statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

and

decision theory Decision theory or the theory of rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, ratio ...

, kurtosis risk is the

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 environ ...

that results when a

statistical model A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repre ...

assumes the

normal distribution In probability theory and 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 ...

, but is applied to

observation Observation in the natural sciences is an act or instance of noticing or perceiving and the acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the percep ...

s that have a tendency to occasionally be much farther (in terms of number of standard deviations) from the average than is expected for a normal distribution.

Overview

Kurtosis In probability theory and statistics, kurtosis (from , ''kyrtos'' or ''kurtos'', meaning "curved, arching") refers to the degree of “tailedness” in the probability distribution of a real-valued random variable. Similar to skewness, kurtos ...

risk applies to any

kurtosis In probability theory and statistics, kurtosis (from , ''kyrtos'' or ''kurtos'', meaning "curved, arching") refers to the degree of “tailedness” in the probability distribution of a real-valued random variable. Similar to skewness, kurtos ...

-related quantitative model that assumes the

for certain of its

independent variable A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...

s when the latter may in fact have kurtosis much greater than does the normal distribution. Kurtosis risk is commonly referred to as " fat tail" risk. The "fat tail" metaphor explicitly describes the situation of having more observations at either extreme than the tails of the normal distribution would suggest; therefore, the tails are "fatter". Ignoring kurtosis risk will cause any model to understate the risk of variables with high kurtosis. For instance,

Long-Term Capital Management Long-Term Capital Management L.P. (LTCM) was a highly leveraged hedge fund. In 1998, it received a $3.6 billion bailout from a group of 14 banks, in a deal brokered and put together by the Federal Reserve Bank of New York. LTCM was founded in ...

, a

hedge fund A hedge fund is a Pooling (resource management), pooled investment fund that holds Market liquidity, liquid assets and that makes use of complex trader (finance), trading and risk management techniques to aim to improve investment performance and ...

cofounded by

Myron Scholes Myron Samuel Scholes ( ; born July 1, 1941) is a Canadian– American financial economist. Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel Laureate in Economic Sciences, and co-ori ...

, ignored kurtosis risk to its detriment. After four successful years, this hedge fund had to be bailed out by major investment banks in the late 1990s because it understated the kurtosis of many financial securities underlying the fund's own trading positions.

Research by Mandelbrot

Benoit 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 ...

, a French mathematician, extensively researched this issue. He felt that the extensive reliance on the normal distribution for much of the body of modern finance and

investment theory Investment is traditionally defined as the "commitment of resources into something expected to gain value over time". If an investment involves money, then it can be defined as a "commitment of money to receive more money later". From a broade ...

is a serious flaw of any related models including the Black–Scholes option model developed by Myron Scholes and

Fischer Black Fischer Sheffey Black (January 11, 1938 – August 30, 1995) was an American economist, best known as one of the authors of the Black–Scholes equation. Working variously at the University of Chicago, the Massachusetts Institute of Technology, ...

, and the

capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a Diversification (finance), well-diversified Portfolio (f ...

developed by William F. Sharpe. Mandelbrot explained his views and alternative finance theory in his book: ''The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward'' published on August 3, 2004.

Notes

References

*{{cite book, last2=Hudson, first2=Richard L., last1=Mandelbrot, first1=Benoit, authorlink1=Benoit Mandelbrot, title=The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward, url=https://archive.org/details/misbehaviorofmar00beno, url-access=registration, year=2004, publisher=

Basic Books Basic Books is a book publisher founded in 1950 and located in New York City, now an imprint of Hachette Book Group. It publishes books in the fields of psychology, philosophy, economics, science, politics, sociology, current affairs, and his ...

, location=New York, isbn=0-465-04355-0 *Premaratne, G., Bera, A. K. (2000). Modeling Asymmetry and Excess Kurtosis in Stock Return Data. Office of Research Working Paper Number 00-0123, University of Illinois Normal distribution Investment Risk analysis Mathematical finance

Overview

Research by Mandelbrot

See also

Notes

References