Some proposals about multivariate risk measurement
In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied. In our work we propose a characterization of some particular classes of multivariate and bivariate risk measures. Given two random variables we can define an univariate integral stochastic ordering by considering a set of functions that, through their peculiar properties, originate different stochastic orderings. These stochastic order relations of integral form may be extended to cover also the case of random vectors. It is, in fact, proposed a kind of stop-loss premium, and then a stop-loss order in the multivariate setting and some equivalent conditions. We propose an axiomatic approach based on a minimal set of properties which characterizes an insurance premium principle. In the univariate case we know that Conditional Value at Risk can be represented through distortion risk measures and a distortion risk measure can be viewed as a combination of CVaRs, we propose a generalization of this result in a multivariate framework. In the bivariate case we want to compare the concept of risk measure to that one of concordance measure when the marginals are given.
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- Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
- Wu, Xianyi & Zhou, Xian, 2006. "A new characterization of distortion premiums via countable additivity for comonotonic risks," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 324-334, April.
- Denuit, Michel & Lefevre, Claude & Mesfioui, M'hamed, 1999. "A class of bivariate stochastic orderings, with applications in actuarial sciences," Insurance: Mathematics and Economics, Elsevier, vol. 24(1-2), pages 31-50, March.
- Scarsini, Marco, 1985. "Stochastic dominance with pair-wise risk aversion," Journal of Mathematical Economics, Elsevier, vol. 14(2), pages 187-201, April.
- Muller, Alfred, 1997. "Stop-loss order for portfolios of dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 21(3), pages 219-223, December.
- Müller, Alfred & Scarsini, Marco, 2000. "Some Remarks on the Supermodular Order," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 107-119, April.
- Marta Cardin & Graziella Pacelli, 2006. "On the characterization of convex premium principles," Working Papers 142, Department of Applied Mathematics, Università Ca' Foscari Venezia.
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