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.

中文翻译：

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尾部依赖下风险价值的渐近次可加性/超可加性

本文提出了一种讨论多重风险的风险价值（VaR）渐近次可加性/超可加性的新方法。我们考虑多重风险的 VaR 的渐近次可加性和超可加性，其连接函数允许稳定的尾部依赖函数（STDF）。为此，通过多重风险的边缘分布定义边缘区域，并提出一种称为尾凹阶的随机顺序来比较各个尾部风险。我们证明，当个体风险小于尾凹阶下指数为 -1 的正则变化 (RV) 随机变量时，VaR 的渐近次可加性成立。我们还为 VaR 渐近超加性提供了充分条件。对于共享相同 copula 函数并满足尾凹顺序的两个多重风险，给出了VaR渐近次可加性/超可加性的比较结果。获得具有特定 copula 结构的 RV 和对数规则变化 (LRV) 边缘的渐近多样化比率。提供对财务数据的实证分析是为了强调我们的结果。