BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

16 ACTA WASAENSIA2.4 Heavy TailsThe kurtosis κ of a random variable X is a measure of its dispersion around the twovalues μ ± σ, whereμ and σ stand for the expected value and standard deviation of X,respectively. 9 This implies that κ grows with probability mass both in the center andthe tails, and declines with probability mass in the shoulders. For risk-mangagementpurposes, however, it is desirable to have a measure of fat-tailedness only.Extreme value theory 10 provides such a measure through its classification of the limitingdistributions of sample extremes of iid random variables with continuous distributions.Denoting with M n =max{x 1 ,x 2 ,...,x n } the maximum of n sample observations ofthe iid random variables X 1 ,X 2 , ..., X n , it has been shown by Fisher & Tippett (1928),that there exist only three classes of non-degenerate limiting distributions for suitablyshifted and rescaled sample maxima M n in the limit n →∞, called Generalized ExtremeValue (GEV) distributions:1. Gumbel (GEV Type I): G I (x) =exp{−e −x }, x ∈ R, (2.5)2. Fréchet (GEV Type II): G II,α (x) =exp{−x −α }I x>0 , (2.6)3. Weibull (GEV Type II): G III,α (x) =exp{−(−x) α }I x≤0 + I x>0 . (2.7)where I x>0 and I x≤0 denote the corresponding indicator functions and α is a positiveshape parameter often denoted as Tail Index for reasons that will become apparentbelow. Their representation may be unified within the so called von Mises parametrizationasG ξ (x) =exp{−(1 + ξx) −1/ξ }, (2.8)where the sign of the shape parameter ξ determines the type of the limiting distribution:ξ > 0forFréchet (II), ξ < 0forWeibull(III)andξ → 0 for Gumbel (I). ξ is relatedto α by ξ =1/α in the type II (Fréchet) case and ξ = −1/α inthetypeIII(Weibull)case. 119 see Moors (1988).10 Recent expositions of extreme value theory include Adler, Feldman & Taqqu (1998); Embrechts,Klüppelberg & Mikosch (1997) and Reiss & Thomas (1997).11 Some studies denote the parameter ξ rather than α as tail index. We shall use this term forthe parameter α as it has the more intuitive interpretation as the highest defined moment of X i indistributions with infinite support (see below).