Worst-case risk with unspecified risk preferences

In this paper, we study the worst-case distortion risk measure for a given risk when information about distortion functions is partially available. We obtain the explicit forms of the worst-case distortion functions for several different sets of plausible distortion functions. When there is no concavity constraint on distortion functions, the worst-case distortion function is independent of the risk to be measured and the corresponding worst-case distortion risk measure is the weighted average of the VaR's of the risk for all decision makers. When the concavity constraint is imposed on distortion functions and the set of concave distortion functions is defined by the riskiness of one single risk, the explicit form of the worst-case distortion function is obtained, which depends the risk to be measured. When the set of concave distortion functions is defined by the riskiness of multiple risks, we reduce the infinite-dimensional optimization problem to a finite-dimensional optimization problem which can be solved numerically. Finally, we apply the worst-case risk measure to optimal decision making in reinsurance.