Properties of Distortion Risk Measures

Abstract

The current literature does not reach a consensus on which risk measures should be used in practice. Our objective is to give at least a partial solution to this problem. We study properties that a risk measure must satisfy to avoid inadequate portfolio selections. The properties that we propose for risk measures can help avoid the problems observed with popular measures, like Value at Risk (VaR α ) or Conditional VaR α (CVaR α ). This leads to the definition of two new families: complete and adapted risk measures. Our focus is on risk measures generated by distortion functions. Two new properties are put forward for these: completeness, ensuring that the distortion risk measure uses all the information of the loss distribution, and adaptability, forcing the measure to use this information adequately.

Beyond Value‐at‐Risk: GlueVaR Distortion Risk Measures

“…There are two key elements to define a distortion risk measure: first, the associated distortion function; and, second, the concept of the Choquet (14) Integral. A detailed literature review of distortion risk measures is available in Denuit et al (20) and Balbás et al (6) . The distortion function, Choquet Integral and the distortion risk measure can be defined as follows:…”

“…In some situations, coherence of risk measures is a requirement (see, for instance, Cox (15) ) but, nonetheless, some criticisms can be found, for example, in Dhaene et al (23) . Additional properties for distortion risk measures are provided by Jiang (41) and Balbás et al (6) . As shown in the previous section, GlueVaR risk measures may be interpreted as a linear combination of VaR and TVaR risk measures.…”

The connection between distortion risk measures and ordered weighted averaging operators

“…Concavity of the distortion function is the key element to define risk measures that belong to both groups (Wang and Dhaene, 1998). Suggestions on new desirable properties for distortion risk measures are proposed in Balbas et al (2009), while generalizations of this kind of risk measures can be found, among others, in Hürlimann (2006) and Wu and Zhou (2006).…”

Second-Order Asymptotics of the Risk Concentration of a Portfolio with Deflated Risks

“…The distortion risk measure has several useful properties such as positive homogeneity, translation invariance, additivity for comonotonic risks, and monotonicity. For more details, see Denuit et al [13], Dhaene et al [20], and Balbás et al [21]. Several popular risk measures belong to the family of distortion risk measures.…”