Asymptotic subadditivity/superadditivity of Value‐at‐Risk under tail dependence
提出新方法研究多个风险在尾部相依下的VaR渐近次可加性与超可加性，定义边际区域和尾部凹序，给出充分条件，并基于金融数据实证分析。
Abstract This paper presents a new method for discussing the asymptotic subadditivity/superadditivity of Value‐at‐Risk (VaR) for multiple risks. We consider the asymptotic subadditivity and superadditivity properties of VaR for multiple risks whose copula admits a stable tail dependence function (STDF). For the purpose, a marginal region is defined by the marginal distributions of the multiple risks, and a stochastic order named tail concave order is presented for comparing individual tail risks. We prove that asymptotic subadditivity of VaR holds when individual risks are smaller than regularly varying (RV) random variables with index −1 under the tail concave order. We also provide sufficient conditions for VaR being asymptotically superadditive. For two multiple risks sharing the same copula function and satisfying the tail concave order, a comparison result on the asymptotic subadditivity/superadditivity of VaR is given. Asymptotic diversification ratios for RV and log regularly varying (LRV) margins with specific copula structures are obtained. Empirical analysis on financial data is provided for highlighting our results.