Some new properties of risk measures

Risk measures are more and more important in financial management. The risk measure most widely adopted in nowadays practice is VaR (Value at Risk). However, within specialized literature some doubts about some properties of VaR originated a large interest in axiomatic definition of risk measures. Coherent risk measures were the first class of risk measures to be proposed in this context and they are also the most well known within the specialized literature. Coherent risk measures are characterized by the following four axioms: monotonicity, positive homogeneity, translation invariance and subadditivity. Recently, some new axioms have been proposed to emphasize particular features of the financial risks. In this paper we are mainly interested in conservatism, i.e. in the property that requires that a risk measure should depend only on the losses, and not also on the gains of an investment. We consider two classes of conservative risk measures: the robust risk measures and the conservative coherent risk measures. While the robust risk measures are based on some characteristic axioms quite different from those of coherent risk measures, conservative coherent risk measures take into account conservatism, trying to maintain as much as possible the properties of coherent risk measures. We give also the financial interpretations of all the axioms considered and we discuss the specific relations between the robust risk measures and the Sugeno integral.