The Misbehavior of Markets: A Fractal View of Financial Turbulence

by Benoît B. Mandelbrot

Three states of matter—solid, liquid, and gas—have long been known. An analogous distinction between three states of randomness—mild, slow, and wild—arises from the mathematics of fractal geometry. Conventional financial theory assumes that variation of prices can be modeled by random processes that, in effect, follow the simplest “mild” pattern, as if each uptick or downtick were determined by the toss of a coin. What fractals show, and this book describes, is that by that standard, real prices “misbehave” very badly. A more accurate, multifractal model of wild price variation paves the way for a new, more reliable type of financial theory.

Understanding fractally wild randomness, also exemplified by such diverse phenomena as turbulent flow, electrical “flicker” noise, and the track of a stock or bond price, will not bring personal wealth. But the fractal view of the market is alone in facing the high odds of catastrophic price changes.

He has been premature, contrary to fashion, trouble-making, in virtually every field he has touched: statistical physics, cosmology, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, and, of course, mathematics. In economics he is especially controversial.

Mandelbrot, like Prime Minister Churchill before him, promises us not utopia but blood, sweat, toil and tears. If he is right, almost all of our statistical tools are obsolete—least squares, spectral analysis, workable maximum-likelihood solutions, all our established sample theory, closed distributions. Almost without exception, past econometric work is meaningless.

In fact, by the conventional wisdom, August 1998 simply should never have happened; it was, according to the standard models of the financial industry, so improbable a sequence of events as to have been impossible. The standard theories, as taught in business schools around the world, would estimate the odds of that final, August 31, collapse at one in 20 million—an event that, if you traded daily for nearly 100,000 years, you would not expect to see even once. The odds of getting three such declines in the same month were even more minute: about one in 500 billion. Surely, August had been supremely bad luck, a freak accident, an “act of God” no one could have predicted. In the language of statistics, it was an “outlier” far, far, far from the normal expectation of stock trading. Or was it? The seemingly improbable happens all the time in financial markets. A year earlier, the Dow had fallen 7.7 percent in one day. (Probability: one in 50 billion.) In July 2002, the index recorded three steep falls within seven trading days. (Probability: one in four trillion.) And on October 19, 1987, the worst day of trading in at least a century, the index fell 29.2 percent. The probability of that happening, based on the standard reckoning of financial theorists, was less than one in 1050—odds so small they have no meaning. It is a number outside the scale of nature.

For more than a century, financiers and economists have been striving to analyze risk in capital markets, to explain it, to quantify it, and, ultimately, to profit from it. I believe that most of the theorists have been going down the wrong track. The odds of financial ruin in a free, global-market economy have been grossly underestimated.

The precise market mechanism that links news to price, cause to effect, is mysterious and seems inconsistent. Threat of war: Dollar falls. Threat of war: Dollar rises. Which of the two will actually happen? After the fact, it seems obvious; in hindsight, fundamental analysis can be reconstituted and is always brilliant.

From 1916 to 2003, the daily index movements of the Dow Jones Industrial Average do not spread out on graph paper like a simple bell curve. The far edges flare too high: too many big changes.

Theory suggests that over that time, there should be fifty-eight days when the Dow moved more than 3.4 percent; in fact, there were 1,001. Theory predicts six days of index swings beyond 4.5 percent; in fact, there were 366. And index swings of more than 7 percent should come once every 300,000 years; in fact, the twentieth century saw forty-eight such days. Truly, a calamitous era that insists on flaunting all predictions. Or, perhaps, our assumptions are wrong.

Why such reluctance to change? The old methods are easy and convenient. They work fine, it is argued, for most market conditions. It is only in the infrequent moments of high turbulence that the theory founders—and at such moments, who can guard against a hostile takeover, a bankruptcy or other financial act of God? Such reasoning, of course, is little comfort to those wiped out on one of those “improbable,” violent trading days.

Mild randomness, then, is like the solid phase of matter: low energies, stable structures, well-defined volume. It stays where you put it. Wild randomness is like the gaseous phase of matter: high energies, no structure, no volume. No telling what it can do, where it will go. Slow randomness is intermediate between the others, the liquid state.

They amount to two different ways of seeing the world: one in which big changes are the result of many small ones, or another in which major events loom disproportionately large. “Mild” and “wild” chance, described earlier, are my generalizations from Gauss and Cauchy.

Suppose you are asked to calculate the average size of companies in the software industry. So you go down a list, counting the firms, adding up their reported revenues, and dividing one number into the other to get a simple average. But how long should the list be? Just the top fifty publicly traded firms? Every company in an industry directory? Every firm that files a tax return and says it is in software? Impossible to say: Each time you lengthen the list and add more, smaller firms, your calculated average drops. And what about Microsoft? It is the colossus of the industry, dwarfing every other firm. Try to survey the industry: If you include Microsoft in the sample, it grotesquely inflates what the survey suggests the “typical” company value is. But if you exclude it, you ignore the most important company in the industry. In short, the distribution of company size is wild—Wild West, in the view of Microsoft’s critics.

The amount by which the stock reacts to the market is the stock’s “beta” or β,

Prominent features: In the Brownian chart, most changes—in fact, about 68 percent—are small. They are within one standard deviation of the average index change, zero. Mathematicians use the Greek letter sigma, σ, for standard deviation. About 95 percent of the changes are within 2σ, 98 percent within 3σ, and very, very few values are any larger. Next look at the Dow variations. The spikes are huge. Some are 10σ; one, in 1987, is 22σ. The odds of that are something less than one in 1050—so minute that the standard Gaussian tables do not even contemplate it. In other words, virtually impossible. Yet there it is.

They spill out beyond the narrow confines of the Brownian model. It has many changes beyond 5σ—and one at 22σ. This is the “fat tail” to which statisticians refer. And it means the standard model of finance is wrong.

He found the same, disturbing pattern: Big price changes were far more common than the standard model allowed. Large changes, of more than five standard deviations from the average, happened two thousand times more often than expected. Under Gaussian rules, you should have encountered such drama only once every seven thousand years; in fact, the data showed, it happened once every three or four years.

A Citigroup study in 2002 found unpleasantly sharp price swings in several currencies—dollar, euro, yen, pound, peso, zloty, even the Brazilian real. On one day, the dollar vaulted over the yen by 3.78 percent. That is 5.1 standard deviations, or 5.1σ, from the average. If exchange rates were Gaussian that would be expected to happen once in a century. But the biggest fall was a heart-stopping 7.92 percent, or 10.7σ. The normal odds of that: Not if Citigroup had been trading dollars and yen every day since the Big Bang 15 billion years ago should it have happened, not once.

If this were astronomy, the argument would have ended long ago. Imagine observatories suddenly finding a new planet where, the standard theory says, none should be. And then another, and another and another. Astronomers, after checking their instruments, would not ignore the data; they would question their understanding of celestial mechanics and a new and fruitful episode in astronomy would dawn. But it does not work that way in economics, even though the equivalent of countless new planetary sightings have been recorded.

They assume that the “average” stock-market profit means something to a real person; in fact, it is the extremes of profit or loss that matter most. Just one out-of-the-average year of losing more than a third of capital—as happened with many stocks in 2002—would justifiably scare even the boldest investors away for a long while. The problem also assumes wrongly that the bell curve is a realistic yardstick for measuring the risk. As I have said often, real prices gyrate much more wildly than the Gaussian standards assume.