The mind-bending laws of markets

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The mind-bending laws of markets

Like the devastating Japanese earthquake of 2011, the stock market crash of Oct. 19, 1987, came as a total shock to most people. Yet the crash wasn’t entirely without warning. Five days before, the Dow Jones Industrial Average dropped 95 points, which was then an all-time record. Two days later, it closed down another 108 points. Just like others crashes - 1929, for example - and all major earthquakes, the 1987 crash was preceded by significant rumblings.

The comparison of tectonic and market shocks goes far beyond metaphor and analogy. Consider, for example, how much the prices of stocks and other financial instruments change over a certain time interval - say a few minutes, a single day, or a week. In the early 1960s, French mathematician Benoit Mandelbrot carried out a landmark study of such changes in the prices of cotton and found that the statistics of large market returns follow an inverse power law very much like the Gutenberg-Richter law for earthquakes. More than 30 years later, physicists found that this law-like pattern holds for intervals varying from a second up to a month and in different kinds of markets - stocks, foreign exchange, futures - as well as in many different countries.

The close correspondence with the Gutenberg-Richter law raises some intriguing questions. The behavior of markets, we tend to believe, rests on the thoughts and emotions and actions of innumerable people, firms and governments. Equilibrium theory - the crystallization of decades of academic thought about how the economy works - insists it reflects the rational nature of human beings.

Yet somehow all this thinking and psychology and individual free will don’t get in the way of a law-like pattern. It turns up in the marketplace just as readily as it does in the purely mechanical workings of the earth’s crust.

What does the power law mean? In the case of earthquakes, it implies that large and small temblors work the same way, as continental plates slide past one another, releasing energy as heat and vibration. How much simply depends on how far the sliding goes and the surface area involved. The point is that large and small earthquakes aren’t essentially different in the way that, say, stars and Ping-Pong balls are. They differ only in degree, not quality.

Power-law mathematics causes trouble for our intuition, which is attuned to another way of thinking. The average weight of an adult male in the U.S. is about 190 pounds. There is a small fraction of men who weigh more than 300 pounds, some more than 400, and a handful pushing the record books at 600 or more. If peoples’ weights worked like earthquakes, then some people would weigh tens of millions of pounds, as much as 10 Boeing 767 jetliners put together. Such people would be big, but not in any sense freakishly big, just big in an expected and ordinary way.

Power laws reveal the profound and generally underestimated importance of extremes. According to exponential, “normal” statistics, extreme events are so rare they can be disdainfully ignored. Power-law statistics, which are ubiquitous in everything from the sizes of meteors to the earnings of books or films, mean that extremes aren’t so rare and they matter most. Things like the movements of continental plates and markets aren’t really driven by a gradual accumulation of the normal and routine, but by the singular, disproportionate impact of a few great tumultuous earthquakes and crises.

Unfortunately, centuries of science and mathematics tradition, focusing on the normal statistics of things like weights, heights and test scores, has taught us to see the world incorrectly. It was a telling moment on April 27, 2010, when Goldman Sachs Chief Financial Officer David Viniar testified to the Senate Permanent Subcommittee on Investigations, which was exploring his company’s role in the financial crisis.

“We were seeing things,” Viniar said, recalling the tumult of the worst days, “that were 25-standard-deviation events, several days in a row.”

In Gaussian mathematics, even an eight-standard-deviation event is expected only about once in the entire history of the universe. A 25-standard-deviation event should be expected about once every 10 to the 135th power years - one followed by 135 zeros. Stocks over a single day typically change less than about 2 percent, so a movement of even 10 standard deviations means a movement of at least 20 percent.

While normal statistics says this should happen once every 10 to the 22nd power days, market data show that it happens essentially every week for at least one of the few thousand stocks in the market. So perhaps we should reexamine our assumptions.

Power laws are immensely important for proper risk management, for assessing the likelihood of large market upheavals with at least some accuracy. But what’s of even greater importance is that the power law of market returns helps illuminate a path to theories of finance with greater explanatory power than we’ve previously seen. It is a core fact that any theory of markets really has to explain.

*The author, a theoretical physicist, is the author of “The Social Atom: Why the Rich Get Richer, Cheaters Get Caught and Your Neighbor Usually Looks Like You.”

by Mark Buchanan
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