Expected Shortfall dominates Value-at-Risk

Expected Shortfall is at least Value-at-Risk, at matching depth. Value-at-Risk asks for the least capital leaving at most $m$ states in strict loss; Expected Shortfall averages the loss across the worst $m+1$ states. The average of the worst $m+1$ losses is at least the smallest of them, which is exactly the capital Value-at-Risk demands.

The proof does not compute either quantity. It shows directly that the capital level $\mathrm{ES}(X,m)$ already leaves at most $m$ states in strict loss, so it belongs to the set over which Value-at-Risk takes its infimum, and the infimum is therefore no larger. That membership is where the work is: if $m+1$ or more states were still in strict loss at that level, any $m+1$ of them would form a group whose total loss strictly exceeds $(m+1)$ times $\mathrm{ES}(X,m)$ -- but $(m+1)\,\mathrm{ES}(X,m)$ is the worst total, which by definition no group of that size can exceed. The infimum is genuine rather than a junk value because the constraint set is bounded below: any admissible capital level leaves some state out of loss, which bounds it by the negative of the largest payoff.

This is the result that ties the development together. Value-at-Risk fails subadditivity and Expected Shortfall does not, but that alone would not single out Expected Shortfall -- the worst-case measure is coherent too, and far more conservative. What makes Expected Shortfall the replacement adopted in the Basel market-risk framework is that it is coherent and sits directly above Value-at-Risk at the same depth, so adopting it is a tightening rather than a change of scale.

As throughout CoherentRisk, the development counts states and refers to no probability measure, so $m$ is an integer depth rather than a confidence level, and the two measures are compared at the same integer depth rather than at a common tail probability.