Risk Measures at Risk- Are we missing the point? Discussions around sub-additivity and distortion.

Summary To measure the major risks experienced by financial institutions, for instance Market, Credit and Operational, regarding the risk measures, the distributions used to model them and the level of confidence, the regulation either offers a limited choice or demands the implementation of a particular approach. In this paper, we review, highlight and illustrate the paradoxes and issues observed when implementing an approach over another, the inconsistencies between the methodologies suggested and the problems related to their interpretation. Starting with a discussion on the intrinsic properties of two classical risk measures: the Value-at-Risk and the Expected Shortfall, at each step we illustrate our proposal through an example based on real data using estimates of the risk measures themselves. \footnote{In this paper, we discuss the theoretical foundations of the risk measures as well as their estimates, applying them to real data for risk management purpose. Our objective is not a statistical discussion of the properties of the estimates used, for that we refer to a companion paper: (Guegan2016). Thus, this exercise provides practitioners and supervisors with some recommendations to assess, manage and control risks in a financial institution relying on alternative approaches, for instance, spectral, spectrum, distortion and Spatial risk measures.