Generalized Shortfall Risk Measure Based on Insurance Premium

Abstract

Utility-based shortfall risk (SR) measure proposed by (F\”ollmer and Schied, 2002) has been well studied in risk management and finance. In this paper, we revisit the concept from insurance premium perspective. We show under some moderate conditions that the indifference equation-based insurance premium calculation can be equivalently formulated as an optimization problem similar to the definition of SR and subsequently call the premium functional as a generalized shortfall risk measure (GSR). We then use the latter formulation to investigate the properties of the GSR with a focus on the case that the preference functional is a distorted expected value function based on prospect theory. Specifically, we exploit Weber's methods and techniques (Weber, 2006) for characterization of the shortfall risk measure to derive a relationship between properties of GSR based on the prospect theory and the underlying value function and weighting function in terms of convexity/concavity and positive homogeneity. We also investigate the GSR as a functional of cumulative distribution function of random loss/liability and derive local and global Lipschitz continuity of the function under Wasserstein metric, a property which is related to statistical robustness of the GSR. The results cover the premium risk measures based on the von Neumann-Morgenstern's expected utility and Yaari's dual theory of choice as special cases. Finally, we propose a computational scheme for calculating GSR.