Coherent estimation of risk measures

Battle Count: 86

Highly relevant for quantitative risk management in trading desks and regulatory compliance. The paper directly addresses how VaR and ES estimators used in practice may violate coherence properties, which affects capital allocation, position sizing, and portfolio construction. The finding that empirical VaR violates subadditivity with >20% probability even under Gaussian assumptions is critical for trading risk management. The CRE framework provides a principled basis for selecting estimation methods that maintain diversification benefits. However, the paper is primarily theoretical and does not directly propose trading strategies.

The theoretical framework is mathematically sophisticated, requiring knowledge of convex analysis, order statistics, and axiomatic risk theory. However, the practical estimators (empirical ES, sample conditional mean ES, Type 6 quantile-based ES) are straightforward to implement as weighted combinations of sorted sample values. The main complexity lies in: (1) understanding which estimators are coherent vs. non-coherent, (2) selecting appropriate weight structures, (3) implementing the supremum over L-estimators for general CREs, and (4) extending to non-i.i.d. settings. The numerical validation requires substantial Monte Carlo simulation.

The paper provides complete mathematical proofs, explicit formulas for all estimators, specific simulation parameters (n=250, alpha=1%/2.5%, N=10000 MC runs, K=10^7 replications), and uses publicly available Fama-French data. However, no code repository or software implementation is provided. The theoretical results are fully specified but practical implementation requires significant effort. Distributional parameters for NIG and Student's t are explicitly given.