Submodular Risk Measures

Battle Count: 62

The paper is primarily theoretical but has direct relevance to quantitative trading through its analysis of risk measures (VaR, ES, AES) used in portfolio risk management. The finding that ES is submodular while VaR is not has practical implications for risk aggregation. The empirical violation rates for VaR (10.29% in pair-based, 4.52% mean daily in sector-based) inform practitioners about when VaR-based risk aggregation may fail. The AES characterization (submodular only when it reduces to ES) is relevant for regulatory capital calculations. However, the paper does not directly propose trading strategies or signal generation methods.

The theoretical results require advanced mathematical tools: lattice theory, Choquet integration, implicit function theorem, Dini derivatives, and approximation on simple subspaces. The empirical implementation is moderate: computing rolling VaR/ES from order statistics, constructing X∧Y and X∨Y at the loss-series level, and testing the submodularity inequality. The AES two-level specification is straightforward to implement. The main complexity lies in the theoretical proofs rather than computational implementation.

The theoretical results are fully self-contained with complete proofs. The empirical section uses publicly available data (Yahoo Finance, Stooq) with clearly specified parameters (rolling windows of 250 and 500 days, confidence levels, AES parameters). However, no code repository is provided. The stock selection methodology (S&P 500 constituents by sector) is documented in Table 4. The submodularity test procedure is precisely defined with tolerance epsilon=10^-8.