This paper studies multivariate Value-at-Risk (VaR) for financial portfolios with a focus on modeling dependence structures through Archimedean copulas. Using the generator representation of Archimedean copulas, we derive explicit analytical expressions for the marginal lower-tail multivariate VaR in arbitrary dimensions. Closed-form formulas are obtained for several commonly used copula families, including Clayton, Frank, Gumbel-Hougaard, Joe and Ali--Mikhail--Haq copulas, allowing a direct assessment of the impact of dependence on multivariate risk. These results complement existing approaches, which largely rely on numerical or simulation-based methods, by providing tractable alternatives for theoretical and applied risk analysis. Monte Carlo simulations are conducted to evaluate the finite-sample performance of the proposed VaR estimator and to illustrate the role of different dependence structures. The proposed analytical setting offers transparent tools for multivariate risk measurement and systemic risk assessment.

翻译：本文研究金融投资组合的多变量风险价值（VaR），重点通过阿基米德Copula对相依结构进行建模。利用阿基米德Copula的生成元表示，我们推导出任意维度下边缘下尾多元VaR的显式解析表达式。针对包括Clayton、Frank、Gumbel-Hougaard、Joe和Ali--Mikhail--Haq在内的常用Copula族获得了闭式公式，从而能够直接评估相依性对多元风险的影响。这些结果为理论和应用风险分析提供了可处理的替代方法，对当前主要依赖数值或模拟方法的研究形成了重要补充。通过蒙特卡洛模拟评估了所提VaR估计量的有限样本性能，并阐明了不同相依结构的作用机制。所建立的解析框架为多元风险度量和系统性风险评估提供了透明化的分析工具。