A tale of two allocations: Risk capital contributions versus risk contributions in the tail

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While primarily focused on actuarial science and insurance capital allocation, the concepts of tail risk decomposition, Expected Shortfall, and comonotonicity are highly relevant to quantitative risk management in trading desks, particularly for portfolio stress testing and capital efficiency analysis.

Implementing the theoretical framework requires advanced knowledge of stochastic calculus, extreme value theory, and numerical methods for saddle-point approximations. It is not a simple plug-and-play model but a rigorous analytical tool.

The paper is purely theoretical with complete mathematical proofs provided in the appendix. All derivations are self-contained and rely on standard statistical and actuarial definitions.