Fat Tails and Limitations of Normal Distirbutions

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Fat Tails and Limitations of Normal Distirbutions

14 June 2007 4 Comments

Normal distributions (a bell curve) of asset returns is a key assumption made by many financial models, including the capital asset pricing model(CAPM) and the Black-Scholes option pricing model(BSM). However, actual asset returns may not be so normal.

Normal Distributions Overestimate the Improbability of Unlikely Market Events
Using a normal distribution, events that diverge from the mean by five or more standard deviations, known as a five-sigma event, are very rare and ten-sigma events are nearly impossible. For example, the 1987 market plunge represents a change equalling 22 standard deviations. The odds of such a 22 standard deviation event occurring are 10^50. However, events deemed nearly impossible by models assuming normal asset return distributions are possible in the financial markets and do occur. In fact, there have been multiple fluctuations greater than five standard deviations. Thus, dramatic market swings do occur in a greater frequency than is possible assuming normal distributions suggesting distributions are not normal.

Fat Tail Definition
Fat tails are statistical irregularities, in which very low and high values are more frequent than a normal distribution predicts. In a normal distribution, the tails to the extreme left and extreme right of the mean become smaller, ultimately reaching zero occurrences. However, some real life statistical series demonstrate occurrences of low and high values that are greater than theoretically expected by a normal distribution. These irregular occurrences or extreme events are described as fat tails.

Possible Reasons Why Fat Tails Exist in Financial Markets
The randomness associated with normal distributions may not completely reflect the financial markets. The central reasons why fat tails exist is a result of interdependence during market extremes. People’s decisions are not always fully independent or logical. At extreme market highs, investors become irrationally exuberant. At extreme lows, investors become fearfull and less risk tolerant. Because of interdependence and influence of aspects of behavioral finance, people buy at illogical highs and sell and ludicrous lows. Illogical and non random events push markets to extremes more frequently than models assuming complete randomness and normal distributions would predict.

Fat tails are an important concept for modeling returns and estimating risk. Fat tails reinforce the idea that past results do not guarentee future results. The risk of extreme price changes are often higher than they appear. Many investors do not take into account these unsuspected risks.