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2015 | OriginalPaper | Buchkapitel

1. Introduction

verfasst von : Marat Ibragimov, Rustam Ibragimov, Johan Walden

Erschienen in: Heavy-Tailed Distributions and Robustness in Economics and Finance

Verlag: Springer International Publishing

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Abstract

The empirical and theoretical study of heavy-tailed distributions within economics and finance is by now a mature area of research, dating back more than 50 years. The first empirical study is usually attributed to Mandelbrot (1963), who noted that the changes of cotton prices seem to be well approximated by heavy-tailed so-called stable distributions. Loosely speaking, this means that rare events tend to happen much more often than they would if risk distributions had standard Gaussian (or other) thin tails. For example, the approximately 20 % drop of the stock market on the so-called Black Monday of October 19, 1987 would occur much less often than once in a billion years under standard assumptions of Gaussian distributions, and has been taken as evidence that stock market returns are heavy-tailed (see, for instance, the striking examples in Chap. 2 in Stock and Watson 2007 that illustrate inappropriateness of Gaussian distributions as models for financial returns based on their behavior during the Black Monday crisis).

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Fußnoten
1
In the terminology introduced by Rothschild and Stiglitz (1970), any other portfolio is risker than the uniformly diversified one in that its distribution is a mean preserving spread of that of the uniformly diversified portfolio. We also note that independence of risks considered is crucial for the result, and cannot be replaced by the weaker condition of uncorrelated risks, as shown in Brumelle (1974).
 
2
Throughout the book, the term “risk” is used as a synonym for the term “random variable,” if this does not lead to a confusion. So that, here, for instance, we mean, in particular, that the risk (r.v. or loss) \(\frac{1} {n}\sum _{i=1}^{n}Z_{i}\) of the portfolio of i.i.d. Cauchy risks (r.v.’s or losses) Z 1, Z 2, , Z n with equal weights has the same Cauchy distribution as does each of the r.v.’s Z i . 
 
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Metadaten
Titel
Introduction
verfasst von
Marat Ibragimov
Rustam Ibragimov
Johan Walden
Copyright-Jahr
2015
DOI
https://doi.org/10.1007/978-3-319-16877-7_1