Risk measures beyond quantiles

Résumé

The use of quantiles forms the basis of the overwhelming majority of current risk management procedures. Yet, there exist alternative instruments of risk protection that are not (unlike quantiles) based solely on the frequency of tail observations and instead take their severity into account, while adhering to axiomatic requirements. These alternative risk measures have seen increasing interest in the past decade. The current state of development of risk measures beyond quantiles is discussed with a particular focus on three prominent classes: (i) Expected Shortfall (ES) and extremiles, part of the class of spectral and distortion risk measures, (ii) expectiles, which constitute a particular case of generalized M-quantiles, and (iii) systemic risk measures including Marginal Expected Shortfall (MES). A structured overview of their strengths and weaknesses with respect to axiomatic theory, estimation properties, and ease-of-use by risk practitioners will be given. In addition, challenges arising in the asymptotics and mathematical developments will be discussed and the use of each of the ES, extremile, expectile and MES risk measures will be illustrated with real data applications to storm losses in China, tornado losses in the United States, and financial returns series.