Coherence and Uniqueness in Risk Capital Allocation: Linking Exposure Curves, Expected Shortfall, and Aumann–Shapley Values

This paper develops a unified framework for risk capital allocation, centered on Ex-pected Shortfall (ES). We establish the structural equivalence between Complementary Risk Exposure (CRE) and ES, showing that CRE is not an independent risk measure but a distributional representation of ES. Under continuous differentiability, ES ensures analytical uniqueness: capital allocation is uniquely determined via Euler decomposi-tion and strictly coincides with the Aumann–Shapley allocation. Incorporating Denault’s (2001) axioms further secures axiomatic uniqueness, demonstrating that ES-based allocation is stable both mathematically and institutionally. For non-smooth or atomic distributions, Rockafellar–Uryasev optimization and subgradient methods are applied, with portfolio-based CRE proposed as a canonical selection, preserving uniqueness in practice. The paper’s contribution is twofold: it is the first to integrate coherent risk measures with the dual uniqueness of capital allocation (analytical and axiomatic) within a sin-gle framework, and it positions the CRE–ES equivalence as the key nexus. This result provides both mathematical rigor and practical applicability, offering a robust foun-dation for theory, regulation, and practice in risk capital allocation.