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

ACTA WASAENSIA 17The survival or tail probabilities ¯F (x) =P (X > x) of a random variable X whosemaxima are described by one of the GEV distributions, is connected to G(x) throughthe relation:¯F (x) =− ln G(x), if ln G(x) > −1. (2.9)This implies the following tail probabilities for the random variables X:Type I: Medium-tailed ¯F (x) =exp(−x)Ix≥0 , (2.10)Type II: Fat-tailed ¯F (x) =x −α I x≥1 , (2.11)Type III: Thin-tailed ¯F (x) =(−x) α I −1≤x≤0 . (2.12)The labels medium-, fat-, and thin-tailed in (2.10) to (2.12) refer to the decay of ¯F (x).We see that random variables whose extremes may be described by Gumbel (typeI) or Fréchet (type II) distributions are characterized by exponentially respectivelyhyperbolically declining tails, whereas distributions with extremal behavior of type III(Weibull) have finite endpoints. Any distribution with limiting extremal behavior maythen be classified into one of the three types according to the asymptotical decay of itstails. Note that the tail index α coincides in the case of fat-tailed distributions (typeII) with the exponent of the hyperbolic decay, implying non-existence of any momentshigher than α for such distributions.A unifying representation of (2.10) to (2.12) is given by the survival or tail probabilityof the Generalized Pareto Distribution (GDP):¯F ξ (x) =(1+ξx) −1/ξ (2.13)where the sign of ξ classifies the distribution into type I (ξ → 0), type II (ξ > 0) andtype III (ξ < 0), and the tail index α is related to ξ by the identity α =1/|ξ| as in(2.8) above.Hill (1975) provides the following maximum likelihood estimator for ξ conditional onthe tail size:ˆξ = 1 kk{ln x (n−i+1) − ln x (n−k) } (2.14)i=1