# Fat-tailed distribution

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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.

## Contents

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. 

## The extreme case: a power-law distribution

The most extreme case of a fat tail is given by a distribution whose tail decays like a power law. A variety of Cauchy distributions for various location and scale parameters. Cauchy distributions are examples of fat-tailed distributions.

That is, if the complementary cumulative distribution of a random variable X can be expressed as[ citation needed ]

$\Pr[X>x]\sim x^{-\alpha }{\text{ as }}x\to \infty ,\qquad \alpha >0.\,$ then the distribution is said to have a fat tail if $\alpha$ is small. For instance, if $\alpha <3$ , the variance and the skewness of the tail is mathematically undefined (a special property of the power-law distribution), and hence larger than any normal or exponential distribution. For values of $\alpha >3$ , the claim of a fat tail is more ambiguous, because in this parameter range, the variance, skewness, and kurtosis can be finite, depending on the precise value of $\alpha >3$ , and thus potentially smaller than a high-variance normal or exponential tail. This ambiguity often leads to disagreements about precisely what is or is not a fat-tailed distribution. For $k>\alpha -1$ , the $k^{th}$ moment is infinite, so for every power law distribution, some moments are undefined.[ citation needed ]

Note: here the tilde notation "$\sim$ " refers to the asymptotic equivalence of functions, meaning that their ratio tends to a constant. In other words, asymptotically, the tail of the distribution decays like a power law.[ citation needed ]

## Fat tails and risk estimate distortions Lévy 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". Lévy flight from a normal distribution (Brownian motion).

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. 

## 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 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. 

## 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,  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.

## Related Research Articles

In probability theory and statistics, kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Different measures of kurtosis may have different interpretations. In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price given the risk of the security and its expected return. The equation and model are named after economists Fischer Black and Myron Scholes; Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited. Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose, given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses. In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.

In mathematics, the moments of a function are quantitative measures related to the shape of the function's graph. If the function represents mass, then the first moment is the center of the mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.

In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors, in which case the mixture distribution is a multivariate distribution. In statistics, a bimodaldistribution is a probability distribution with two different modes, which may also be referred to as a bimodal distribution. These appear as distinct peaks in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form bimodal distributions.

In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.

The normal-inverse Gaussian distribution (NIG) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997.

The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The distribution was introduced in the financial literature by Madan and Seneta. The variance-gamma distributions form a subclass of the generalised hyperbolic distributions.

In statistics and decision theory, kurtosis risk is the risk that results when a statistical model assumes the normal distribution, but is applied to observations that have a tendency to occasionally be much farther from the average than is expected for a normal distribution.

Skewness risk in financial modeling is the risk that results when observations are not spread symmetrically around an average value, but instead have a skewed distribution. As a result, the mean and the median can be different. Skewness risk can arise in any quantitative model that assumes a symmetric distribution but is applied to skewed data. In finance, volatility is the degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns. In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

Financial models with long-tailed distributions and volatility clustering have been introduced to overcome problems with the realism of classical financial models. These classical models of financial time series typically assume homoskedasticity and normality cannot explain stylized phenomena such as skewness, heavy tails, and volatility clustering of the empirical asset returns in finance. In 1963, Benoit Mandelbrot first used the stable distribution to model the empirical distributions which have the skewness and heavy-tail property. Since -stable distributions have infinite -th moments for all , the tempered stable processes have been proposed for overcoming this limitation of the stable distribution. The seven states of randomness in probability theory, fractals and risk analysis are extensions of the concept of randomness as modeled by the normal distribution. These seven states were first introduced by Benoît Mandelbrot in his 1997 book Fractals and Scaling in Finance, which applied fractal analysis to the study of risk and randomness. This classification builds upon the three main states of randomness: mild, slow, and wild.

Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. See Quantitative analyst.

In statistics and probability theory, the nonparametric skew is a statistic occasionally used with random variables that take real values. It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean. Its calculation does not require any knowledge of the form of the underlying distribution—hence the name nonparametric. It has some desirable properties: it is zero for any symmetric distribution; it is unaffected by a scale shift; and it reveals either left- or right-skewness equally well. In some statistical samples it has been shown to be less powerful than the usual measures of skewness in detecting departures of the population from normality.

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