  The robustness of risk measures to changes in underlying loss distributions
(distributional uncertainty) is of crucial importance in making well-informed
decisions. In this paper, we quantify, for the class of distortion risk
measures with an absolutely continuous distortion function, its robustness to
distributional uncertainty by deriving its largest (smallest) value when the
underlying loss distribution has a known mean and variance and, furthermore,
lies within a ball - specified through the Wasserstein distance - around a
reference distribution. We employ the technique of isotonic projections to
provide for these distortion risk measures a complete characterisation of sharp
bounds on their value, and we obtain quasi-explicit bounds in the case of
Value-at-Risk and Range-Value-at-Risk. We extend our results to account for
uncertainty in the first two moments and provide applications to portfolio
optimisation and to model risk assessment.
