Insurance premium-based shortfall risk measure induced by cumulative prospect theory

The risk measure of utility-based shortfall risk (SR) proposed by Föllmer and Schied (Finance Stoch 6:429–447, 2002) has been well studied in risk management and finance. In this paper, we revisit the concept from an insurance premium perspective. Under some moderate conditions, we show that the indifference equation-based insurance premium calculation can be equivalently formulated as an optimization problem similar to the definition of SR. Subsequently, we call the premium functional as an insurance premium-based shortfall risk measure (IPSR) defined over non-negative random variables. We then use the latter formulation to investigate the properties of the IPSR with a focus on the case that the preference functional is a distorted expected value function based on the cumulative prospect theory (CPT). Specifically, we exploit Weber’s approach (Weber in Math Finance Int J Math Stat Financ Econ 16:419–441, 2006) for characterization of the shortfall risk measure to derive a relationship between properties of IPSR induced by the CPT (IPSR-CPT) and the underlying value function in terms of convexity/concavity and positive homogeneity. We also investigate the IPSR-CPT 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 IPSR-CPT. 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 the IPSR-CPT.