Comparing Higher-Order Co-Moment Functionals with Conditional Tail Risk Measures

This paper compares higher-order co-moment functionals (co-skewness and co-kurtosis) with conditional tail-risk measures, namely Co-Expected Shortfall (CoES) and Co-Value at Risk (CoVaR), within a unified coherence-based framework. On the theoretical side, we present explicit counterexamples showing that co-skewness violates subadditivity and co-kurtosis violates monotonicity, confirming that higher-order co-moments are descriptive diagnostics rather than admissible risk measures. By contrast, CoES inherits the coherence of Expected Shortfall in a conditional joint-tail setting, while CoVaR remains non-coherent and captures tail events only at a quantile level without accounting for loss severity. Empirically, we adopt a predictive, single-index, lagged-conditioning design to examine temporal conditional tail dependence in S&P 500 daily losses from 2007 to 2023. This framework measures the persistence and amplification of market-wide tail risk rather than cross-sectional contagion across institutions. Conditional tail-risk estimates are reported only when the joint tail is sufficiently populated to ensure reliable identification. When these conditions are met, CoES delivers stable and economically interpretable signals of conditional tail fragility, with pronounced elevations during prolonged stress episodes such as the Lehman collapse and the COVID-19 crisis. Robustness analysis at a less extreme tail level confirms that the qualitative ordering of stress regimes is preserved. CoVaR captures sharp conditional stress episodes but exhibits greater sensitivity to sample size, while higher-order co-moments, both raw and standardized, remain sign-unstable and weakly informative. Overall, the results support a clear hierarchy: co-moments as descriptive supplements, CoVaR as a scenario-based stress indicator, and CoES as the coherent benchmark for conditional tail-risk measurement.