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Robust measurement of economic tail risk

We prove that the only tail risk measure that satisfies a set of economic axioms proposed by Schmeidler (1989, Econometrica) and a statistical requirement called elicitability (i.e. there exists an objective function such that a reasonable estimator must be a solution of minimizing the expected objective function) is the median shortfall, which is the median of the tail loss distribution and is also the VaR at a high confidence level.