文章基本信息

标题：On quantile based co-risk measures and their estimation
其他标题：On quantile based co-risk measures and their estimation
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作者：Sebastian Fuchs ; Wolfgang Trutschnig
期刊名称：Dependence Modeling
电子版ISSN：2300-2298
出版年度：2020
卷号：8
期号：1
页码：396-416
DOI：10.1515/demo-2020-0021
出版社：Walter de Gruyter GmbH
摘要：Conditional Value-at-Risk (CoVaR) is defined as the Value-at-Risk of a certain risk given that the related risk equals a given threshold (CoVaR = ) or is smaller/larger than a given threshold (CoVaR /CoVaR ≥ ). We extend the notion of Conditional Value-at-Risk to quantile based co-risk measures that are weighted mixtures of CoVaR at different levels and hence involve the stochastic dependence that occurs among the risks and that is captured by copulas. We show that every quantile based co-risk measure is a quantile based risk measure and hence fulfills all related properties. We further discuss continuity results of quantile based co-risk measures from which consistent estimators for CoVaR and CoVaR ≥ based risk measures immediately follow when plugging in empirical copulas. Although estimating co-risk measures based on CoVaR = is a nontrivial endeavour since conditioning on events with zero probability is necessary we show that working with so-called empirical checkerboard copulas allows to construct strongly consistent estimators for CoVaR = and related co-risk measures under very mild regularity conditions. A small simulation study illustrates the performance of the obtained estimators for special classes of copulas.
关键词：Co-Value-at-Risk ; co-risk measure ; checkerboard copula ; weak convergence ; estimation ; 62H05 ; 60E05 ; 91G70