Tail mutual exclusivity and Tail-VaR lower bounds

Abstract

In this paper, we extend the concept of mutual exclusivity proposed by [Dhaene, J. & Denuit, M. (1999). The safest dependence structure among risks. Insurance: Mathematics and Economics 25, 11–21] to its tail counterpart and baptize this new dependency structure as tail mutual exclusivity. Probability levels are first specified for each component of the random vector. Under this dependency structure, at most one exceedance over the corresponding Value-at-Risks (VaRs) is possible, the other components being zero in such a case. No condition is imposed when all components stay below the VaRs. Several properties of this new negative dependence concept are derived. We show that this dependence structure gives rise to the smallest value of Tail-VaR (TVaR) of a sum of risks within a given Fréchet space, provided that the probability level of the TVaR is close enough to one.

Keywords:

• mutual exclusivity
• stop-loss transform
• tail convex order
• risk measures

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. Ka Chun Cheung acknowledges the support by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU701213P), and the CAE 2013 research grant from the Society of Actuaries. Any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA. Michel Denuit acknowledges the financial support from the contract ‘Projet d’Actions de Recherche Concertées’ No 12/17-045 of the ‘Communauté française de Belgique’, granted by the ‘Académie universitaire Louvain’. Jan Dhaene acknowledges the financial support of Onderzoeksfonds KU Leuven (GOA/13/002).

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. HKU701213P]; the Society of Actuaries [CAE 2013 Research Grant]; Académie universitaire Louvain; Onderzoeksfonds KU Leuven [GOA/13/002].