Shortfall Risk Models When Information on Loss Function Is Incomplete

The utility-based shortfall risk (SR) measure effectively captures a decision maker’s risk attitude on tail losses by an increasing convex loss function. In this paper, we consider a situation where the decision maker’s risk attitude toward tail losses is ambiguous and introduce a robust version of SR, which mitigates the risk arising from such ambiguity. Specifically, we use some available partial information or subjective judgement to construct a set of utility-based loss functions and define a so-called preference robust shortfall risk (PRSR) through the worst loss function from the (ambiguity) set. We then apply the PRSR to optimal decision-making problems and demonstrate how the latter can be reformulated as tractable convex programs when the underlying exogenous uncertainty is discretely distributed.