By Christopher M. Wright

NAME: Benoit Mandelbrot TITLE: Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus at IBM's Thomas J. Watson Laboratory. BORN: Warsaw, 1924 HIS LATEST BOOK: "The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward," by Benoit B. Mandelbrot and Richard L. Hudson (Basic Books 2004) WEB SITE: www.math.yale. edu/mandelbrot/ EXPERIENCE: A self-described maverick, Professor Mandelbrot is a Polish-born Frenchman with a Ph.D. in mathematics. His prior postings include Harvard, M.I.T., the Princeton Institute for Advanced Study, and universities abroad. He has written several other books, including "The Fractal Geometry of Nature" (1982). He is the winner of numerous awards including the Wolf Prize for Physics (1993) and the Japan Prize in Science and Technology (2003). He has received more than a dozen honorary doctorate degrees in mathematics from universities in the U.S. and abroad.

Forget Euclidean geometry with its smooth lines and planes. Now comes Benoit Mandelbrot, the inventor of fractal geometry, who recently wrote an entertaining and challenging book, "The (mis)Behavior of Markets," in which he argues that his study of roughness, already applied to topography, meteorology, the compression of computer files, and many other fields, will rewrite the canon on finance. Portfolio asked the Yale University mathematics professor, among other things, how real estate prices look under the fractal microscope.

Portfolio: What do you mean by roughness and how does it apply to financial markets?
Mandelbrot: Nobody would describe a cloud as a perfect sphere. Clouds are like billows upon billows upon billows. They look the same whether you view a single cloud close up or a cloud bank far away. This property is called self-similarity and happens to also hold in financial market records. Look at price changes over different time frames—days, months, years, or even a century—the ups and downs are statistically the same at every scale, except when you get down to a few minutes or less.

Portfolio: Can your theories be used to predict prices or beat the market?
Mandelbrot: I'm not primarily concerned with predicting prices, although others may use my work that way. My concern is with the prediction of risk. Large price changes tend to cluster and follow one another. If there was a large price change yesterday, then today is a risky day.

Portfolio: You give as examples in the book the three large declines in the market in August 1998 where the odds of that happening were one in 500 billion.
Mandelbrot: My work can tell you that large changes tend to follow one another, but not their direction. In records of any financial price series, you will see long periods of relatively stable prices and short periods of extraordinary variation I call "storms" or "clusters." They don't average out as modern portfolio theory assumes. Portfolio: Where did conventional theory go wrong, in your view?
Mandelbrot: In 1900, the French mathematician Louis Bachelier came up with the idea that price changes follow the normal distribution, that most variations are small and plot nicely on a relatively narrow bell curve. It was a great contribution to science at the time.

But financial theorists after him have stuck to the normal distribution and disregarded extreme events, what statisticians call "outliers" and I call "fat tails."

Portfolio: One example you give is the crash of October 1987 where the Dow Jones went down more than 20 percent in a single day, a 20-standard deviation event.
Mandelbrot: Yes, the odds of that happening were less than one in 10 to the 50th power. It was not supposed to happen under conventional theory. A key point is that Bachelier assumed price changes to be continuous—the change over a short time interval would be small. But that's completely against the evidence.

Price changes can be brutally sudden and large, as the crash of '87 shows. In 2005, Merck went down by a third immediately after it announced it would take Vioxx off the market. Modern portfolio theorists need ad-hoc fixes to try to account for large price changes and bring their conventional models closer to the evidence. In my theories, large discontinuities are recognized from the outset.

Portfolio: You also zero in on other simplifying assumptions underlying today's financial theories.
Mandelbrot: Bachelier assumed that "every day the market is born again" and what happened yesterday doesn't matter. But think of the Merck example again: If the stock went down a third yesterday, is it really just an ordinary day for Merck today? Not likely. Prices do depend on what people remember. So I criticize the Efficient Markets Hypothesis that holds that price changes are solely driven by new information.

Portfolio: Your book says that markets are far riskier than people have wanted to believe. Why is that?
Mandelbrot: First let me clarify whom I was talking about. The general population views the stock market as risky, but financial theorists are working from models that assume price changes are small and normally distributed. The view of the market taught in business schools ignores "outliers"— extreme events that, if included, would change how the price series would have to be viewed.

It's true that price changes are small 95 percent of the time. But most of the change in price over a 10-year period doesn't come from the accumulation of small changes but from a few large events—the 5 percent of "outliers" in the "fat tail" on statistical charts. Money managers could go fishing most days if they only knew when to be at their desk to catch the big moves. You can't dismiss the outliers. They are very important, the essential element of risk.

Portfolio: You criticize VAR (value at risk, a common risk measurement tool) for being predicated on normal distributions, but risk management has moved beyond plain vanilla VAR to incorporate semi-variance (where catastrophes are scored exponentially), stress testing, worst-case scenarios, and Extreme Value Theory (the latter accounting for wild price changes and fat-tail distribution). What's still wrong with the state of the art in risk management today, in your view?
Mandelbrot: Risk management still does not take into account long-term price dependence, the tendency of bad news to come in flocks, as we say in the book. A bank might weather one crisis but not a second or third that follows in quick succession.

It's like the engineer we wrote about in the book who was smart enough to realize that the Nile might only flood a handful of times every hundred years but that the years of heavy rain might bunch together and designed his dams accordingly.

Portfolio: What could people learn from studying the price history of commercial real estate or any individual real estate stock using your methods?
Mandelbrot: Modern portfolio theory assumes that all financial instruments–stocks, bonds, foreign exchange–have the same properties. But there's no reason to think that such is the case.

The first stage of analysis should be to classify markets in a gross taxonomy as to how erratic they are. This goes well beyond the common measure of volatility. The second step would be to examine why one market behaves differently from another. For example, you get many more big price changes on individual properties when liquidity is low and transactions are few. The evidence that real estate price changes are not normally distributed is overwhelming. My models allow for behavioral differences between markets—for a taxonomy that has many species.

Running the REIT index price series under my models would tell you about the tail exponent. It replaces the more familiar "kurtosis" as an indication of how wild the price behavior of the index is. A score of 1.5 indicates that behavior is wilder than 1.6, etc. In addition, my analysis can give you an H-score, which is the exponent for long-term price dependence—how much of a "memory" the series seems to retain. And there are further intrinsic indicators that allow finer tuning.

Portfolio: Would the fact that REITs pay high dividends affect the outcome?
Mandelbrot: I can't answer that. Let me tell you why. I'm a mathematician who has worked in many fields—physics, economics, geomorphology, among others. As an outsider, I always skate on thin ice and must be extremely cautious in what I say.

Portfolio: Your models for finance are still embryonic, but are leading to some interesting ideas, like the construction of portfolios that disregard normal risk and construct efficient frontiers that minimize catastrophic risk instead. What do you hope for your models eventually?
Mandelbrot: Nobody believes in modern portfolio theory any longer. Everybody uses proprietary fixes. But that's like designing an airplane in a great hurry, then trying to fix all the problems as they show up. A theory that has too many fixes won't fly. Discontinuity is essential in the stock market. You can't neglect it and survive.

It's my hope that my work will become the basis of the next stage of financial analysis. It only took six or eight months for the world to respond to my Mandelbrot Set [popularized on posters and T-shirts], but some other ideas of mine have taken many years to find acceptance. I hope I'm still around to see my ideas in finance developed further. One thing is certain—change is unavoidable everywhere—including the quants' toolbox [referring to quantitative analysts on Wall Street].