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This paper studies the optimal insurance policy that maximizes the decision maker (DM)'s expected utility under distortion risk constraints. To alleviate the ex post moral hazard issues arising from the discontinuity of the indemnity functions in Huang (Geneva Risk Insur Rev 31(2):91–110, 2006) and Bernard and Tian (Geneva Risk Insur Rev 35(1):47–80, 2010) we re-visit their problems under the so called incentive compatibility condition, which requires that both the ceded and retained loss functions are non-decreasing. In addition, we generalize the value-at-risk (VaR) constraints used in the literature to the distortion-risk-measure-based constraints. We first implicitly characterize the optimal indemnity function when the risk constraints are defined in terms of the general distortion risk measure and then provide explicit solutions for the VaR and tail value-at-risk (TVaR) cases. The effect of the risk constraints on the optimal indemnity function are analyzed in great detail. Our results show that under the VaR risk constraints, the DM chooses to ignore the risk which does not contribute to its VaR value and only manages the risk that influences its VaR value. This problem is alleviated under the TVaR risk constraints. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper considers distributionally robust formulations of a two stage stochastic programming problem with the objective of minimizing a distortion risk of the minimal cost incurred at the second stage. We carry out a stability analysis by looking into variations of the ambiguity set under the Wasserstein metric, decision spaces at both stages and the support set of the random variables. In the case when the risk measure is risk neutral, the stability result is presented with the variation of the ambiguity set being measured by generic metrics of ζ -structure, which provides a unified framework for quantitative stability analysis under various metrics including total variation metric and Kantorovich metric. When the ambiguity set is structured by a ζ -ball, we find that the Hausdorff distance between two ζ -balls is bounded by the distance of their centers and difference of their radii. The findings allow us to strengthen some recent convergence results on distributionally robust optimization where the center of the Wasserstein ball is constructed by the empirical probability distribution. [ABSTRACT FROM AUTHOR]

This paper studies the Pareto-optimal reinsurance policies, where both the insurer’s and the reinsurer’s risks and returns are considered. We assume that the risks of the insurer and the reinsurer, as well as the reinsurance premium, are determined by some distortion risk measures with different distortion operators. Under the constraint that a reinsurance policy is feasible only if the resulting risk of each party is below some pre-determined values, we derive explicit expressions for the optimal reinsurance polices. Methodologically, we show that the generalized Neyman-Pearson method, the Lagrange multiplier method, and the dynamic control methods can be utilized to solve our problem. Special cases when both parties’ risks are measured by Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) are studied in great details. Numerical examples are provided to illustrate practical implications of the results. [ABSTRACT FROM AUTHOR]

Coherent risk measures (Artzner et al. in Math. Finance 9:203-228, 1999) and convex risk measures (Föllmer and Schied in Finance Stoch. 6:429-447, 2002) are characterized by desired axioms for risk measures. However, concrete or practical risk measures could be proposed from different perspectives. In this paper, we propose new risk measures based on behavioural economics theory. We use rank-dependent expected utility (RDEU) theory to formulate an objective function and propose the smallest solution that minimizes the objective function as a risk measure. We also employ cumulative prospect theory (CPT) to introduce a set of acceptable regulatory capitals and define the infimum of the set as a risk measure. We show that the classes of risk measures derived from RDEU theory and CPT are equivalent, and they are all monetary risk measures. We present the properties of the proposed risk measures and give sufficient and necessary conditions for them to be coherent and convex, respectively. The risk measures based on these behavioural economics theories not only cover important risk measures such as distortion risk measures, expectiles and shortfall risk measures, but also produce new interesting coherent risk measures and convex, but not coherent risk measures. [ABSTRACT FROM AUTHOR]

In this note we establish some appropriate conditions for stochastic equality of two random variables/ vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result are also considered. [ABSTRACT FROM AUTHOR]

This paper discusses optimal reinsurance strategy by minimizing insurer's risk under one general risk measure: Distortion risk measure. The authors assume that the reinsurance premium is determined by the expected value premium principle and the retained loss of the insurer is an increasing function of the initial loss. An explicit solution of the insurer's optimal reinsurance problem is obtained. The optimal strategies for some special distortion risk measures, such as value-at-risk (VaR) and tail value-at-risk (TVaR), are also investigated. [ABSTRACT FROM AUTHOR]

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel's classical notion of qualitative robustness is not suitable for risk measurement, and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz spaces. This concept captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence. [ABSTRACT FROM AUTHOR]

Distorted expectations can be expressed as weighted averages of quantiles. In this note, we show that this statement is essentially true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of quantiles. [ABSTRACT FROM AUTHOR]