Power-law distributions: are they better than normal?

Everybody in the investment world surely knows by now about fat tails and the shortcomings of the normal distribution as the basis of a model of investment behavior. But where should we turn? One suggestion is power-law distributions.

Not normal at all

We learned in high school about the Central Limit Theorem and how normal distributions naturally arise when lots of independent variables are added together. Power-law distributions, on the other hand, tend to arise when there’s a more complex system at work, most notably when there’s a self-reinforcing dynamic. The distribution of wealth, for example, tends to follow a power-law distribution, a natural consequence of the old saying that “the rich get richer”. Web page hits; book sales; the size of cities: these have all been found to approximate to a power-law distribution, as have natural phenomena such as earthquakes and even the flooding of the Nile river.

That power laws can offer a good fit when modeling the tails of the distributions of financial outcomes was a cause initiated by Benoit Mandelbrot, and given a boost by Nassim Nicholas Taleb’s The Black Swan. Paul Kaplan of Morningstar has written about power-law-like distributions on a number of occasions, including a recent article “The Tails That Wag the Dog.”¹ In it, he describes how Eugene Fama—when a doctoral student way back in 1964—applied Mandelbrot’s model to stock prices for his dissertation. While not a perfect description of the range of possible future returns (it is still just a model), the log-stable distribution (whose tails have power-law qualities) seems to offer a better fit of actual past market behavior than most alternatives, and certainly a much better fit than the normal distribution.

Silly math

But Kaplan also provides reasons that these models have not been more widely adopted. One reason he puts forward is that the variance (and hence standard deviation) can be infinite.

“An infinite standard deviation,” you may be thinking, “that’s silly.”

And indeed it is, if you’re talking about the standard deviation of a sample of observations drawn from a distribution. But when we move from a sample to the full distribution of potential outcomes (which is not the same thing), there really can be an infinite standard deviation. And if that distribution is a power-law distribution, there generally is. So if actual stock market returns do indeed have power-law-like tails, then things become awkward: as Kaplan puts it “the lack of a finite variance means that most portfolio theories and most portfolio construction techniques are invalid.”

A second practical challenge that Kaplan highlights is that it’s difficult to estimate the parameters of a power-law distribution. The characteristics of observed results converge very slowly to the characteristics of the underlying distribution from which they are drawn: even with several thousand observations, the sample may not provide an accurate estimate. So switching from a normally-distributed world view to a power-law world view is far from simple in practice.

Yet another reminder of just how difficult it is to accurately model investment market behavior.