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Motivated by the Basel 3 regulations, recent studies have considered jointforecasts of Value-at-Risk and Expected Shortfall. A large family of scoringfunctions can be used to evaluate forecast performance in this context.However, little intuitive or empirical guidance is currently available, whichrenders the choice of scoring function awkward in practice. We thereforedevelop graphical checks (Murphy diagrams) of whether one forecast methoddominates another under a relevant class of scoring functions, and propose anassociated hypothesis test. We illustrate these tools with simulation examplesand an empirical analysis of S&P 500 and DAX returns.

Murphy Diagrams: Forecast Evaluation of Expected Shortfall