A theory for combinations of risk measures

Abstract

We study combinations of risk measures under no restrictive assumption on the set of alternatives. The main result is the representation for resulting risk measures from the properties of both alternative functionals and combination functions. To that, we develop a representation for mixture of convex risk measures. As an application, we address the context of probability-based risk measurements for functionals on the set of distribution functions. We develop results related to this specific context. We also explore features of individual interest generated by our framework, such as the preservation of continuity properties, the representation of worst-case risk measures, stochastic dominance and elicitability.

Spectral Risk Measures and Uncertainty

“…We intend to address the above formulated questions by adapting a consistent framework to the class of spectral risk measures. In this sense, our paper contributes to the existent literature recently addressing this issue, see [Wang and Ziegel, 2018], [Jokhadze and Schmidt, 2018], [Righi, 2018b] and the references therein. Moreover, we intend to propose an alternative approach for measuring uncertainty itself.…”

“…Remark 3.7. According to the properties of ρ φ and R, and from Corollary 1 [Righi, 2018b] (for f (•) = R(−•)), we provide results regarding dual representation of the superposed robust risk measure R o ρ φ . Let ρ φ be a scenario-based SRM and R be a monetary risk measure.…”

“…where, m µ ∈ cl(M E µ φ ). For V the set of probability measures in (S, S), and δ R a penalty term; see [Righi, 2018b].…”

Minkowski gauges and deviation measures

“…Now, we analyze how some operations on a deviation measure are reflected on its corresponding acceptance set. For a comprehensive theory on combinations of monetary risk measures, see Righi (2020b).…”

Scenario-Based Risk Evaluation

“…The class of scenario-based risk measures is quite general. In addition to classic law-invariant risk measures, it also includes various forms of risk evaluation procedures such as the ones studied in Delbaen (2002), Cherny and Madan (2009), Adrian and Brunnermeier (2016), Kou and Peng (2016) and Righi (2018); see Sections 2 and 3 for details. For recent developments of risk measures, including various practical issues of statistical analysis, robustness, model uncertainty, and optimization, we refer to Fissler and Ziegel (2016), Cambou and Filipovic (2017), Krätschmer et al (2017), Du and Escanciano (2017), Embrechts et al (2018) and the references therein.…”