Distortion risk measures for sums of dependent losses

Abstract. We discuss two distinct approaches, for distorting risk measures of sums of dependent random variables, which preserve the property of coherence. The first, based on distorted expectations, operates on the survival function of the sum. The second, simultaneously applies the distortion on the survival function of the sum and the dependencestructure of risks, represented by copulas. Our goal is to propose risk measures that take into account the fluctuations of losses and possible correlations between risk components.Resume. Nous discutons deux approches distinctes, de distortion des mesures de risque de la somme de variables al´eatoires d´ependantes, qui conservent la propri´et´e de coh´erence. La premi`ere, bas´ee sur les esp´erances distordues, agit sur la fonction de survie de la somme. La seconde, applique des d´eformations simultan´ees sur la fonction de survie de la somme et sur la structure de d´ependance des risques, repr´esent´ee par une copule. Notre objectif est de proposer des mesures qui prennent en compte les fluctuations des pertes et des corr´elations ´eventuelles entre les composantes d’un risque multivari´e.Key words: Coherence; Dependence structure; Distortion function; Risk measure; Risk theory; insurance; Wang transform