Shortfall Risk Models When Information of Loss Function Is Incomplete

Utility-based shortfall risk measure (SR) has received increasing attentions over the past few years for its potential to quantify more effectively the risk of large losses than conditional value at risk. In this paper we consider the case that the true loss function is unavailable either because it is difficult to be identified or the decision maker is ambiguous about disutility of the losses. We propose a preference robust SR (PRSR) model where it is possible to construct a set of utility-based loss functions from empirical data or subjective judgements and define PRSR through the worst loss function from the set in order to mitigate the effect from the ambiguity. We then demonstrate that the worst of a set of convex utility-based shortfall risk measure with some specified characteristics such as positive homogeneity and pairwise comparisons can be represented as PRSR by choosing an appropriate class of utility loss functions. The representation enables us to develop tractable formulations for optimization problem with the objective of minimizing the worst convex utility-based shortfall risk measures when the underlying probability distribution is discrete. In the case when the probability distribution is continuous, we propose a sample average approximation scheme and show that it converges to the true problem in terms of the optimal value and the optimal solutions as the sample size increases.



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