Explicit Finite-Sum Tail Risk Measures for Hierarchical Market--Credit Copula Aggregation

Hierarchical copula models are widely used for aggregating market and credit risks across multi-level portfolio structures. Existing approaches often rely on Monte Carlo simulation, recursive sample reordering, or high-dimensional numerical procedures. This paper develops an analytical framework for hierarchical copula aggregation based on explicit finite-sum representations of aggregate loss distributions and tail-risk measures for two-level market-credit aggregation with Gaussian and mirrored Clayton copulas under empirical leaf-level marginals. We derive deterministic formulas for aggregate distribution functions, quantiles, expected tail losses, and market-credit tail contributions, together with a compatibility condition linking local hierarchical copulas and global leaf-level dependence structures. Estimation-error expressions for quantiles and expected tail losses are also obtained in finite-sum form. The paper further analyzes the qualitative asymptotic behavior of hierarchical aggregation under increasing tree depth. Gaussian hierarchies are argued to remain diversification-dominated, whereas mirrored Clayton hierarchies propagate upper-tail dependence recursively and may amplify extreme losses across aggregation levels. The resulting formulas provide a deterministic alternative to simulation-based aggregation methods and clarify how tail dependence propagates through hierarchical market-credit portfolios, with direct relevance for portfolio risk measurement and capital assessment.