Fat Tail

A fat tail is a probability distribution that more commonly forecasts movements of 3 or more standard deviations than a normal distribution

What is a fat tail?

A fat tail is a probability distribution that more commonly forecasts movements of three or more standard deviations than a normal distribution. Periods of financial stress had resulted in market conditions with broader tails even before the financial crisis. Distributions with high kurtosis are also termed fat-tailed distributions.

A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed. Different research communities favor one or the other largely for historical reasons, and may have differences in the precise definition of either.

Fat-tailed distributions have been empirically encountered in a variety of areas: physics, earth sciences, economics and political science. The class of fat-tailed distributions includes those whose tails decay like a power law, which is a common point of reference in their use in the scientific literature. However, fat-tailed distributions also include other slowly-decaying distributions, such as the log-normal.

Fat tails and risk estimate distortions

Central events are more common and rare events more extreme in the Cauchy distribution than in Brownian motion. A single event may comprise 99% of total variation, hence the “undefined variance”.

Compared to fat-tailed distributions, in the normal distribution, events that deviate from the mean by five or more standard deviations (“5-sigma events”) have lower probability, meaning that in the normal distribution extreme events are less likely than for fat-tailed distributions. Fat-tailed distributions such as the Cauchy distribution (and all other stable distributions with the exception of the normal distribution) have “undefined sigma” (more technically, the variance is undefined).

As a consequence, when data arise from an underlying fat-tailed distribution, shoehorning in the “normal distribution” model of risk—and estimating sigma based (necessarily) on a finite sample size—would understate the true degree of predictive difficulty (and of risk). Many—notably Benoît Mandelbrot as well as Nassim Taleb—have noted this shortcoming of the normal distribution model and have proposed that fat-tailed distributions such as the stable distributions govern asset returns frequently found in finance.

The Black–Scholes model of option pricing is based on a normal distribution. If the distribution is actually a fat-tailed one, then the model will under-price options that are far out of the money, since a 5- or 7-sigma event is much more likely than the normal distribution would predict.[6]

Applications in economics

In finance, fat tails often occur but are considered undesirable because of the additional risk they imply. For example, an investment strategy may have an expected return, after one year, that is five times its standard deviation. Assuming a normal distribution, the likelihood of its failure (negative return) is less than one in a million; in practice, it may be higher. Normal distributions that emerge in finance generally do so because the factors influencing an asset’s value or price are mathematically “well-behaved”, and the central limit theorem provides for such a distribution. However, traumatic “real-world” events (such as an oil shock, a large corporate bankruptcy, or an abrupt change in a political situation) are usually not mathematically well-behaved.

Historical examples include the Wall Street Crash of 1929, Black Monday (1987), Dot-com bubble, Late-2000s financial crisis, 2010 flash crash, the 2020 stock market crash and the unpegging of some currencies.

Fat tails in market return distributions also have some behavioral origins (investor excessive optimism or pessimism leading to large market moves) and are therefore studied in behavioral finance.

In marketing, the familiar 80-20 rule frequently found (e.g. “20% of customers account for 80% of the revenue”) is a manifestation of a fat tail distribution underlying the data.

The “fat tails” are also observed in commodity markets or in the record industry, especially in phonographic markets. The probability density function for logarithm of weekly record sales changes is highly leptokurtic and characterized by a narrower and larger maximum, and by a fatter tail than in the normal distribution case. On the other hand, this distribution has only one fat tail associated with an increase in sales due to promotion of the new records that enter the charts

Why are fat tails important in finance?

Periods of financial stress had resulted in market conditions with broader tails even before the financial crisis. This is significant because normal distributions understate asset prices, stock returns, and risk management measures.

Managing Fat Tail Risk

Even though fat tail risk is difficult or perhaps even impossible to calculate, it can still be lessened by taking appropriate measures. Here are two ideas for curtailing fat tails.

Use Stop Losses

Stop losses are orders placed to purchase or sell a security when it reaches a designated price. Since your position sizing should be based on the distance of your stop from your entry price, and your stop loss should be placed in an area that invalidates your trade signal if price touches, then exiting a losing trade with a stop loss can help you cut off the ends of those fat tails. 

However, it is worth remembering that since stop losses are triggered as market orders, their price is not guaranteed and can sometimes result in slippage. This is usually not a huge problem and results in only a few points lost in most cases, however during a genuine fat tail event, prices could gap significantly. It is worth bearing this in mind when calculating position sizing, perhaps trading at a slightly smaller size for cover for such an event.

Profiting from Fat Tails

As mentioned many times throughout this post, the fat tailed nature of market results can mean extreme losses or extreme profits, depending on the structure of the trader and how risk was managed. Therefore, traders can try to capture the positive side of fat tails to profit from them.

Other than options as mentioned before, the other method is to follow the trend.

Use Trend-Following Strategies

Trend following strategies simply trade in the direction of price, buying higher highs and selling lower lows. Such strategies are often based on breakouts or moving averages.

In combination with the use of strict stop losses, trend following strategies essentially cut off the left-side tail of a strategy’s returns by exposing itself to the tails on both sides of the assets distribution with the aid of long and short positions. This, combined with letting winners run, means trend following strategies can capture and profit from fat tails.