The (Mis)Behavior of Markets

by Benoît B. Mandelbrot

post-doc to study with the great Hungarian-born mathematician, John von Neumann;

power-law distributions.

After the fact, it seems obvious; in hindsight, fundamental analysis can be reconstituted and is always brilliant. But before the fact, both outcomes may seem equally likely. So how can one base an investment strategy and a risk profile entirely on this one dubious principle: I can know more than anybody else?

It beggars belief that vast sums can change hands on the basis of such financial astrology.

Instead, they learned to think of the world in the second way, as a black box. We can see what goes into the box and what comes out of it, but not what happens inside; we can only draw inferences about the odds of input A producing output Z. Seeing nature through the lens of probability theory is what mathematicians call the stochastic view. The word comes from the Greek stochastes, a diviner, which in turn comes from stokhos, a pointed stake used as a target by archers. We

MIT economist Paul A. Samuelson.

Thus, Markowitz and others transformed investing from a game of stock tips and hunches to an engineering of means, variances, and “risk aversion” indices. In fact, the term “financial engineering” has been popular on Wall Street ever since. There were problems, of course. First, as Markowitz himself pointed out, it is not certain that using the bell curve is the best way to measure stock-market risk; it is easy, but not necessarily right. Second, to build efficient portfolios you need good forecasts of earnings, share prices, and volatility for thousands of stocks. Otherwise, garbage in, garbage out. Finally, for each stock, you must laboriously calculate its “covariance” with, or how it fluctuates against, every other stock. For a thirty-stock portfolio, about the minimum needed to make the numbers work well, that means 495 different calculations of mean, variance, and covariance.

pockets. Still, a by-now substantial body of economics research suggests that there is, indeed, money to be made in such a “trend-following” strategy; how much, and whether it is worth the risk and expense, is a matter of debate. But clearly, the market pros have already voted: More than half of currency speculators play some form of trend-following game, market analysts estimate.

They can flip a coin to win $200 for heads and nothing for tails, or they can skip the toss and collect $100 immediately. Most people, researchers have found, will take the sure thing. Now alter the game: They can flip a coin to lose $200 for heads and nothing for tails, or they can skip the toss and pay $100 immediately. Most people will take the gamble.

For instance, assume just two types of investors, instead of one: fundamentalists who believe that each stock or currency has its own, intrinsic value and will eventually sell for that value, and chartists who ignore the fundamentals and only watch the price trends so they can jump on and off bandwagons. In computer simulations by economists Paul De Grauwe and Marianna Grimaldi at the Catholic University of Leuven, in Belgium, the two groups start interacting in unexpected ways, and price bubbles and crashes arise, spontaneously. The market switches from a well-behaved “linear” system in which one factor adds predictably to the next, to a chaotic “non-linear” system in

First, independence: Each change in price—whether a five-cent uptick or a $26 collapse—appears independently from the last, and price changes last week or last year do not influence those today. That means any information that could be used to predict tomorrow’s price is contained in today’s price, so there is no need to study the historical charts. A second assumption: statistical stationarity of the price changes. That means the process generating price changes, whatever it may be, stays the same over time. If you assume coin tosses decide prices, the coin does not get switched or weighted in the middle of the game. All that changes are the number of heads or tails as the coin is tossed; not the coin itself. And a third assumption: the normal distribution. Price changes follow the proportions of the bell curve—most changes are small, an extremely few are large, in predictable and rapidly declining frequency. Reality: Life is more complex. This third set of assumptions is the one most clearly contradicted by the facts. Because it underpins almost every tool of modern finance, it gets special attention in the following chapter-in-a-chapter. Pictorial Essay: Images of the Abnormal PICTURES

ARE UNDERVALUED in science. They are not trusted. That is partly the 200-year-old legacy of the French mathematicians Lagrange and Laplace, who scrupulously labored to reduce all logical thought to precise formulae and carefully chosen words; sloppy diagrams were suspect. Their motivation was, I believe, partly technological: At that time drawings were imprecise and costly, a product of human hands. But in our lifetime the computer has changed all that. A modern diagram or chart can be as precise as desired, and is no more costly than the computer that draws it.

The most-studied evidence, by the greatest number of economists, concerns what is called short-term dependence. This refers to the way price levels or price changes at one moment can influence those shortly afterwards—an hour, a day, or a few years, depending on what you consider “short.” A “momentum” effect is at work, some economists theorize: Once a stock price starts climbing, the odds are slightly in favor of it continuing to climb for a while longer.

Stock prices are not independent. Today’s action can, at least slightly, affect tomorrow’s action. The standard model is, again, wrong.

P/E should be meaningless. In fact, several studies have found, stocks with high P/E ratios tend to perform worse than stocks with low ratios.

That something, though expressed in a neat equation, is harsh and Darwinian, in Pareto’s view. At the very bottom of the wealth curve, he wrote, men and women starve and children die young. In the broad middle of the curve all is turmoil and motion: people rising and falling, climbing by talent or luck and falling by alcoholism, tuberculosis, or other forms of unfitness. At the very narrow top sit the elite of the elite, who control wealth and power for a time—until they are unseated through revolution or upheaval by a new aristocratic class. There is no progress in human history. Democracy is a fraud. Human nature is primitive, emotional, unyielding. The smarter, abler, stronger, and shrewder take the lion’s share. The weak starve, lest society become degenerate:

British philosopher Karl Popper called him the “theoretician of totalitarianism.”

serendipity,

Maybe, but the formula works. For instance, suppose you want to keep New York in steady water supply for a century: How big should you make the reservoirs? Hurst’s formula gave the answer. From 1826 to 1945, Hurst observed, it had rained an average of forty-two inches a year in New York; and from year to year, the standard deviation was 6.3 inches. According to Hurst’s formula, to store enough water for one hundred-year extremes of drought or flood, you would need reservoirs deep enough to store up to 16.7 times the standard deviation: 105 inches, or two and a half years’ supply. Other hydrologists on other rivers have since confirmed his basic findings. His

In the 1960s, some old-timers on Wall Street—the men who remembered the trauma of the 1929 Crash and the Great Depression—gave me a warning: “When we fade from this business, something will be lost. That is the memory of 1929.” Because of that personal recollection, they said, they acted with more caution than they otherwise might. Collectively, their generation provided an in-built brake on the wildest forms of speculation, an insurance policy against financial excess and consequent catastrophe. Their memories provided a practical form of long-term dependence in the financial markets. Is it any wonder that in 1987, when most of those men were gone and their wisdom forgotten, the market encountered its first crash in nearly sixty years? Or that, two decades later, we would see the biggest bull market, and the worst bear market, in generations? Yet standard financial theory holds that, in modeling markets, all that matters is today’s news and the expectation of tomorrow’s news.

Brownian motion is a bank economist’s best friend. When asked by his boss to predict the dollar-sterling rate a year from now, he can smartly sidestep the question. Working from today’s rate of $1.65 to the pound, he gives, not a specific forecast of $1.70, but a vague Brownian range: “The pound will trade between $1.55 and $1.75, and there is a good case to make for it trending up within that range—if the U.S. economy stutters, if inflation in Britain rises moderately, if ...” Of course, he is only staking his job on the vague range, not on the what-ifs—so he survives to forecast another year. But what happens if the exchange rate wanders farther than the square-root-of-time law forecasts? Trouble for the economist, obviously. How could that happen? Easily, if exchange rates exhibit long dependence. A rate move in one direction will tend to continue on the next day, and a few days later. The rate will still bounce around from day to day. But in the long run it will drift farther and farther from its starting point. The rates are no longer fluctuating by the blindest form of chance. Now the game is rigged, just like the Nile floods running in sequence.

Such forceful editors are the salt of the earth, but rare in scientific publishing. More common is the risk-avoiding bureaucrat, nailed to an influential editorial chair.

1990s, the first tests of the model’s applicability were conducted in the doctoral dissertations of two of my Yale students, Laurent Calvet and Adlai Fisher, now teaching at, respectively, Harvard University and the University of British Columbia. We

Zurich consulting firm, Olsen & Associates,

What economists call market “microstructure” starts to kick in.

Market “Timing” Matters Greatly. Big Gains and Losses Concentrate into Small Packages of Time. Concentration is common. Look at a map of gold deposits around the world: You see clusters of gold veins—in South Africa and Zimbabwe, in the far reaches of Siberia and elsewhere. This is not total chance; millennia of real tectonic forces gradually worked it that way. Understanding concentration is crucial to many businesses, especially insurance. A recent study of tornado damage in Texas, Louisiana, and Mississippi found 90 percent of the claims came from just 5 percent of the insured land area. In a financial market, volatility is concentrated, too; and it is no mystery why. News events—corporate earnings releases, inflation reports, central bank pronouncements—help drive prices. Orthodox economists often model them as a long series of random events spread out over time. While they can be of varying importance and size, their assumed distribution follows the bell curve so that no single one is preeminent. What sense is this? The terrorist attack on the World Trade Center was, by anyone’s reckoning, far and away the most important event in years for world stability and, consequently, for financial markets. It forced the closure of the New York Stock Exchange for an unprecedented five days, and when trading reopened caused a 7.5 percent fall. It was one titanic event, not the sum of many small ones. Big news causes big market action. And that action concentrates in small slices ...

Now, very few of us are in that league, but we can in our modest way take cognizance of concentration. Suppose big news has inflated a stock price by 40 percent in a week, more than twice its normal volatility. What are the odds that, anytime soon, yet another 40 percent run will occur? Not impossible, of course, but certainly not large. A prudent investor would do as the Wall Street pros: Take a profit.

The mathematics of Bachelier, Markowitz, Sharpe, and Black-Scholes all assume continuous change from one price to the next. Without that, their formulae simply do not work. Alas, the assumption is false and so the math is wrong. Financial prices certainly jump, skip, and leap—up and down.

Time is different for every investor. Each time-scale you consider, each holdingperiod for a stock or bond, has its own kind of risks. Under this view, a quick day-trade poses entirely different scales of risk than does a six-month investment—and in most eyes, the day-trader is the more likely to go broke.

We mathematicians and physicists love what we call an invariance. That is a property that remains unchanged, no matter how you transform the data, shape, or object under study. Fractal geometry is the mathematics of one such invariance in the physical world—the study of patterns, in space or time, that remain the same even as the scale of observation changes. Statisticians have a kindred concept, called stationarity: A stationary time series has the same basic statistical properties throughout. Economists argue their field may be different. Economist Jacob Marshak once proclaimed at a meeting I attended that the only economic invariance he could imagine concerned the equality between the number of left and right shoes—and not even that could be trusted. Following that thinking, many recent models of price variation try to explain the obviously shifting pattern of volatility by inserting parameters that change by the day, hour, and second; such are in the GARCH family mentioned earlier. I am an optimist. I would rather not dismiss the existence of invariances but continually look for them hiding in non-obvious places. Invariances make life easier. If you can find some market properties that remain constant over time or place, you can build better, more useful models and make sounder financial decisions. My multifractal model works with just such a set of consistent parameters.

To be sure, I do not argue there is no such thing as intrinsic value. It remains a popular notion, and one that I myself have used in some of my economic models. But the turbulent markets of the past few decades should have taught us, at the least, that value is a slippery concept, and one whose usefulness is vastly over-rated.

existentialist world, a world without absolutes?

For starters, portfolio managers can more frequently resort to what is called stress-testing. It means letting a computer simulate everything that could possibly go wrong, and seeing if any of the possible outcomes seem so unbearable

that you want to rethink the whole strategy. The technology is called a Monte Carlo simulation.

many of the world’s banks. The method is called Value at Risk, or VAR,

Extreme Value Theory and borrowed from the insurance industry, is on the right track: It assumes prices vary wildly, with “fat tails” that scale.