Fat tail

A fat tail is a property of some probability distributions (alternatively referred to as heavy-tailed distributions) exhibiting extremely large kurtosis particularly relative to the ubiquitous normal which itself is an example of an exceptionally thin tail distribution. Fat tail distributions have power law decay. More precisely, the distribution of a random variable "X" is said to have a fat tail if

:$Pr\; [X>x]\; sim\; x^\{-\; (1\; +\; alpha)\}mbox\{\; as\; \}x\; o\; infty,qquad\; alpha\; >\; 0.,$

Some reserve the term "fat tail" for distributions only where 0 < "α" < 2 (i.e. only in cases with infinite variance).

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 &mdash; and an estimate of the corresponding sigma based necessarily on a finite sample size &mdash; would severely understate the true risk. Many &mdash; notably Benoît Mandelbrot &mdash; 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 and under-prices options that are far out of the money since a 5 or 7 sigma event is more likely than the normal distribution predicts.

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.

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.