*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.