Stock markets move logically - except when they don't

Fractal geometry may finally explain financial risks

Don't be surprised while reading "The (Mis)behavior of Markets" if you find yourself asking, "When it comes to risk, does anybody on Wall Street know what they're talking about?"

The answer is a little chilling. Despite decades of research, no one has yet developed a good theory that explains the risks of financial markets. Worse, many investors underestimate those risks, according to Benoit Mandelbrot, a maverick mathematician, and his coauthor Richard Hudson, a financial journalist. Under their analytical microscope, all those traditional and comforting models for reducing investment risk - like "buy for the long term" and "allocate your assets" - suddenly sound hollow.

The reason is that today's accepted models are wrong, Mandelbrot argues.

For example, modern portfolio theory treats price movements much like the average height of a population. Most adults hover around that average. Those who top 7 feet or fall below 4 are too few to change the average much. And there are no 12-foot giants or 6-inch dwarfs. Likewise in the markets, most day-to-day price movements are moderate; a few big ones don't change the averages much, and really huge ups or downs almost never occur.

The problem, Mandelbrot points out, is that giants and dwarfs haunt the market far more frequently than the theory suggests. For example: From 1916 to 2003, according to the conventional mathematical model, there should have been only 58 times when the Dow Jones Industrial Average moved more than 3.4 percent in a single day. But in reality, there were 1,001 such exciting days. In theory, one-day swings of more than 7 percent should come only once every 300,000 years. But so far in the past 87 years, American stock investors have experienced 48 moves like that.

The fallacy extends to other markets. The exchange rate of the dollar vs. the yen once moved 7.9 percent in a single day. Theory suggests that change should never have happened, even if people had been trading currencies every day since the universe was formed, some 15 billion years ago, Mandelbrot argues.

Such revelations come as no surprise to professional traders. They began tweaking the traditional model long ago to try to account for these anomalies. But for the average investor, trusting retirement savings to conventional asset management, Mandelbrot's conclusions come like a bolt from the blue.

If the markets are riskier than most analysts let on, what's the average investor to do?

Regrettably, that's a question this book never quite answers. For all his potent insight into the fallacies of modern portfolio theory - and the engaging way he traces the historical development of its various mathematical underpinnings - Mandelbrot doesn't offer a convincing replacement.

To be fair, he never claims to have one. Stock prices are probably not predictable in a way that can make people rich, he writes. But the risk of the markets can be modeled so that people can avoid big losses - if someone can figure out how to do it.

Mandelbrot, who has made his reputation with fractal geometry (those elegantly complex figures that repeat themselves no matter what the magnification), thinks fractals offer a way forward.

He makes an intriguing case. Unlike, say, Euclidean geometry, fractals measure roughness, whether it's a jagged coastline or the volatility of financial markets. And fractals have been able to simulate the unpredictability and wild swings of markets fairly convincingly. Some 100 people around the world are seriously working on fractal finance now.

Mandelbrot details a few of the more interesting experiments. But the research hasn't progressed very far, he adds, which leaves ordinary investors in an all-too- familiar gap, where cutting-edge researchers know far more about what's wrong than how to fix it.

Readers of this book will probably take a hard look at their own assumptions about the riskiness of financial markets. That's a good thing. But they'll search these pages in vain for an alternative that can keep them from losing their shirts.

Laurent Belsie is the Monitor's personal-finance editor.

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