More Than You Know

by Michael J Mauboussin

  Estimating Printers with Populi

  In the prior example, the aggregation of individuals determined a particular state—the ox’s weight, the number of jellybeans, the location of the bomb—but did not make a prediction of a future state. Is there any difference between estimating what is and estimating what is going to be?

  Well, solid evidence exists to suggest that the vox populi is pretty good at anticipating the future. Scientists at Hewlett-Packard have demonstrated that even small groups can predict results better than individuals. Apparently, the internal market that Hewlett-Packard set up to predict sales was more accurate than its official internal forecasts.6

  And Now, For the Real World

  So collectives have proven adept at matching seekers and solvers and determining current or future states. How does all of this apply to the stock market?

  The stock market is different than the markets I’ve described because there is no answer—stocks have no specified time horizon or value. (The exception is when a company has agreed to be acquired, in which case the stock price tends to very accurately represent the ultimate value.) As a result, stock investors are susceptible to imitation because they can earn excess profits by selling to someone else willing to pay a higher price. Said differently, one or more of the three conditions for proper information aggregation—group heterogeneity—is violated.

  However, I would argue that extraordinary popular delusions and the madness of crowds are exceptions, not the rule. Investors who appreciate how and why markets are efficient will have better insight into how and why markets are inefficient. Further, investors who identify companies intelligently using collectives—the vox populi—may gain an investment edge.

  31

  A Tail of Two Worlds

  Fat Tails and Investing

  [Victor Niederhoffer] looked at markets as a casino where people act as gamblers and where their behavior can be understood by studying gamblers. He regularly made small amounts of money trading on that theory. There was a flaw in his approach, however. If there is a . . . tide . . . he can be seriously hurt because he doesn’t have a proper fail-safe mechanism.

  —George Soros, Soros on Soros

  In statistical terms, I figure I have traded about 2 million contracts . . . with an average profit of $70 per contract. This average profit is approximately 700 standard deviations away from randomness, a departure that would occur by chance alone about as frequently as the spare parts in an automotive salvage lot might spontaneously assemble themselves into a McDonald’s restaurant.

  —Victor Niederhoffer, The Education of a Speculator

  On Wednesday Niederhoffer told investors in three hedge funds he runs that their stakes had been “wiped out” Monday by losses that culminated from three days of falling stock prices and big hits earlier this year in Thailand.

  —David Henry, USA Today, October 30, 1997

  Much of the real world is controlled as much by the “tails” of distributions as by means or averages: by the exceptional, not the mean; by the catastrophe, not the steady drip; by the very rich, not the “middle class.” We need to free ourselves from “average” thinking.

  —Philip Anderson, Nobel Prize recipient in physics, “Some Thoughts About Distribution in Economics”

  Experience Versus Exposure

  In his 2001 letter to shareholders, Warren Buffett distinguishes between experience and exposure. Although Buffett’s comments are in the context of Berkshire Hathaway’s insurance business, his point is valid for any exercise with subjective probabilities. Experience, of course, looks to the past and considers the probability of future outcomes based on occurrence of historical events. Exposure, on the other hand, considers the likelihood—and potential risk—of an event that history (especially recent history) may not reveal. Buffett argues that in 2001 the insurance industry assumed huge terrorism risk without commensurate premiums because it was focused on experience, not exposure.

  Investors, too, must discern between experience and exposure. The high-profile failures of Long Term Capital Management and Victor Niederhoffer give witness to this point. Remarkably, however, standard finance theory does not easily accommodate extreme events. Financial economists generally assume that stock price changes are random, akin to the motion of pollen in water as molecules bombard it.1

  In a triumph of modeling convenience over empirical results, finance theory treats price changes as independent, identically distributed variables and generally assumes that the distribution of returns is normal, or lognormal. The virtue of these assumptions is that investors can use probability calculus to understand the distribution’s mean and variance and can therefore anticipate various percentage price changes with statistical accuracy. The good news is that these assumptions are reasonable for the most part. The bad news, as physicist Phil Anderson notes above, is that the tails of the distribution often control the world.

  Tell Tail

  Normal distributions are the bedrock of finance, including the random walk, capital asset pricing, value-at-risk (VaR), and Black-Scholes models. Value-at-risk models, for example, attempt to quantify how much loss a portfolio may suffer with a given probability. While there are various forms of VaR models, a basic version relies on standard deviation as a measure of risk. Given a normal distribution, it is relatively straightforward to measure standard deviation, and hence risk. However, if price changes are not normally distributed, standard deviation can be a very misleading proxy for risk.2

  The research, some done as far back as the early 1960s, shows that price changes do not follow a normal distribution. Exhibit 31.1 shows the frequency distribution of S&P 500 daily returns from January 1, 1978, to March 30, 2007, and a normal distribution derived from the data. Exhibit 31.2 highlights the difference between the actual returns and the normal distribution. Analyses of different asset classes and time horizons yield similar results.3 The figures show that:• Small changes appear more frequently than the normal distribution predicts

  • There are fewer medium-sized changes than the model implies (roughly 0.5 to 2.0 standard deviations)

  • There are fatter tails than what the standard model suggests. This means that there is a greater-than-expected number of large changes

  The fat tails, in particular, warrant additional comment. These extreme value changes happen considerably more frequently than the standard model implies and can have a substantial influence on portfolio performance—especially for leveraged portfolios. For example, during the October 1987 crash, which I excluded from my figures for presentation purposes, the *S&P 500 plummeted over 20 percent, a change that is twenty standard deviations from the mean. Roger Lowenstein notes:Economists later figured that, on the basis of the market’s historical volatility, had the market been open every day since the creation of the Universe, the odds would still have been against its falling that much in a single day. In fact, had the life of the Universe been repeated one billion times, such a crash would still have been theoretically “unlikely.”4

  EXHIBIT 31.1 Frequency Distribution of the S&P 500 Daily Returns, January 1978-March 2007

  Source: FactSet, author analysis.

  The pattern of many small events and few large events is not unique to asset prices. Indeed, it is a signature of systems in the state of “self-organized criticality.” Self-organization is the result of interaction between individual agents (in this case, investors) and requires no leadership. A critical state is one where small perturbations can lead to events of many types. Self-organized criticality marks systems as varied as earthquakes, extinction events, and traffic jams.5

  Is there a mechanism that can help explain these episodic lunges? I think so. As I have noted in other essays, markets tend to function well when a sufficient number of diverse investors interact.6 Conversely, markets tend to become fragile when this diversity breaks down and investors act in a similar way (this can also result from some investors withdrawing). A burgeoning literature on herding addresses thi
s phenomenon. Herding is when many investors make the same choice based on the observations of others, independent of their own knowledge. Information cascades, another good illustration of a self-organized critical system, are closely linked to herding.7

  EXHIBIT 31.2 Frequency Difference: Normal Versus Actual Daily Returns, January 1978-March 2007

  Source: FactSet, author analysis.

  What Fat Tails Mean for Investors

  OK: Big changes in prices appear more frequently than they are supposed to. What does this mean for investors from a practical standpoint? I believe there are a few important implications:• Cause and effect thinking. One of the essential features of self-organized critical systems is that the size of the perturbation and that of the resulting event may not be linearly linked. Sometimes small-scale inputs can lead to large-scale events. This dashes the hope of finding causes for all effects.

  • Risk and reward. The standard model for assessing risk, the capital-asset-pricing model, assumes a linear relationship between risk and reward. In contrast, nonlinearity is endogenous to self-organized critical systems like the stock market. Investors must bear in mind that finance theory stylizes real world data. That the academic and investment communities so frequently talk about events five or more standard deviations from the mean should be a sufficient indication that the widely used statistical measures are inappropriate for the markets.

  • Portfolio construction. Investors that design portfolios using standard statistical measures may understate risk (experience versus exposure). This concern is especially pronounced for portfolios that use leverage to enhance returns. Many of the most spectacular failures in the hedge fund world have been the direct result of fat-tail events. Investors need to take these events into consideration when constructing portfolios.

  A useful means to navigate a fat-tailed world is to first measure the current expectations underlying an asset price and then contemplate various ranges of value outcomes and their associated probabilities. This process allows investors to give some weight to potential fat-tail events.8

  Standard finance theory has advanced our understanding of markets immensely. But some of the theory’s foundational assumptions are not borne out by market facts. Investors must be aware of the discrepancies between theory and reality and adjust their thinking (and portfolios) accordingly.

  32

  Integrating the Outliers

  Two Lessons from the St. Petersburg Paradox

  The risk-reducing formulas behind portfolio theory rely on a number of demanding and ultimately unfounded premises. First, they suggest that price changes are statistically independent from one another. . . . The second assumption is that price changes are distributed in a pattern that conforms to a standard bell curve.

  Do financial data neatly conform to such assumptions? Of course, they never do.

  —Benoit B. Mandelbrot, “A Multifractal Walk down Wall Street”

  The very fact that the Petersburg Problem has not yielded a unique and generally acceptable solution to more than 200 years of attack by some of the world’s great intellects suggests, indeed, that the growth-stock problem offers no hope of a satisfactory solution.

  —David Durand, “Growth Stocks and the Petersburg Paradox”

  Bernoulli’s Challenge

  Competent investors take great pride in their ability to place an appropriate value on a financial claim. This ability is the core of investing: markets are just vehicles to trade cash for future claims, and vice versa.

  OK, here’s a cash-flow stream for you to value: Say the house flips a fair coin. If it lands on heads, you receive two dollars and the game ends. If it lands on tails, the house flips again. If the second flip lands on heads, you get four dollars; if it lands on tails, the game continues. For each successive round, the payoff for landing on heads doubles (i.e., $2, $4, $8, $16, etc.) and you progress to the next round until you land on heads. How much would you pay to play this game?

  Daniel Bernoulli, one of a family of distinguished mathematicians, first presented this problem to the Russian Imperial Academy of Sciences in 1738.1 Bernoulli’s game, known as the St. Petersburg Paradox, challenges classical theory, which says that a player should be willing to pay the game’s expected value to participate. The expected value of this game is infinite. Each round has a payoff of one dollar (probability of 1/2n and a payoff of $2n , or 1/2 x $2, 1/4 x $4, 1/8 x $8, etc.) So,Expected value = 1 + 1 + 1 + 1 . . . = ∞

  Naturally, very few people would be willing to pay even twenty dollars to play the game. Bernoulli tried to explain the paradox with the marginal utility of money. He argued that the amount you would be willing to pay is a function of your resources—the greater your resources, the more you would be willing to pay. Still, Bernoulli’s explanation is not altogether satisfactory. The St. Petersburg Paradox has kept philosophers, mathematicians, and economists thinking for over two and a half centuries.2

  Philosophical discourse aside, the St. Petersburg Paradox helps illuminate two very concrete ideas for investors. The first is that the distribution of stock market returns does not follow the pattern that standard finance theory assumes. This deviation from theory is important for risk management, market efficiency, and individual stock selection.

  The second idea relates to valuing growth stocks. What do you pay today for a business with a low probability of an extraordinarily high payoff? This question is more pressing than ever in a world with violent value migrations and increasing returns.

  What’s Normal?

  Asset price distributions are of great practical significance for portfolio managers. Standard finance theory assumes that asset price changes follow a normal distribution—the well-known bell curve. That this assumption is roughly accurate most of the time allows analysts to use very robust probability statistics. For example, for a sample that follows a normal distribution, you can identify the population average and characterize the likelihood of variance from that average.

  However, much of nature—including the manmade stock market—is not normal.3 Many natural systems have two defining characteristics: an ever-larger number of smaller pieces and similar-looking pieces across the different size scales. For example, a tree has a large trunk and a number of ever-smaller branches, and the small branches resemble the big branches. These systems are fractal. Unlike a normal distribution, no average value adequately characterizes a fractal system. Exhibit 32.1 contrasts normal and fractal systems visually and shows the probability functions that represent the data. Fractal systems follow a power law.4

  EXHIBIT 32.1 Probability Density Functions for Normal and Fractal Systems

  Source: Liebovitch and Scheurle, “Two Lessons from Fractals and Chaos.” Reproduced with permission.

  Using the statistics of normal distributions to characterize a fractal system like financial markets is potentially very hazardous. Yet theoreticians and practitioners do it daily.5 The distinction between the two systems boils down to probabilities and payoffs. Fractal systems have few, very large observations that fall outside the normal distribution. The classic example is the crash of 1987. The probability (assuming a normal distribution) of the market’s 20-plus percent plunge in one day was so infinitesimally low it was practically zero. And still the losses were a staggering $2 trillion-plus.

  A comparison of a normal coin-toss game and the St. Petersburg game illustrates the point. Assume that you toss a coin and receive $2 if it lands heads and nothing if it lands tails. The expected value of the game is $1, the amount you would be willing to pay to play the game in a fair casino. I simulated 1 million rounds of 100 tosses each, and plotted the payoffs in exhibit 32.2. Just as you would expect, I got a well-defined normal distribution.6

  EXHIBIT 32.2 Standard Coin Toss Game

  Source: Author analysis.

  I then simulated the St. Petersburg game 1 million times and plotted that distribution (see exhibit 32.3). While the underlying process is stochastic, the outcome is a power law. For exampl
e, half the time the game only pays two dollars, and three-quarters of the time it pays four dollars or less. However, a run of thirty provides a $1.1 billion payoff, but this is only a 1-in-1.1 billion probability. Lots of small events and a few very large events characterize a fractal system. Further, the average winnings per game is unstable with the St. Petersburg game, so no average accurately describes the game’s long-term outcome.

  Are stock market returns fractal? Benoit Mandelbrot shows that by lengthening or shortening the horizontal axis of a price series—effectively speeding up or slowing down time—price series are indeed fractal. Not only are rare large changes interspersed with lots of smaller ones, the price changes look similar at various scales (e.g., daily, weekly, and monthly returns). Mandelbrot calls financial time series multifractal, adding the prefix “multi” to capture the time adjustment.

  EXHIBIT 32.3 Fractal Coin Toss Game

  Source: Author analysis.

  In an important and fascinating book, Why Stock Markets Crash, geophysicist Didier Sornette argues that stock market distributions comprise two different populations, the body (which you can model with standard theory) and the tail (which relies on completely different mechanisms). Sornette’s analysis of market drawdowns convincingly dismisses the assumption that stock returns are independent, a key pillar of classical finance theory. His work provides fresh and thorough evidence of finance theory’s shortcomings.7