On the Excess Distribution of Sums of Random Variables in Bivariate EV Models

Abstract

Let (U,V ) be a random vector following a bivariate extreme value distribution (EVD) with reverse exponential margins. It is known that the excess distribution F_c(t) = P(U+V >ct | U+V > c) of U+V converges to F(t)= t² as the threshold c increases if U,V are independent, and to F(t) = t, t ∈ [0, 1], elsewhere. We investigate the limit of the excess distribution of aU + bV in case of an EVD with arbitrary margins and with arbitrary scale parameters a,b > 0. It turns out that the limiting excess df may have a different behavior. For Fr´echet margins, independence of U,V does not affect the limit excess distribution, whereas for Gumbel and reverse Weibull margins it does. Unless for Gumbel margins, the limit excess distribution is independent of a,b. Interpreting a,b as weights and U,V  as risks, aU + bV can be viewed as a (short) linear portfolio. The fact that the limiting excess distribution of aU + bV does not depend on a,b, unless for Gumbel margins, implies that risk measures such as the expected shortfall E (aU + bV | aU + bV < c) might fail for multivariate extreme value models.