Newsletter: The Fractal Market Hypothesis, the Infinite Variance Syndrome and the Nature of Risk
Markowitz and Mandelbrot were both right. Previously unpublished research.
Benoit Mandelbrot, the father of fractals, was always controversial. But one of his more misunderstood theories dealt with the variance of market returns. He equated the fat-tailed, high-peak-at-the-mean distribution we see in realized market returns with Stable Paretian distributions. In these distributions the population variance is undefined, or infinite. But the idea of infinite variance seemed bizarre to those of us schooled in traditional statistics, so it's often dismissed. I've written extensively about this in my first two books, and it's an important aspect of the Fractal Market Hypothesis. If the "infinite variance syndrome" is true, then the entire structure of market risk as we know it is out the window. It would mean that the sample standard deviation we calculate to represent risk (the population standard deviation) as postulated by Markowitz does not really exist.
Back in 2009 I discussed what infinite variance meant for Modern Portfolio Theory (MPT) with Harry Markowitz. He told me that he had shown Mandelbrot that the sum of a series of normal distributions would result in a distribution that looked much like Stable Paretian distributions. So you could have fat-tails, but variance was still finite, and measurable. This is the argument of choice for rejecting the infinite variance syndrome.
In December 2014 I published a paper called "Stable and Unstable Markets: A Tail of 2 States" at First Quadrant that used the VIX to show that the stock market had two states: a low volatility state and a high volatility state. These states lasted for years and were measurable. I also found that the low vol state did have a finite variance, but the high vol state did not. That meant that our nice measure of market risk would fail when downside risk was the highest.
But in 2018 I did further research parsing vol into 4 states based upon levels of the VXO, or the implied volatility of S&P 100 index options. This research has not been published, and I am describing it for the first time in this newsletter. I found that Markowitz was right, but unfortunately he was only right on a technicality. For investors, the infinite variance syndrome holds in the high vol state because the market moves between the states so rapidly that investors cannot use them.