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 dependence structure 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.Résumé. Nous discutons deux approches distinctes, de distortion des mesures de risque de la somme de variables aléatoires dépendantes, qui conservent la propriété de cohérence. La première, basée sur les espérances distordues, agit sur la fonction de survie de la somme. La seconde, applique des déformations simultanées sur la fonction de survie de la somme et sur la structure de dépendance des risques, représentée par une copule. Notre objectif est de proposer des mesures qui prennent en compte les fluctuations des pertes et des corrélationséventuelles entre les composantes d'un risque multivarié.MSC 2010: 60B05, 62H20, 91B30.

Involving copula functions in Conditional Tail Expectation

“…Again, CCTE satisfies all the desirable properties of a coherent risk measure (Artzner et al, 1999). The notion of copula in risk measure filed has recently been considered by several authors, see for instance Embrechts et al (2003a), Di Clemente andRomano (2004), Dalla Valle (2009), Brahimi et al (2010) and the references therein.…”

“…In actuarial science literature a several risk measures have been proposed, namely: the Value-at-Risk (VaR), the expected shortfall or the conditional tail expectation (CTE), the distorted risk measures (DRM), and recently the copula distorted risk measure (CDRM) as risk measure which takes into account the fluctuations dependence between random variables (rv). See Brahimi et al (2010).…”

Association measures and copula estimation

“…An exhaustive list of copula applications can be found in, for instance, Balakrishnan and Lai (2009), pages 55-58, with full details. The notion of copula is used by Brahimi et al (2010) to analyze the distortion risk measures of the sum of two or more insurance losses, where the dependence structure is a very significant factor. A copula is a mean of linking a multivariate distribution function (df) with its margins.…”