Back in grade school, I was the kind of kid who got excited about things like fractal geometry. I even went so far as to attend math camp one summer on the Eastern Shore. I learned back then about what is still a relatively unknown branch of mathematics.

Everyone in school learns about Euclidean geometry, which describes perfect shapes that are never actually observed in nature. Yet few learn about fractal geometry, even though it is the best tool we have to describe many types of complex natural phenomena, e.g. weather patterns, turbulence, the location of oil and other natural deposits in the earth, irregularity in the rhythm of heartbeats, etc. (With a very simple formula, it also produces the infinitely complex object pictured above known as the Mandelbrot set.)

Until a few weeks ago, I hadn't given much thought to these ideas for many years. However, I stumbled on a book by the man who discovered fractal geometry, Benoit Mandelbrot, while browsing through the section of the store devoted to finance. I was surprised to see that Mandelbrot had applied this new branch of mathematics not just to purely natural phenomena, but also to the world of finance. After reading The (Mis)behavior of Markets, I also discovered that Mandelbrot was—perhaps ironically—the dissertation advisor of Eugene Fama, the father of the now much disputed efficient markets hypothesis.

Some months ago, Myron Scholes (a disciple of Fama) argued that we don't have an alternative paradigm to replace the efficient markets hypothesis. Mandelbrot makes it clear that we do have an alternative, and one that demands our attention. He points out two fallacies at the heart of the efficient markets hypothesis that are much better dealt with by fractal geometry: (1) price changes are not independent, but rather depend on earlier price changes; and (2) the distibution of price changes in financial markets does not exhibit a bell curve shape, but instead has fat tails governed by power laws.

So what does this have to do with the Aswan Dam? For some types of natural phenomena, one piece of data is independent from others. This is true of the heights of people in the general population—my height doesn't depend on the height of the guy sitting next to me. It turns out that for some natural phenomena, though, this doesn't hold. One example is the flooding of the Nile. If floods were independently distributed, then each year would face about the same probability of flooding as the last. In fact, empirical data show this is not true—years of flooding are likely to follow each other. Likewise, years of drought are also likely to be clustered together. The consequence is that the Aswan Dam had to be built considerably higher than would have been true if the probability of flooding each year were independently distributed.

Mandelbrot shows that the same is true for price fluctutations—in certain markets, price changes exhibit dependence wherein volatility is likely to cluster. In other words, markets do not follow a random walk, as posited by Fama. The consequence? Financial institutions need to hold more reserves than would be predicted by the efficient markets hypothesis.

As for the second point, Mandelbrot demonstrates that large price changes are much more frequent than the mild volatility predicted by the bell-curve shape that undergirds the efficient markets hypothesis. He first discovered this by analyzing price changes in the cotton market. The bell curve predicts that the more extreme a price change, the less likely it will occur. In fact, price changes are governed by a power law, whereby a few giant price changes may be more important for the average price than all other price changes combined. Anyone who lost multiple years' worth of gains from their portfolio in 2008 will understand what Mandelbrot is getting at.

The consequence? Again, a much more cautious approach to finance is called for, as markets are far riskier than predicted by the efficient markets hypothesis. Hopefully, the folks responsible for Basel II will take note.

(*Photo Credit: Eugene Regis*)

## Comments

## Good thinking! I expected you to mention black swans as well - a

## @Alberto: I almost thought to mention Nassim Taleb...

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