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Multivariate extremal dependence and risk measures Hua, Lei


Overlooking non-Gaussian and tail dependence phenomena has emerged as an important reason of underestimating aggregate financial or insurance risks. For modeling the dependence structures between non-Gaussian random variables, the concept of copula plays an important role and provides practitioners with promising quantitative tools. In order to study copula families that have different tail patterns and tail asymmetry than multivariate Gaussian and t copulas, we introduce the concepts of tail order and tail order functions. These provide a unified way to study three types of dependence in the tails: tail dependence, intermediate tail dependence and tail orthant independence. Some fundamental properties of tail order and tail order functions are obtained. For multivariate Archimedean copulas, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed by the Laplace transform of the random variable. Quantitative risk measurements pay more attention on large losses. A good statistical approach for the whole data does not guarantee a good way for risk assessments. We use tail comonotonicity as a conservative dependence structure for modeling multivariate dependent losses. By this way, we do not lose too much accuracy but gain reasonable conservative risk measures, especially when we consider high-risk scenarios. We have conducted a thorough investigation on the properties and constructions of tail comonotonicity, and found interesting properties such as asymptotic additivity properties of risk measures. Sufficient conditions have also been obtained to justify the conservativity of tail comonotonicity. For large losses, tail behavior of loss distributions is more critical than the whole distributions. Asymptotic study assuming that each marginal risk goes to infinity is more mathematically tractable. However, the asymptotic study that leads to a first order approximation is only a crude way and may not be sufficient. To this end, we study the second order conditions for risk measures of sub-extremal multiple risks. Some relationships between Value at Risk and Conditional Tail Expectation have been obtained under the condition of Second Order Regular Variation. We also find that the second order parameter determines whether a higher order approximation is necessary.

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