A fat-tailed distribution is a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
that exhibits a large
skewness
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.
For a unimodal d ...
or
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
, relative to that of either a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
or an
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
. In common usage, the terms fat-tailed and
heavy-tailed
In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distr ...
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
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
, 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
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a 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
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
.
![Cauchy pdf](https://upload.wikimedia.org/wikipedia/commons/8/8c/Cauchy_pdf.svg)
That is, if the complementary
cumulative distribution
In statistics, the frequency (or absolute frequency) of an event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabular form.
Types
The cumul ...
of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
''X'' can be expressed as
:
then the distribution is said to have a fat tail if
. For such values the variance and the skewness of the tail are mathematically undefined (a special property of the power-law distribution), and hence larger than any normal or exponential distribution. For values of
, 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
, 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
, the
moment is infinite, so for every power law distribution, some moments are undefined.
Note: here the tilde notation "
" 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.
Fat tails and risk estimate distortions
Compared to fat-tailed distributions, in the normal distribution events that deviate from the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
by five or more
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
s ("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
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
(and all other
stable distributions
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stab ...
with the exception of the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
) have "undefined sigma" (more technically, the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
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
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
as well as
Nassim Taleb
Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness, ...
—have noted this shortcoming of the normal distribution model and have proposed that fat-tailed distributions such as the
stable distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stab ...
s govern asset returns frequently found in
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
.
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
In finance, moneyness is the relative position of the current price (or future price) of an underlying asset (e.g., a stock) with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a thr ...
, since a 5- or 7-sigma event is much more likely than the normal distribution would predict.
Applications in economics
In
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
, fat tails often occur but are considered undesirable because of the additional
risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environme ...
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
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
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
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
.
Historical examples include the
Wall Street Crash of 1929
The Wall Street Crash of 1929, also known as the Great Crash, was a major American stock market crash that occurred in the autumn of 1929. It started in September and ended late in October, when share prices on the New York Stock Exchange colla ...
,
Black Monday (1987)
Black Monday is the name commonly given to the global, sudden, severe, and largely unexpected stock market crash on Monday, October 19, 1987. In Australia and New Zealand, the day is also referred to as ''Black Tuesday'' because of the time zo ...
,
Dot-com bubble
The dot-com bubble (dot-com boom, tech bubble, or the Internet bubble) was a stock market bubble in the late 1990s, a period of massive growth in the use and adoption of the Internet.
Between 1995 and its peak in March 2000, the Nasdaq Compo ...
,
Late-2000s financial crisis,
2010 flash crash, the
2020 stock market crash
On 20 February 2020, stock markets across the world suddenly crashed after growing instability due to the COVID-19 pandemic. It ended on 7 April 2020.
Beginning on 13 May 2019, the yield curve on U.S. Treasury securities inverted, and rema ...
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
Behavioral economics studies the effects of psychological, cognitive, emotional, cultural and social factors on the decisions of individuals or institutions, such as how those decisions vary from those implied by classical economic theory.
...
.
In
marketing
Marketing is the process of exploring, creating, and delivering value to meet the needs of a target market in terms of goods and services; potentially including selection of a target audience; selection of certain attributes or themes to emph ...
, the familiar
80-20 rule
The Pareto principle states that for many outcomes, roughly 80% of consequences come from 20% of causes (the "vital few"). Other names for this principle are the 80/20 rule, the law of the vital few, or the principle of factor sparsity.
Manage ...
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 market
A commodity market is a market that trades in the primary economic sector rather than manufactured products, such as cocoa, fruit and sugar. Hard commodities are mined, such as gold and oil. Futures contracts are the oldest way of investin ...
s or in the
record industry
The music industry consists of the individuals and organizations that earn money by writing songs and musical compositions, creating and selling recorded music and sheet music, presenting concerts, as well as the organizations that aid, train, ...
, especially in
phonographic markets. The probability density function for logarithm of weekly record sales changes is highly
leptokurtic
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
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.
See also
*
Tail risk
Tail risk, sometimes called "fat tail risk," is the financial risk of an asset or portfolio of assets moving more than three standard deviations from its current price, above the risk of a normal distribution. Tail risks include low-probability ev ...
*
Black swan theory
The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight. The term is based on ...
*
Seven states of randomness
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 boo ...
*
Taleb distribution
In economics and finance, a Taleb distribution is the statistical profile of an investment which normally provides a payoff of small positive returns, while carrying a small but significant risk of catastrophic losses. The term was coined by jo ...
References
External links
Examples of Fat Tails in Financial Time Series
{{DEFAULTSORT:Fat Tail
Tails of probability distributions
Behavioral finance