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

TL;DR

This paper derives explicit analytical formulas for multivariate Value-at-Risk using Archimedean copulas, enabling more direct and transparent risk assessment in financial portfolios with dependence structures.

Contribution

It provides closed-form expressions for multivariate VaR under various Archimedean copulas, advancing beyond numerical methods for risk measurement.

Findings

Explicit formulas for Clayton, Frank, Gumbel, Joe, and Ali-Mikhail-Haq copulas.

Monte Carlo simulations validate the finite-sample performance of the VaR estimator.

Dependence structures significantly influence multivariate risk estimates.

Abstract

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.