Explicit Expressions for Multidimensional Value-at-Risk under Archimedean Copulas

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.

💡 Research Summary

This paper addresses the longstanding challenge of computing multivariate Value‑at‑Risk (VaR) in a tractable, analytical manner by exploiting the generator representation of Archimedean copulas. After a concise review of copula theory and the definition of multivariate VaR as the conditional expectation on the α‑level set of the joint distribution, the authors invoke Hürlimann’s (2017) theorem, which expresses each marginal VaR component as an integral involving the copula generator ϕ and its derivative ϕ′. Specifically, for a d‑dimensional Archimedean copula the marginal VaR satisfies

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