth or falsity of the Efficient Market Hypothesis is not immutable but depends critically on the beliefs of investors. Furthermore, as the percentage of investors who believe in the hypothesis itself varies, the truth of the hypothesis varies inversely with it.
On the whole, most investors, professionals on Wall Street, and amateurs everywhere, disbelieve in it, so for this reason I think it holds, but only approximately and only most of the time.
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The Prisoner’s Dilemma and the Market
So you don’t believe in the Efficient Market Hypothesis. Still, it’s not enough that you discover simple and effective investing rules. Others must not find out what you’re doing, either by inference or by reading your boastful profile in a business magazine. The reason for secrecy, of course, is that without it, simple investing rules lead to more and more complicated ones, which eventually lead to zero excess returns and a reliance on chance.
This inexorable march toward increased complexity arises from the actions of your co-investors, who, if they notice (or infer, or are told) that you are performing successfully on the basis of some simple technical trading rule, will try to do the same. To take account of their response, you must complicate your rule and likely decrease your excess returns. Your more complicated rule will, of course, also inspire others to try to follow it, leading to further complications and a further decline in excess returns. Soon enough your rule assumes a near-random complexity, your excess returns are reduced essentially to zero, and you’re back to relying on chance.
Of course, your behavior will be the same if you learn of someone else’s successful performance. In fact, a situation arises that is clarified by the classic “prisoner’s dilemma,” a useful puzzle originally framed in terms of two people in prison.
Suspected of committing a major crime, the two are apprehended in the course of committing some minor offense. They’re then interrogated separately, and each is given the choice of confessing to the major crime and thereby implicating his partner or remaining silent. If they both remain silent, they’ll each get one year in prison. If one confesses and the other doesn’t, the one who confesses will be rewarded by being set free, while the other one will get a five-year term. If they both confess, they can both expect to spend three years
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in prison. The cooperative option (cooperative with the other prisoner, that is) is to remain silent, while the non-cooperative option is to confess. Given the payoffs and human psychology, the most likely outcome is for both to confess; the best outcome for the pair as a pair is for both to remain silent; the best outcome for each prisoner as an individual is to confess and have one’s partner remain silent.
The charm of the dilemma has nothing to do with any interest one might have in prisoners’ rights. (In fact, it has about as much relevance to criminal justice as the four-color-map theorem has to geography.) Rather, it provides the logical skeleton for many situations we face in everyday life. Whether we’re negotiators in business, spouses in a marriage, or nations in a dispute, our choices can often be phrased in terms of the prisoner’s dilemma. If both (all) parties pursue their own interests exclusively and do not cooperate, the outcome is worse for both (all) of them; yet in any given situation, any given party is better off not cooperating. Adam Smith’s invisible hand ensuring that individual pursuits bring about group well-being is, at least in these situations (and some others), quite arthritic.
The dilemma has the following multi-person market version: Investors who notice some exploitable stock market anomaly may either act on it, thereby diminishing its effectiveness (the non-cooperative option) or ignore it, thereby saving themselves the trouble of keeping up with developments (the cooperative option). If some ignore it and others act on it, the latter will receive the biggest payoffs, the former the smallest. As in the standard prisoner’s dilemma, the logical response for any player is to take the non-cooperative option and act on any anomaly likely to give one an edge. This response leads to the “arms race” of ever more complex technical trading strategies. People search for special knowledge, the result eventually becomes common knowledge, and the dynamic between the two generates the market.
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This searching for an edge brings us to the social value of stock analysts and investment professionals. Although the recipients of an abundance of bad publicity in recent years, they provide a most important service: By their actions, they help turn special knowledge into common knowledge and in the process help make the market relatively efficient. Absent a draconian rewiring of human psychology and an accompanying draconian rewiring of our economic system, this accomplishment is an impressive and vital one. If it means being “noncooperative” with other investors, then so be it. Cooperation is, of course, generally desirable, but cooperative decisionmaking among investors seems to smack of totalitarianism.
Pushing the Complexity Horizon
The complexity of trading rules admits of degrees. Most of the rules to which people subscribe are simple, involving support levels, P/E ratios, or hemlines and Super Bowls, for example. Others, however, are quite convoluted and conditional. Because of the variety of possible rules, I want to take an oblique and abstract approach here. The hope is that this approach will yield insights that a more pedestrian approach misses. Its key ingredient is the formal definition of (a type of) complexity. An intuitive understanding of this notion tells us that someone who remembers his eight-digit password by means of an elaborate, long-winded saga of friends’ addresses, children’s ages, and special anniversaries is doing something silly. Mnemonic rules make sense only when they’re shorter than what is to be remembered.
Let’s back up a bit and consider how we might describe the following sequences to an acquaintance who couldn’t see them. We may imagine the Is to represent upticks in the
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price of a stock and the Os downticks or perhaps up-and- down days.
1. 0101010101010101010101010...
2. 0101101010 101101010101011...
3. 1000101101101100010101100...
The first sequence is the simplest, an alternation of 0s and Is. The second sequence has some regularity to it, a single 0 alternating sometimes with a 1, sometimes with two Is, while the third sequence doesn’t seem to manifest any pattern at all. Observe that the precise meaning of “ ...” in the first sequence is clear; it is less so in the second sequence, and not at all clear in the third. Despite this, let’s assume that these sequences are each a trillion bits long (a bit is a 0 or a 1) and continue on “in the same way.”
Motivated by examples like this, the American computer scientist Gregory Chaitin and the Russian mathematician A. N. Kolmogorov defined the complexity of a sequence of 0s and Is to be the length of the shortest computer program that will generate (that is, print out) the sequence in question.
A program that prints out the first sequence above can consist simply of the following recipe: print a 0, then a 1, and repeat a half trillion times. Such a program is quite short, especially compared to the long sequence it generates. The complexity of this first trillion-bit sequence may be only a few hundred bits, depending to some extent on the computer language used to write the program.
A program that generates the second sequence would be a translation of the following: Print a 0 followed by either a single 1 or two Is, the pattern of the intervening Is being one, two, one, one, one, two, one, one, and so on. Any program that prints out this trillion-bit sequence would have to be
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quite long so as to fully specify the “and so on” pattern of the intervening Is. Nevertheless, because of the regular alternation of Os and either one or two Is, the shortest such program will be considerably shorter than the trillion-bit sequence it generates. Thus the complexity of this second sequence might be only, say, a quarter trillion bits.
With the third sequence (the commonest type) the situation is different. This sequence, let us assume, remains so disorderly throughout its trillion-bit length that no program we might use to generate it would be any shorter than the sequence itself. It never repeats, never exhibits a pattern. All any program can do in this case is dumbly list the bits in the sequence: print 1, then 0, then 0, then 0, then 1, then 0, then 1, . ... There is no way the . . . can be compressed or the program shortened. Such a program will be as long as the sequence it’s supposed to print out, and thus the third sequence has a complexity of approximately a trillion.
A sequence like the third one, which requires a program as long as itself to be generated, is said to be random. Random sequences manifest no regularity or order, and the programs that print them out can do nothing more than direct that they be copied: print 10001011011.... These programs cannot be abbreviated; the complexity of the sequences they generate is equal to the length of these sequences. By contrast, ordered, regular sequences like the first can be generated by very short programs and have complexity much less than their length.
Returning to stocks, different market theorists will have different ideas about the likely pattern of 0s and Is (downs and upticks) that can be expected. Strict random walk theorists are likely to believe that sequences like the third characterize price movements and that the market’s movements are therefore beyond the “complexity horizon” of human forecasters (more complex than we, or our brains, are, were we
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expressed as sequences of Os and Is). Technical and fundamental analysts might be more inclined to believe that sequences like the second characterize the market and that there are pockets of order amidst the noise. It’s hard to imagine anyone believing that price movements follow sequences as regular as the first except, possibly, those who send away “only $99.95 for a complete set of tapes that explain this revolutionary system.”
I reiterate that this approach to stock price movements is rather stark, but it does nevertheless “locate” the debate. People who believe there is some pattern to the market, whether exploitable or not, will believe that its movements are characterized by sequences of complexity somewhere between those of type two and type three above.
A rough paraphrase of Kurt Godel’s famous incompleteness theorem of mathematical logic, due to the aforementioned Gregory Chaitin, provides an interesting sidelight on this issue. It states that if the market were random, we might not be able to prove it. The reason: encoded as a sequence of Os and Is, a random market would, it seems plausible to assume, have complexity greater than that of our own were we also so encoded; it would be beyond our complexity horizon. From the definition of complexity it follows that a sequence can’t generate another sequence of greater complexity than itself. Thus if a person were to predict the random market’s exact gyrations, the market would have to be less complex than the person, contrary to assumption. Even if the market isn’t random, there remains the possibility that its regularities are so complex as to be beyond our complexity horizons.
In any case, there is no reason why the complexity of price movements as well as the complexity of investor/computer blends cannot change over time. The more inefficient the market is, the smaller the complexity of its price movements, and the more likely it is that tools from technical and fundamental
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analysis will prove useful. Conversely, the more efficient the market is, the greater the complexity of price movements, and the closer the approach to a completely random sequence of price changes.
Outperforming the market requires that one remain on the cusp of our collective complexity horizon. It requires faster machines, better data, improved models, and the smarter use of mathematical tools, from conventional statistics to neural nets (computerized learning networks, the connections between the various nodes of which are strengthened or weakened over a period of training). If this is possible for anyone or any group to achieve, it’s not likely to remain so for long.
Game Theory and
Supernatural Investor/Psychologists
But what if, contrary to fact, there were an entity possessing sufficient complexity and speed that it was able with reasonably high probability to predict the market and the behavior of individuals within it? The mere existence of such an entity leads to Newcombe’s paradox, a puzzle that calls into question basic principles of game theory.
My particular variation of Newcombe’s paradox involves the World Class Options Market Maker (WCOMM), which (who?) claims to have the power to predict with some accuracy which of two alternatives a person will choose. Imagine further that WCOMM sets up a long booth on Wall Street to demonstrate its abilities.
WCOMM explains that it tests people by employing two portfolios. Portfolio A contains a $1,000 treasury bill, whereas portfolio B (consisting of either calls or puts on WCOM stock) is either worth nothing or $1,000,000. For each person in line at the demonstration, WCOMM has reserved a portfolio of
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each type at the booth and offers each person the following choice: He or she can choose to take portfolio B alone or choose to take both portfolios A and B. However, and this is crucial, WCOMM also states that it has used its unfathomable powers to analyze the psychology, investment history, and trading style of everyone in line as well as general market conditions, and if it believes that a person will take both portfolios, it has ensured that portfolio B will be worthless. On the other hand, if WCOMM believes that a person will trust its wisdom and take only portfolio B, it has ensured that portfolio B will be worth $1,000,000. After making these announcements, WCOMM leaves in a swirl of digits and stock symbols, and the demonstration proceeds.
Investors on Wall Street see for themselves that when a person in the long line chooses to take both portfolios, most of the time (say with probability 90 percent) portfolio B is worthless and the person gets only the $1,000 treasury bill in portfolio A. They also note that when a person chooses to take the contents of portfolio B alone, most of the time it’s worth $1,000,000.
After watching the portfolios placed before the people in line ahead of me and seeing their choices and the consequences, I’m finally presented with the two portfolios prepared for me by WCOMM. Despite the evidence I’ve seen, I see no reason not to take both portfolios. WCOMM is gone, perhaps to the financial district of London or Frankfurt or Tokyo, to make similar offers to other investors, and portfolio B is either worth $1,000,000 or not, so why not take both portfolios and possibly get $1,001,000. Alas, WCOMM read correctly the skeptical smirk on my face and after opening my portfolios, I walk away with only $1,000. My portfolio B contains call options on WCOM with a strike price of 20, when the stock itself is selling at $1.13.
The paradox, due to the physicist William Newcombe (not the Newcomb of Benford’s Law, but the same mocking four
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letters WCOM) and made well-known by the philosopher Robert Nozick, raises other issues. As mentioned, it makes problematical which of two game-theoretic principles one should use in making decisions, principles that shouldn’t conflict.
The “dominance” principle tells us to take both portfolios since, whether portfolio B contains options worth $1,000,000 or not, the value of two portfolios is at least as great as the value of one. (If portfolio B is worthless, $1,000 is greater than $0, and if portfolio B is worth $1,000,000, $1,001,000 is greater than $1,000,000.)
On the other hand, the “maximization of expected value” principle tells us to take only portfolio B since the expected value of doing so is greater. (Since WCOMM is right about 90 percent of the time, the expected value of taking only portfolio B is (.90 x $1,000,000) + (.10 x $0), or $900,000, whereas the expected value of taking both is (.10 x $1,001,000) + (.90 x $1,000), or $101,000.) The paradox is that both principles seem reasonable, yet they counsel different choices.
This raises other general philosophical matters as well, but it reminds me of my resistance to following the WCOM- fleeing crowd, most of whose B portfolios contained puts on the stock worth $1,000,000.
One conclusion that seems to follow from the above is that such supernatural investor/psychologists are an impossibility. For better or worse, we’re on our own.
Absurd Emails and the WorldCom Denouement
A natural reaction to the vagaries of chance is an attempt at control, which brings me to emails regarding WorldCom that, Herzog-like, I sent to various influential people. I had grown tired of carrying on one-sided arguments with CNBC’s always
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perky Maria Bartiromo and always apoplectic James Cramer as they delivered the relentlessly bad news about WorldCom. So in fall 2001, five or six months before its final swoon, I contacted a number of online business commentators critical of WorldCom’s past performance and future prospects. Having spent too much time in the immoderate atmosphere of WorldCom chatrooms, I excoriated them, though mildly, for their shortsightedness and exhorted them to look at the company differently.
Finally, out of frustration with the continued decline of WCOM stock, I emailed Bernie Ebbers, then the CEO, in early February 2002 suggesting that the company was not effectively stating its case and quixotically offering to help by writing copy. I said I’d invested heavily in WorldCom, as did family and friends at my suggestion, that I could be a persuasive wordsmith when I believed in something, and WorldCom, I believed, was well positioned but dreadfully undervalued. UUNet, the “backbone” of much of the Internet, was, I fatuously informed the CEO of the company, a gem in and of itself.
I knew, even as I was writing them, that sending these electronic epistles was absurd, but it gave me the temporary illusion of doing something about this recalcitrant stock other than dumping it. Investing in it had originally seemed like a no-brainer. The realization that doing so had indeed been a no-brainer was glacially slow in arriving. During the 2001-2002 academic year, I took the train once a week from Philadelphia to New York to teach a course on “numbers in the news” at the Columbia School of Journalism. Spending the two and a half hours of the commute out of contact with WCOM’s volatile movements was torturous, and upon emerging from the subway, I’d run to my office computer to check what had happened. Not exactly the behavior of a sage long-term investor; my conduct even then suggested to me a rather dim-witted addict.
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Recalling the two or three times I almost got out of the stock is dispiriting as well. The last time was in April 2002. Amazingly, I was even then still somewhat in thrall to the idea of averaging down, and when the price dipped below $5, I bought more WCOM shares. Around the middle of the month, however, I did firmly and definitively resolve to sell. By Friday, April 19, WCOM had risen to over $7, which would have allowed me to recoup at least a small portion of my losses, but I didn’t have time to sell that morning. I had to drive to northern New Jersey to give a long-promised lecture at a college there. When it was over, I wondered whether to return home to sell my shares or simply use the college’s computer to log onto my Schwab account to do so. I decided to go home, but there was so much traffic on the cursed New Jersey turnpike that afternoon that I didn’t arrive until 4:05, after the market had closed. I had to wait until Monday.
Investors are often nervous about holding volatile stocks over the weekend, and I was no exception. My anxiety was well-founded. Later that evening there was news about impending cuts in WorldCom’s bond ratings and another announcement from the SEC regarding its comprehensive investigation of the company. The stock lost more than a third of its value by Monday, when I did finally sell the stock at a huge loss. A few months later the stock completely collapsed to $.09 upon revelations of massive accounting fraud.
Why had I violated the most basic of investing fundamentals: Don’t succumb to hype and vaporous enthusiasm; even if you do, don’t put too many eggs in one basket (especially with the uncritical sunny-side up); even if you do this, don’t forget to insure against sudden drops (say with puts, not calls); and even if you do this too, don’t buy on margin. After selling my shares, I felt as if I were gradually and groggily coming out of a self-induced trance. I’d long known about one of the earliest “stock” hysterias on record, the sevenA
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teenth century tulip bulb craze in Holland. After its collapse, people also spoke of waking up and realizing that they were stuck with nearly worthless bulbs and truly worthless options to buy more of them. I smiled ruefully at my previous smug dismissal of people like the tulip bulb “investors.” I was as vulnerable to transient delirium as the dimmest bulb-buying bulbs.
I’ve followed the ongoing drama of the WorldCom story— the fraud investigations, various prosecutions, new managers, promised reforms, and court settlements—and, oddly perhaps, the publicity surrounding the scandals and their aftermath has distanced me from my experience and lessened its intensity. My losses have become less a small personal story and more (a part of) a big news story, less a result of my mistakes and more a consequence of the company’s behavior. This shifting of responsibility is neither welcome nor warranted. For reasons of fact and of temperament, I continue to think of myself as having been temporarily infatuated rather than deeply victimized. Remnants of my fixation persist, and I still sometimes wonder what might have happened if WorldCom’s deal with Sprint hadn’t been foiled, if Ebbers hadn’t borrowed $400 million (or more), if Enron hadn’t imploded, if this or that or the other event hadn’t occurred before I sold my shares. My recklessness might then have been seen as daring. Post hoc stories always seem right, whatever the pre-existing probabilities.
One fact remains incontrovertible: Narratives and numbers coexist uneasily on Wall Street. Markets, like people, are largely rational beasts occasionally provoked and disturbed by their underlying animal spirits. The mathematics discussed in this book is often helpful in understanding (albeit not beating) the market, but I’d like to end with a psychological caveat. The basis for the application of the mathematical tools discussed herein is the sometimes shifty and always shifting attitudes of investors. Since these psychological states
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are to a large extent imponderable, anything that depends on them is less exact than it appears.
The situation reminds me a bit of the apocryphal story of the way cows were weighed in the Old West. First the cowboys would find a long, thick plank and place the middle of it on a large, high rock. Then they’d attach the cow to one end of the plank with ropes and tie a large boulder to the other end. They’d carefully measure the distance from the cow to the rock and from the boulder to the rock. If the plank didn’t balance, they’d try another big boulder and measure again. They’d keep this up until a boulder exactly balanced the cattle. After solving the resulting equation that expresses the cow’s weight in terms of the distances and the weight of the boulder, there would be only one thing left for them to do: They would have to guess the weight of the boulder. Once again the mathematics may be exact, but the judgments, guesses, and estimates supporting its applications are anything but.
More apropos of the self-referential nature of the market would be a version in which the cowboys had to guess the weight of the cow whose weight varied depending on their collective guesses, hopes, and fears. Bringing us full circle to Keynes’s beauty contest, albeit in a rather forced, more bovine mode, I conclude that despite rancid beasts like WorldCom, I’m still rather fond of the pageant that is the market. I just wish I had a better (and secret) method for weighing the cows.
Warped perhaps by my study of mathematical logic and its emphasis on paradoxes and self-reference, I’m naturally interested in the paradoxical and self-referential aspects of the market, particularly of the Efficient Market Hypothesis. Can it be proved? Can it be disproved? These questions beg a deeper question. The Efficient Market Hypothesis is, I think, neither necessarily true nor necessarily false.
The Paradoxical Efficient Market Hypothesis
If a large majority of investors believe in the hypothesis, they would all assume that new information about a stock would quickly be reflected in its price. Specifically, they would affirm that since news almost immediately moves the price up or
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down, and since news can’t be predicted, neither can changes in stock prices. Thus investors who subscribe to the Efficient Market Hypothesis would further believe that looking for trends and analyzing companies’ fundamentals is a waste of time. Believing this, they won’t pay much attention to new developments. But if relatively few investors are looking for an edge, the market will not respond quickly to new information. In this way an overwhelming belief in the hypothesis ensures its falsity.
To continue with this cerebral somersault, recall now a rule of logic: Sentences of the form “H implies I” are equivalent to those of the form “not I implies not H.” For example, the sentence “heavy rain implies that the ground will be wet” is logically equivalent to “dry ground implies the absence of heavy rain.” Using this equivalence, we can restate the claim that overwhelming belief in the Efficient Market Hypothesis leads to (or implies) its falsity. Alternatively phrased, the claim is that if the Efficient Market Hypothesis is true, then it’s not the case that most investors believe it to be true. That is, if it’s true, most investors believe it to be false (assuming almost all investors have an opinion and each either believes it or disbelieves it).
Consider now the inelegantly named Sluggish Market Hypothesis, the belief that the market is quite slow in responding to new information. If the vast majority of investors believe the Sluggish Market Hypothesis, then they all would believe that looking for trends and analyzing companies is well worth their time and, by so exercising themselves, they would bring about an efficient market. Thus, if most investors believe the Sluggish Market Hypothesis is true, they will by their actions make the Efficient Market Hypothesis true. We conclude that if the Efficient Market Hypothesis is false, then it’s not the case that most investors believe the
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Sluggish Market Hypothesis to be true. That is, if the Efficient Market Hypothesis is false, then most investors believe it (the EMH) to be true. (You may want to read over the last few sentences in a quiet corner.)
In summary, if the Efficient Market Hypothesis is true, most investors won’t believe it, and if it’s false, most investors will believe it. Alternatively stated, the Efficient Market Hypothesis is true if and only if a majority believes it to be false. (Note that the same holds for the Sluggish Market Hypothesis.) These are strange hypotheses indeed!
Of course, I’ve made some big assumptions that may not hold. One is that if an investor believes in one of the two hypotheses, then he disbelieves in the other, and almost all believe in one or the other. I’ve also assumed that it’s clear what “large majority” means, and I’ve ignored the fact that it sometimes requires very few investors to move the market. (The whole argument could be relativized to the set of knowledgeable traders only.)
Another gap in the argument is that any suspected deviations from the Efficient Market Hypothesis can always be attributed to mistakes in asset pricing models, and thus the hypothesis can’t be conclusively rejected for this reason either. Maybe some stocks or kinds of stock are riskier than our pricing models allow for and that’s why their returns are higher. Nevertheless, I think the point remains: The truth or falsity of the Efficient Market Hypothesis is not immutable but depends critically on the beliefs of investors. Furthermore, as the percentage of investors who believe in the hypothesis itself varies, the truth of the hypothesis varies inversely with it.
On the whole, most investors, professionals on Wall Street, and amateurs everywhere, disbelieve in it, so for this reason I think it holds, but only approximately and only most of the time.
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The Prisoner’s Dilemma and the Market
So you don’t believe in the Efficient Market Hypothesis. Still, it’s not enough that you discover simple and effective investing rules. Others must not find out what you’re doing, either by inference or by reading your boastful profile in a business magazine. The reason for secrecy, of course, is that without it, simple investing rules lead to more and more complicated ones, which eventually lead to zero excess returns and a reliance on chance.
This inexorable march toward increased complexity arises from the actions of your co-investors, who, if they notice (or infer, or are told) that you are performing successfully on the basis of some simple technical trading rule, will try to do the same. To take account of their response, you must complicate your rule and likely decrease your excess returns. Your more complicated rule will, of course, also inspire others to try to follow it, leading to further complications and a further decline in excess returns. Soon enough your rule assumes a near-random complexity, your excess returns are reduced essentially to zero, and you’re back to relying on chance.
Of course, your behavior will be the same if you learn of someone else’s successful performance. In fact, a situation arises that is clarified by the classic “prisoner’s dilemma,” a useful puzzle originally framed in terms of two people in prison.
Suspected of committing a major crime, the two are apprehended in the course of committing some minor offense. They’re then interrogated separately, and each is given the choice of confessing to the major crime and thereby implicating his partner or remaining silent. If they both remain silent, they’ll each get one year in prison. If one confesses and the other doesn’t, the one who confesses will be rewarded by being set free, while the other one will get a five-year term. If they both confess, they can both expect to spend three years
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in prison. The cooperative option (cooperative with the other prisoner, that is) is to remain silent, while the non-cooperative option is to confess. Given the payoffs and human psychology, the most likely outcome is for both to confess; the best outcome for the pair as a pair is for both to remain silent; the best outcome for each prisoner as an individual is to confess and have one’s partner remain silent.
The charm of the dilemma has nothing to do with any interest one might have in prisoners’ rights. (In fact, it has about as much relevance to criminal justice as the four-color-map theorem has to geography.) Rather, it provides the logical skeleton for many situations we face in everyday life. Whether we’re negotiators in business, spouses in a marriage, or nations in a dispute, our choices can often be phrased in terms of the prisoner’s dilemma. If both (all) parties pursue their own interests exclusively and do not cooperate, the outcome is worse for both (all) of them; yet in any given situation, any given party is better off not cooperating. Adam Smith’s invisible hand ensuring that individual pursuits bring about group well-being is, at least in these situations (and some others), quite arthritic.
The dilemma has the following multi-person market version: Investors who notice some exploitable stock market anomaly may either act on it, thereby diminishing its effectiveness (the non-cooperative option) or ignore it, thereby saving themselves the trouble of keeping up with developments (the cooperative option). If some ignore it and others act on it, the latter will receive the biggest payoffs, the former the smallest. As in the standard prisoner’s dilemma, the logical response for any player is to take the non-cooperative option and act on any anomaly likely to give one an edge. This response leads to the “arms race” of ever more complex technical trading strategies. People search for special knowledge, the result eventually becomes common knowledge, and the dynamic between the two generates the market.
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This searching for an edge brings us to the social value of stock analysts and investment professionals. Although the recipients of an abundance of bad publicity in recent years, they provide a most important service: By their actions, they help turn special knowledge into common knowledge and in the process help make the market relatively efficient. Absent a draconian rewiring of human psychology and an accompanying draconian rewiring of our economic system, this accomplishment is an impressive and vital one. If it means being “noncooperative” with other investors, then so be it. Cooperation is, of course, generally desirable, but cooperative decisionmaking among investors seems to smack of totalitarianism.
Pushing the Complexity Horizon
The complexity of trading rules admits of degrees. Most of the rules to which people subscribe are simple, involving support levels, P/E ratios, or hemlines and Super Bowls, for example. Others, however, are quite convoluted and conditional. Because of the variety of possible rules, I want to take an oblique and abstract approach here. The hope is that this approach will yield insights that a more pedestrian approach misses. Its key ingredient is the formal definition of (a type of) complexity. An intuitive understanding of this notion tells us that someone who remembers his eight-digit password by means of an elaborate, long-winded saga of friends’ addresses, children’s ages, and special anniversaries is doing something silly. Mnemonic rules make sense only when they’re shorter than what is to be remembered.
Let’s back up a bit and consider how we might describe the following sequences to an acquaintance who couldn’t see them. We may imagine the Is to represent upticks in the
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price of a stock and the Os downticks or perhaps up-and- down days.
1. 0101010101010101010101010...
2. 0101101010 101101010101011...
3. 1000101101101100010101100...
The first sequence is the simplest, an alternation of 0s and Is. The second sequence has some regularity to it, a single 0 alternating sometimes with a 1, sometimes with two Is, while the third sequence doesn’t seem to manifest any pattern at all. Observe that the precise meaning of “ ...” in the first sequence is clear; it is less so in the second sequence, and not at all clear in the third. Despite this, let’s assume that these sequences are each a trillion bits long (a bit is a 0 or a 1) and continue on “in the same way.”
Motivated by examples like this, the American computer scientist Gregory Chaitin and the Russian mathematician A. N. Kolmogorov defined the complexity of a sequence of 0s and Is to be the length of the shortest computer program that will generate (that is, print out) the sequence in question.
A program that prints out the first sequence above can consist simply of the following recipe: print a 0, then a 1, and repeat a half trillion times. Such a program is quite short, especially compared to the long sequence it generates. The complexity of this first trillion-bit sequence may be only a few hundred bits, depending to some extent on the computer language used to write the program.
A program that generates the second sequence would be a translation of the following: Print a 0 followed by either a single 1 or two Is, the pattern of the intervening Is being one, two, one, one, one, two, one, one, and so on. Any program that prints out this trillion-bit sequence would have to be
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quite long so as to fully specify the “and so on” pattern of the intervening Is. Nevertheless, because of the regular alternation of Os and either one or two Is, the shortest such program will be considerably shorter than the trillion-bit sequence it generates. Thus the complexity of this second sequence might be only, say, a quarter trillion bits.
With the third sequence (the commonest type) the situation is different. This sequence, let us assume, remains so disorderly throughout its trillion-bit length that no program we might use to generate it would be any shorter than the sequence itself. It never repeats, never exhibits a pattern. All any program can do in this case is dumbly list the bits in the sequence: print 1, then 0, then 0, then 0, then 1, then 0, then 1, . ... There is no way the . . . can be compressed or the program shortened. Such a program will be as long as the sequence it’s supposed to print out, and thus the third sequence has a complexity of approximately a trillion.
A sequence like the third one, which requires a program as long as itself to be generated, is said to be random. Random sequences manifest no regularity or order, and the programs that print them out can do nothing more than direct that they be copied: print 10001011011.... These programs cannot be abbreviated; the complexity of the sequences they generate is equal to the length of these sequences. By contrast, ordered, regular sequences like the first can be generated by very short programs and have complexity much less than their length.
Returning to stocks, different market theorists will have different ideas about the likely pattern of 0s and Is (downs and upticks) that can be expected. Strict random walk theorists are likely to believe that sequences like the third characterize price movements and that the market’s movements are therefore beyond the “complexity horizon” of human forecasters (more complex than we, or our brains, are, were we
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expressed as sequences of Os and Is). Technical and fundamental analysts might be more inclined to believe that sequences like the second characterize the market and that there are pockets of order amidst the noise. It’s hard to imagine anyone believing that price movements follow sequences as regular as the first except, possibly, those who send away “only $99.95 for a complete set of tapes that explain this revolutionary system.”
I reiterate that this approach to stock price movements is rather stark, but it does nevertheless “locate” the debate. People who believe there is some pattern to the market, whether exploitable or not, will believe that its movements are characterized by sequences of complexity somewhere between those of type two and type three above.
A rough paraphrase of Kurt Godel’s famous incompleteness theorem of mathematical logic, due to the aforementioned Gregory Chaitin, provides an interesting sidelight on this issue. It states that if the market were random, we might not be able to prove it. The reason: encoded as a sequence of Os and Is, a random market would, it seems plausible to assume, have complexity greater than that of our own were we also so encoded; it would be beyond our complexity horizon. From the definition of complexity it follows that a sequence can’t generate another sequence of greater complexity than itself. Thus if a person were to predict the random market’s exact gyrations, the market would have to be less complex than the person, contrary to assumption. Even if the market isn’t random, there remains the possibility that its regularities are so complex as to be beyond our complexity horizons.
In any case, there is no reason why the complexity of price movements as well as the complexity of investor/computer blends cannot change over time. The more inefficient the market is, the smaller the complexity of its price movements, and the more likely it is that tools from technical and fundamental
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analysis will prove useful. Conversely, the more efficient the market is, the greater the complexity of price movements, and the closer the approach to a completely random sequence of price changes.
Outperforming the market requires that one remain on the cusp of our collective complexity horizon. It requires faster machines, better data, improved models, and the smarter use of mathematical tools, from conventional statistics to neural nets (computerized learning networks, the connections between the various nodes of which are strengthened or weakened over a period of training). If this is possible for anyone or any group to achieve, it’s not likely to remain so for long.
Game Theory and
Supernatural Investor/Psychologists
But what if, contrary to fact, there were an entity possessing sufficient complexity and speed that it was able with reasonably high probability to predict the market and the behavior of individuals within it? The mere existence of such an entity leads to Newcombe’s paradox, a puzzle that calls into question basic principles of game theory.
My particular variation of Newcombe’s paradox involves the World Class Options Market Maker (WCOMM), which (who?) claims to have the power to predict with some accuracy which of two alternatives a person will choose. Imagine further that WCOMM sets up a long booth on Wall Street to demonstrate its abilities.
WCOMM explains that it tests people by employing two portfolios. Portfolio A contains a $1,000 treasury bill, whereas portfolio B (consisting of either calls or puts on WCOM stock) is either worth nothing or $1,000,000. For each person in line at the demonstration, WCOMM has reserved a portfolio of
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each type at the booth and offers each person the following choice: He or she can choose to take portfolio B alone or choose to take both portfolios A and B. However, and this is crucial, WCOMM also states that it has used its unfathomable powers to analyze the psychology, investment history, and trading style of everyone in line as well as general market conditions, and if it believes that a person will take both portfolios, it has ensured that portfolio B will be worthless. On the other hand, if WCOMM believes that a person will trust its wisdom and take only portfolio B, it has ensured that portfolio B will be worth $1,000,000. After making these announcements, WCOMM leaves in a swirl of digits and stock symbols, and the demonstration proceeds.
Investors on Wall Street see for themselves that when a person in the long line chooses to take both portfolios, most of the time (say with probability 90 percent) portfolio B is worthless and the person gets only the $1,000 treasury bill in portfolio A. They also note that when a person chooses to take the contents of portfolio B alone, most of the time it’s worth $1,000,000.
After watching the portfolios placed before the people in line ahead of me and seeing their choices and the consequences, I’m finally presented with the two portfolios prepared for me by WCOMM. Despite the evidence I’ve seen, I see no reason not to take both portfolios. WCOMM is gone, perhaps to the financial district of London or Frankfurt or Tokyo, to make similar offers to other investors, and portfolio B is either worth $1,000,000 or not, so why not take both portfolios and possibly get $1,001,000. Alas, WCOMM read correctly the skeptical smirk on my face and after opening my portfolios, I walk away with only $1,000. My portfolio B contains call options on WCOM with a strike price of 20, when the stock itself is selling at $1.13.
The paradox, due to the physicist William Newcombe (not the Newcomb of Benford’s Law, but the same mocking four
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letters WCOM) and made well-known by the philosopher Robert Nozick, raises other issues. As mentioned, it makes problematical which of two game-theoretic principles one should use in making decisions, principles that shouldn’t conflict.
The “dominance” principle tells us to take both portfolios since, whether portfolio B contains options worth $1,000,000 or not, the value of two portfolios is at least as great as the value of one. (If portfolio B is worthless, $1,000 is greater than $0, and if portfolio B is worth $1,000,000, $1,001,000 is greater than $1,000,000.)
On the other hand, the “maximization of expected value” principle tells us to take only portfolio B since the expected value of doing so is greater. (Since WCOMM is right about 90 percent of the time, the expected value of taking only portfolio B is (.90 x $1,000,000) + (.10 x $0), or $900,000, whereas the expected value of taking both is (.10 x $1,001,000) + (.90 x $1,000), or $101,000.) The paradox is that both principles seem reasonable, yet they counsel different choices.
This raises other general philosophical matters as well, but it reminds me of my resistance to following the WCOM- fleeing crowd, most of whose B portfolios contained puts on the stock worth $1,000,000.
One conclusion that seems to follow from the above is that such supernatural investor/psychologists are an impossibility. For better or worse, we’re on our own.
Absurd Emails and the WorldCom Denouement
A natural reaction to the vagaries of chance is an attempt at control, which brings me to emails regarding WorldCom that, Herzog-like, I sent to various influential people. I had grown tired of carrying on one-sided arguments with CNBC’s always
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perky Maria Bartiromo and always apoplectic James Cramer as they delivered the relentlessly bad news about WorldCom. So in fall 2001, five or six months before its final swoon, I contacted a number of online business commentators critical of WorldCom’s past performance and future prospects. Having spent too much time in the immoderate atmosphere of WorldCom chatrooms, I excoriated them, though mildly, for their shortsightedness and exhorted them to look at the company differently.
Finally, out of frustration with the continued decline of WCOM stock, I emailed Bernie Ebbers, then the CEO, in early February 2002 suggesting that the company was not effectively stating its case and quixotically offering to help by writing copy. I said I’d invested heavily in WorldCom, as did family and friends at my suggestion, that I could be a persuasive wordsmith when I believed in something, and WorldCom, I believed, was well positioned but dreadfully undervalued. UUNet, the “backbone” of much of the Internet, was, I fatuously informed the CEO of the company, a gem in and of itself.
I knew, even as I was writing them, that sending these electronic epistles was absurd, but it gave me the temporary illusion of doing something about this recalcitrant stock other than dumping it. Investing in it had originally seemed like a no-brainer. The realization that doing so had indeed been a no-brainer was glacially slow in arriving. During the 2001-2002 academic year, I took the train once a week from Philadelphia to New York to teach a course on “numbers in the news” at the Columbia School of Journalism. Spending the two and a half hours of the commute out of contact with WCOM’s volatile movements was torturous, and upon emerging from the subway, I’d run to my office computer to check what had happened. Not exactly the behavior of a sage long-term investor; my conduct even then suggested to me a rather dim-witted addict.
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Recalling the two or three times I almost got out of the stock is dispiriting as well. The last time was in April 2002. Amazingly, I was even then still somewhat in thrall to the idea of averaging down, and when the price dipped below $5, I bought more WCOM shares. Around the middle of the month, however, I did firmly and definitively resolve to sell. By Friday, April 19, WCOM had risen to over $7, which would have allowed me to recoup at least a small portion of my losses, but I didn’t have time to sell that morning. I had to drive to northern New Jersey to give a long-promised lecture at a college there. When it was over, I wondered whether to return home to sell my shares or simply use the college’s computer to log onto my Schwab account to do so. I decided to go home, but there was so much traffic on the cursed New Jersey turnpike that afternoon that I didn’t arrive until 4:05, after the market had closed. I had to wait until Monday.
Investors are often nervous about holding volatile stocks over the weekend, and I was no exception. My anxiety was well-founded. Later that evening there was news about impending cuts in WorldCom’s bond ratings and another announcement from the SEC regarding its comprehensive investigation of the company. The stock lost more than a third of its value by Monday, when I did finally sell the stock at a huge loss. A few months later the stock completely collapsed to $.09 upon revelations of massive accounting fraud.
Why had I violated the most basic of investing fundamentals: Don’t succumb to hype and vaporous enthusiasm; even if you do, don’t put too many eggs in one basket (especially with the uncritical sunny-side up); even if you do this, don’t forget to insure against sudden drops (say with puts, not calls); and even if you do this too, don’t buy on margin. After selling my shares, I felt as if I were gradually and groggily coming out of a self-induced trance. I’d long known about one of the earliest “stock” hysterias on record, the sevenA
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teenth century tulip bulb craze in Holland. After its collapse, people also spoke of waking up and realizing that they were stuck with nearly worthless bulbs and truly worthless options to buy more of them. I smiled ruefully at my previous smug dismissal of people like the tulip bulb “investors.” I was as vulnerable to transient delirium as the dimmest bulb-buying bulbs.
I’ve followed the ongoing drama of the WorldCom story— the fraud investigations, various prosecutions, new managers, promised reforms, and court settlements—and, oddly perhaps, the publicity surrounding the scandals and their aftermath has distanced me from my experience and lessened its intensity. My losses have become less a small personal story and more (a part of) a big news story, less a result of my mistakes and more a consequence of the company’s behavior. This shifting of responsibility is neither welcome nor warranted. For reasons of fact and of temperament, I continue to think of myself as having been temporarily infatuated rather than deeply victimized. Remnants of my fixation persist, and I still sometimes wonder what might have happened if WorldCom’s deal with Sprint hadn’t been foiled, if Ebbers hadn’t borrowed $400 million (or more), if Enron hadn’t imploded, if this or that or the other event hadn’t occurred before I sold my shares. My recklessness might then have been seen as daring. Post hoc stories always seem right, whatever the pre-existing probabilities.
One fact remains incontrovertible: Narratives and numbers coexist uneasily on Wall Street. Markets, like people, are largely rational beasts occasionally provoked and disturbed by their underlying animal spirits. The mathematics discussed in this book is often helpful in understanding (albeit not beating) the market, but I’d like to end with a psychological caveat. The basis for the application of the mathematical tools discussed herein is the sometimes shifty and always shifting attitudes of investors. Since these psychological states
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are to a large extent imponderable, anything that depends on them is less exact than it appears.
The situation reminds me a bit of the apocryphal story of the way cows were weighed in the Old West. First the cowboys would find a long, thick plank and place the middle of it on a large, high rock. Then they’d attach the cow to one end of the plank with ropes and tie a large boulder to the other end. They’d carefully measure the distance from the cow to the rock and from the boulder to the rock. If the plank didn’t balance, they’d try another big boulder and measure again. They’d keep this up until a boulder exactly balanced the cattle. After solving the resulting equation that expresses the cow’s weight in terms of the distances and the weight of the boulder, there would be only one thing left for them to do: They would have to guess the weight of the boulder. Once again the mathematics may be exact, but the judgments, guesses, and estimates supporting its applications are anything but.
More apropos of the self-referential nature of the market would be a version in which the cowboys had to guess the weight of the cow whose weight varied depending on their collective guesses, hopes, and fears. Bringing us full circle to Keynes’s beauty contest, albeit in a rather forced, more bovine mode, I conclude that despite rancid beasts like WorldCom, I’m still rather fond of the pageant that is the market. I just wish I had a better (and secret) method for weighing the cows.
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