The persistence of kurtosis in financial markets

December 15, 2014 06:40 AM

I can calculate the movement of the stars, but not the madness of people.” 

—Sir Isaac Newton

Newton said this after losing a small fortune in the South Sea Bubble of 1720. I like to think that after this episode Newton realized that markets are fraught with large and quite unexpected price movements and perhaps exploited this phenomenon in his subsequent trades. The propensity of markets to generate what are often called “fat-tails” is the cornerstone of the philosophy of trend-following and the primary generator of its returns. 

A fat-tailed distribution is a probability distribution that has the property, along with the heavy-tailed distributions, that exhibits larger than normal kurtosis. That is to say that the probability of large magnitude events is more likely than a normal distribution would imply. This report highlights the prevalence of this pattern among price movements in many markets and across many time frames and why they are critical to the long-term trend following approach.

“The real normal” (right) shows a histogram of the daily returns for the Dow Jones Industrial Average since 1933. It is clear that daily returns near 0% are common whereas returns of +/–5% are much less frequent. 

The red overlaid curve is the Gaussian or normal distribution function with the same average and standard deviation as the DJIA returns. Such a function is how statisticians try to describe, mathematically, information like “how much more likely am I to observe a return between –3% and –1% than between –1% and +1%?” Much of finance theory is based on the red “normal” curve. The efficient market hypothesis basically implies that asset movements from one time to the next are unpredictable or random and a common model for this randomness is that the returns are normally distributed.

As noted in “The real normal” this assumption grossly underestimates the likelihood of large magnitude market moves. It suggests a single move outside of +/–5% should occur once every 66,000 trading days—that is once every 264 years. In reality it has happened 129 times in the last 80 years. This is the perfect example of fat tails and it is this fat tail phenomenon that trend followers rely on. Unfortunately, because of the way the data is typically presented it is hard to even notice this feature and it is often dismissed as insignificant market inefficiency.

A better view

Perhaps a better way to present the data would be to show how often these events occur as compared to how often they would be predicted to occur under the normal assumption (see “Not so un-normal,” right). The red line at 1.0 is the baseline normal assumption and the blue bars are the actual occurrences. The parallels are clear— the middle high peak from “The real normal,” is reflected in the fact that the middle bar in “Not so un-normal,” shows that small return events happen 1.5 times more often than normal would predict. And the high blue bars on the left and right extremes reflect how poorly the normal distribution underestimates the likelihood of these large magnitude events.

From Newton’s time until now

In our research we find this fat-tailed feature of freely traded assets is the most consistent and robust feature of markets. It exists across a variety of markets (commodities, equities, etc.), across a variety of return periods (daily, weekly, monthly, etc.), and across a variety of time periods studied (1970’s, 80’s, 90’s, 2000’s). “No new phenomena,” (below) shows the “actual to normal” comparison for a variety of markets, for various return periods, and over differing periods of study.

As one can see this sampling encompasses a variety of asset types, industries, time frames, and return frequencies, yet they all have one thing in common: kurtosis or fat tails. The goal of most long-term trend followers is to exploit this inefficiency. The trends our system follows go hand-in-hand with these fat-tailed events. We make no predictions about markets other than to anticipate that markets will continue to exhibit this feature and hence trend in the future, and we choose to invest in this phenomenon via systematic trading.

Further illustration

We only were able to show the results of a small number of markets and timeframes in this article. However, given an adequately sized data sample we were unable to find any markets on any time frames that failed to demonstrate the type of kurtosis (fat tails) illustrated in the charts above. 

For those of you not convinced, take a look at the link we provide in the online version of this article. It contains an Excel spreadsheet that generated some of the plots in this report. The database within the sheet contains market histories on most all of the liquid futures markets as well as approximately 200 equities markets. It will allow you to conduct this analysis on your own. Download the spreadsheet and when opening it choose Enable Editing, Enable Content and “make this a trusted document” or Enable Macros. Once the file is open you will find instructions on how to use it. 

Scot Billington co-founded Covenant Capital Management, a boutique CTA that has been managing client assets for more than 15 years. Rob Matthews is CCM’s Director of Research. Reach them at

About the Author

Rob Matthews develops technical trading strategies as Research Director for Covenant Capital Management.