ABSTRACT

Contents 1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Coherency Axioms and the Shortcomings of VaR . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Objectivist Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Estimability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 The Diversification Principle Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Spectral Measures of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Estimators of Spectral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.8 Optimization of CRMs: Exploiting Convexity . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.9 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

This chapter presents a guided tour of the recent (sometimes very technical) literature on coherent risk measures (CRMs). Our purpose is to overview the theory of CRMs from the perspective of practical risk-management applications. We have tried to single out those results of the theory that help in understanding which CRMs can be considered as realistic candidate alternatives to value at risk (VaR) in the financial risk-management practice. This has also been the spirit of the author’s research line in recent years [1, 4-6] (see Acerbi [2] for a review).