On the Interplay between Distortion, Mean Value and Haezendonck-Goovaerts Risk Measures

On the Haezendonck–Goovaerts risk measure for extreme risks

“…Based on a recent conversation with Bellini and Rosazza Gianin during the 15th International Congress on Insurance: Mathematics and Economics in Trieste, we think that it is more proper to call it the Haezendonck-Goovaerts risk measure in order to acknowledge the contribution of both authors in their seminal paper. This risk measure has recently been studied by Bellini and Rosazza Gianin (2008a, 2008b, Nam et al (2011), Krätschmer and Zähle (2011), Goovaerts et al (2012) and Ahn and Shyamalkumar (2012). We have followed the style of Rosazza Gianin (2008a, 2008b) to define this risk measure.…”

Haezendonck–Goovaerts risk measures and Orlicz quantiles

“…From a mathematical point of view, Haezendonck-Goovaerts risk measures are interesting because they are the simplest example of coherent risk measures 1 that are not comonotonically additive; equivalently, they cannot in general be expressed as distortion risk measures (see Goovaerts et al, 2010, for further interesting discussions on the relationships between Haezendonck-Goovaerts and distortion risk measures). Moreover, as we will discuss in details in the following, Haezendonck-Goovaerts risk measures are naturally defined on Orlicz spaces, that are becoming more and more popular in the field of risk measures because of a well established duality theory that generalize the standard L p theory (see for example Cheridito and Li, 2009, and the references therein).…”

“…Several expressions have been introduced in the literature (see Goovaerts et al, 2010, Ahn and Shyamalkumar, 2011b, Nam et al, 2011. In general, the minimizer x * cannot be characterized by a single equation but only by a couple of inequalities, that are obtained by computing the right and left derivatives with the aid of the Gateaux derivative of the norm in the Orlicz heart (see Kosmol and Müller-Wichards, 2011).…”

Beyond Value‐at‐Risk: GlueVaR Distortion Risk Measures

“…Our reason for defining these GlueVaR risk measures is a response to the concerns expressed by risk managers regarding the choice of risk measures in the case of regulatory capital requirements. However, an axiomatic approach to define or represent risk measures is more frequent in the literature (3,28,29,21,47,12,24,32,36) . Artzner et al (3) established the following set of axioms that a risk measure should satisfy: positive homogeneity, translation invariance, monotonicity and subadditivity.…”