The paper discusses the coherence of Expected Shortfall (ES) as a risk measure compared to Value-at-Risk (VaR). ES is proposed as a remedy for VaR's shortcomings, particularly its lack of sub-additivity. The authors compare various definitions of ES and other related risk measures, such as Tail Conditional Expectation (TCE) and Worst Conditional Expectation (WCE). They highlight that while some definitions of ES may not be coherent, the specific definition used in the paper ensures coherence regardless of the underlying distribution. ES is shown to be a coherent risk measure that is also easy to compute and estimate. The paper also provides a mathematical characterization of ES, showing it can be represented as an integral of quantiles, a limit in a tail law of large numbers, a minimum of a certain functional, and a maximum of WCEs over different probability spaces. The authors emphasize that ES is a robust and practical risk measure, and that it can be effectively estimated even when traditional VaR estimators fail. The paper concludes that ES is a superior risk measure compared to other alternatives, as it is both coherent and computationally feasible.The paper discusses the coherence of Expected Shortfall (ES) as a risk measure compared to Value-at-Risk (VaR). ES is proposed as a remedy for VaR's shortcomings, particularly its lack of sub-additivity. The authors compare various definitions of ES and other related risk measures, such as Tail Conditional Expectation (TCE) and Worst Conditional Expectation (WCE). They highlight that while some definitions of ES may not be coherent, the specific definition used in the paper ensures coherence regardless of the underlying distribution. ES is shown to be a coherent risk measure that is also easy to compute and estimate. The paper also provides a mathematical characterization of ES, showing it can be represented as an integral of quantiles, a limit in a tail law of large numbers, a minimum of a certain functional, and a maximum of WCEs over different probability spaces. The authors emphasize that ES is a robust and practical risk measure, and that it can be effectively estimated even when traditional VaR estimators fail. The paper concludes that ES is a superior risk measure compared to other alternatives, as it is both coherent and computationally feasible.