# Fat-tailed distribution

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This article is about probability distributions. For animals, see Fat-tailed (disambiguation).

A fat-tailed distribution is a probability distribution that has the property, along with the other heavy-tailed distributions, that it exhibits large skewness or kurtosis. This comparison is often made relative to the normal distribution, or to the exponential distribution. Fat-tailed distributions have been empirically encountered in a variety of areas: economics, physics, and earth sciences. Some fat-tailed distributions have power law decay in the tail of the distribution, but do not necessarily follow a power law everywhere.[1]

## Definition

A variety of Cauchy distributions for various location and scale parameters. Cauchy distributions are examples of fat-tailed distributions.

The distribution of a random variable X is said to have a fat tail if

$\Pr[X>x] \sim x^{- \alpha}\text{ as }x \to \infty,\qquad \alpha > 0.\,$

That is, if X has probability density function $f_X(x)$,

$f_X(x) \sim x^{ - (1 + \alpha)} \text{ as }x \to \infty, \qquad \alpha > 0.\,$

Here the tilde notation "$\sim$" refers to the asymptotic equivalence of functions. Some reserve the term "fat tail" for distributions where 0 < α < 2 (i.e. only in cases with infinite variance).

## Fat tails and risk estimate distortions

Levy flight from a Cauchy Distribution compared to Brownian Motion (below). 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".

By contrast 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, thus meaning that in the normal distribution rare events can happen but are likely to be more mild in comparison to fat-tailed distributions . On the other hand, fat-tailed distributions such as the Cauchy distribution (and all other stable distributions with the exception of the normal distribution) are examples of fat-tailed distributions that have "undefined sigma" (more technically, the variance is not bounded).

Thus when data naturally arise from a fat-tailed 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 degree of predictive difficulty. 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.[2][3][4]

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.[5]

## 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.[6]

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.[citation needed]

The "fat tails" are also observed in commodity markets or in the record industry, especially in phonographic market. 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.[7]

## Applications in geopolitics

In The Fat Tail: The Power of Political Knowledge for Strategic Investing, political scientists Ian Bremmer and Preston Keat propose to apply the fat tail concept to geopolitics. As William Safire notes in his etymology of the term,[8] a fat tail occurs when there is an unexpectedly thick end or “tail” toward the edges of a distribution curve, indicating an irregularly high likelihood of catastrophic events. This represents the risks of a particular event occurring that are so unlikely to happen and difficult to predict that many choose to ignore their possibility. One example that Bremmer and Keat highlight in The Fat Tail is the August 1998 Russian devaluation and debt default. Leading up to this event, economic analysts predicted that Russia would not default because the country had both the ability and willingness to continue to make its payments. However, political analysts argued that Russia’s fragmented leadership and lack of market regulation—along with the fact that several powerful Russian officials would benefit from a default—reduced Russia’s willingness to pay. Since these political factors were missing from the economic models, the economists did not assign the correct probability to a Russian default.

## References

1. ^ Bahat; Rabinovich; Frid (2005). Tensile Fracturing in Rocks. Springer.
2. ^ Taleb, N. N. (2007). The Black Swan. Random House and Penguin.
3. ^ Mandelbrot, B. (1997). Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer.
4. ^ Mandelbrot, B. (1963). "The Variation of Certain Speculative Prices" (PDF). The Journal of Business.
5. ^ Steven R. Dunbar,Limitations of the Black-Scholes Model, Stochastic Processes and Advanced Mathematical Finance 2009 http://www.math.unl.edu/~sdunbar1/MathematicalFinance/Lessons/BlackScholes/Limitations/limitations.xml
6. ^ Dash, Jan W. (2004). Quantitative Finance and Risk Management: A Physicist's Approach. World Scientific Pub.
7. ^ Buda, A. (2012). "Does pop music exist? Hierarchical structure in phonographic markets". Physica A.
8. ^ On Language: Fat Tail