Fat-tailed distribution

fat-tailed distribution is a probability distribution that has the property, along with the heavy-tailed distributions, that they exhibit extremely large skewness or kurtosis. This comparison is often made relative to the ubiquitous normal distribution, which itself is an example of an exceptionally thin tail distribution, or to the exponential distribution. Fat tail distributions have been empirically encountered in a fair number of areas: economics, physics, and earth sciences. Fat tail distributions have power law decay in the tail of the distribution, but do not necessarily follow a power law everywhere.

Fat tails and risk estimate distortions

By contrast to fat tail distributions, the normal distribution posits events that deviate from the mean by five or more standard deviations (“5-sigma event”) are extremely rare, with 10- or more sigma being practically impossible. On the other hand, fat tail distributions such as the Cauchy distribution (and all other stable distributions with the exception of the normal distribution) are examples of fat tail distributions that have “infinite sigma” (more technically: “the variance does not exist”).

Thus when data naturally arise from a fat tail distribution, shoehorning the normal distribution model of risk—and an estimate of the corresponding sigma based necessarily on a finite sample size—would severely understate the true risk. Many—notablyBenoît Mandelbrot as well as Nassim Taleb—have noted this shortcoming of the normal distribution model and have proposed that fat tail distributions such as the stable distribution 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.

Applications in economics

In finance, fat tails 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 Black Monday (1987), Dot-com bubble, Late-2000s financial crisis, 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 e.g record industry. 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 Gaussian 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.