Some Results on Measures of Interaction between Paired Risks

Abstract

Co-risk measures and risk contribution measures have been introduced to evaluate the degree of interaction between paired risks in actuarial risk management. This paper attempts to study the ordering behavior of measures on interaction between paired risks. For various co-risk measures and risk contribution measures, we investigate how the marginal distributions and the dependence structure impact on the level of interaction between paired risks. Also, several numerical examples based on Monte Carlo simulation are presented to illustrate the main findings.

Systemic Risk: Conditional Distortion Risk Measures

“…'s and the dependence structure affect the interactions among paired risks under the CoVaR, CoES, ∆CoVaR, and ∆CoES measures. In this section, we shall establish some novel results for our CoD-risk measures and distortion risk contribution measures, which generalize the corresponding ones established in Fang and Li (2018).…”

“…Recently, Fang and Li (2018) studied how the marginal d.f. 's and the dependence structure affect the interactions among paired risks under the CoVaR, CoES, ∆CoVaR, and ∆CoES measures.…”

“…In the context of comparisons of these co-risk measures and risk contribution measures, an interesting paper by Sordo et al (2018) provides sufficient conditions to stochastically order two random vectors in terms of their CoVaR, CoES, ∆CoVaR, and ∆CoES, where the conditions are expressed using conventional stochastic orders for the marginals under some assumptions of positive dependence. Furthermore, Fang and Li (2018) investigate how the marginal distributions and the dependence structure affect the interactions among paired risks under the above co-risk measures and risk contribution measures. It is well known that the VaR and ES arise as two special cases within the rich class of distortion risk measures (Yaari, 1987;Denuit et al, 2005Denuit et al, , 2006Goovaerts et al, 2010;Föllmer and Schied, 2011).…”

“…The interactions between paired risks are also investigated. Existing results in Mainik and Schaanning (2014), Sordo et al (2018), and Fang and Li (2018) are generalized and extended.…”