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AbstractIn this article, we study the optimal insurance-reinsurance problem from the perspectives of both insurers and reinsurers. Namely, we study the optimal insurance-reinsurance treaties by minimizing the convex combination of the risk measurements of the insurer and the reinsurer, and taking into account the reinsurer’s default risk. We assume that the risks of the insurer and the reinsurer, as well as the insurance premium and the reinsurance premium, are calculated by some distortion risk measures with different distortion functions. Optimal insurance-reinsurance treaties are provided. Finally, an illustrating numerical example is given. [ABSTRACT FROM AUTHOR]

Extreme-case risk measures provide an approach for quantifying the upper and lower bounds of risk in situations where limited information is available regarding the underlying distributions. Previous research has demonstrated that for popular risk measures, such as value-at-risk and conditional value-at-risk, the worst-case counterparts can be evaluated in closed form when only the first two moments of the underlying distributions are known. In this study, we extend these findings by presenting closed-form solutions for a general class of distortion risk measures, which consists of various popular risk measures as special cases when the first and certain higher-order (i.e., second or more) absolute center moments, alongside the symmetry properties of the underlying distributions, are known. Moreover, we characterize the extreme-case distributions with convex or concave envelopes of the corresponding distributions. By providing closed-form solutions for extreme-case distortion risk measures and characterizations for the corresponding distributions, our research contributes to the understanding and application of risk quantification methodologies. Funding: H. Shao acknowledges support from the Yangtze River Delta Science and Technology Innovation Community Joint Research Program [Grant 2022CSJGG0800]. Z. G. Zhang acknowledges support from the Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2019-06364]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2022.0156. [ABSTRACT FROM AUTHOR]

Increasing rearrangement is a rewarding instrument in financial risk management. In practice, risks must be managed from different perspectives. A common example is the portfolio risk, which often can be seen from at least two perspectives: market value and book value. Different perspectives with different distributions can be coupled by increasing rearrangement. One distribution is regarded as underlying, and the other distribution can be expressed as an increasing rearrangement of the underlying distribution. Then, the risk measure for the latter can be expressed in terms of the underlying distribution. Our first objective is to introduce increasing rearrangement for application in financial risk management and to apply increasing rearrangement to the class of distortion risk measures. We derive formulae to compute risk measures in terms of the underlying distribution. Afterwards, we apply our results to a series of special distortion risk measures, namely the value at risk, expected shortfall, range value at risk, conditional value at risk, and Wang's risk measure. Finally, we present the connection of increasing rearrangement with inverse transform sampling, Monte Carlo simulation, and cost-efficient strategies. Butterfly options serve as an illustrative example of the method. [ABSTRACT FROM AUTHOR]

Abstract.There are several ways to establish the asymptotic normality of L-statistics, which depend on the choice of the weights-generating function and the cumulative distribution selection of the underlying model. In this study, we focus on establishing computational formulas for the asymptotic variance of two robust L-estimators: the method of trimmed moments (MTM) and the method of winsorized moments (MWM). We demonstrate that two asymptotic approaches for MTM are equivalent for a specific choice of the weights-generating function. These findings enhance the applicability of these estimators across various underlying distributions, making them effective tools in diverse statistical scenarios. Such scenarios include actuarial contexts, such as payment-per-payment and payment-per-loss data scenarios, as well as in evaluating the asymptotic distributional properties of distortion risk measures. The effectiveness of our methodologies depends on the availability of the cumulative distribution function, ensuring broad usability in various statistical environments. [ABSTRACT FROM AUTHOR]

Distortion risk measures have a significant effect on the fields of finance and risk management. In this article, we consider two optimal reinsurance designs under a new distortion risk measure with mixed methods, which was proposed by Zhu and Yin (Communications in Statistics - Theory and Methods 2023, 4151–4164), one with the reinsurer's default risk and another one with the context of determining the Pareto-optimal reinsurance policies. The closed-form solutions of optimal reinsurance policies in both setting are obtained. The GlueVaR and generalized GlueVaR are considered in the application of designing optimal reinsurance contracts with reinsurer's default risk as two special cases. Finally, we give two numerical examples, one with default risk and another one without default risk, to illustrate our results. [ABSTRACT FROM AUTHOR]

Distortion risk measures play an essential role in the fields of finance and risk management. In this paper, we present a new distortion risk measure with mixed methods. We then investigate the optimal reinsurance problem under the new risk measure and the closed-form solutions of optimal reinsurance policies are obtained. As special cases of the new distortion risk measure, VaR and GlueVaR are considered in the application of risk management. [ABSTRACT FROM AUTHOR]

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]

Bowley reinsurance solutions are reinsurance contracts for which the reinsurer optimally sets the pricing density while anticipating that the insurer will choose the optimal reinsurance indemnity given this pricing density. This Bowley solution concept of equilibrium reinsurance strategy has been revisited in the modern risk management framework by Boonen et al. [(2021). Bowley reinsurance with asymmetric information on the insurer's risk preferences. Scandinavian Actuarial Journal2021, 623–644], where the insurer and reinsurer are both endowed with distortion risk measures but there is asymmetric information on the distortion risk measure of the insurer. In this article, we continue to study this framework, but we allow the premium principle to be more flexible. We call this solution the first-best Bowley solution. We provide first-best Bowley solutions in closed form under very general assumptions. We implement some numerical examples to illustrate the findings and the comparisons with the second-best solution. The main result is further extended to the case when both the reinsurer and the insurers have heterogeneous beliefs on the distribution functions of the underlying risk. [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]

Extreme losses are specially relevant in finance and insurance. In this work, we estimate the distorted tail value-at-risk measure that is attributable to extreme losses using the semi-parametric approache with an application on insurance data. [ABSTRACT FROM AUTHOR]

Distortion risk measure, defined by an integral of a distorted tail probability, has been widely used in behavioral economics and risk management as an alternative to expected utility. The sensitivity of the distortion risk measure is a functional of certain distribution sensitivities. We propose a new sensitivity estimator for the distortion risk measure that uses generalized likelihood ratio estimators for distribution sensitivities as input and establish a central limit theorem for the new estimator. The proposed estimator can handle discontinuous sample paths and distortion functions. [ABSTRACT FROM AUTHOR]

The Bowley solution refers to the optimal pricing density for the reinsurer and optimal ceded loss for the insurer when there is a monopolistic reinsurer. In a sequential game, the reinsurer first sets the pricing kernel, and thereafter the insurer selects the reinsurance contract given the pricing kernel. In this article, we study Bowley solutions under asymmetric information on the insurer's risk preferences where the identity of the insurer is unknown to the reinsurer. By assuming that the insurer adopts a Value-at-Risk measure or a convex distortion risk measure, the optimal pricing kernel for the insurer and the optimal ceded loss function for the reinsurer are determined. Numerical examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we study two classes of optimal reinsurance models from perspectives of both insurers and reinsurers by minimizing their convex combination of the total losses where the risk is measured by a distortion risk measure and the reinsurance premium is calculated according to a distortion premium principle. In the first place, we show how to formulate the unconstrained optimization problem and constrained optimization problem in a unified way. Then, we propose a geometric approach to solve optimal reinsurance problems directly. This paper considers a class of increasing convex ceded loss functions and derives the explicit solutions of the optimal reinsurance, which can be in forms of quota-share, stop-loss, change-loss, the combination of quota-share and change-loss or the combination of change-loss and change-loss with different retentions. Finally, we consider two specific cases of the distortion risk measures: Value at Risk (VaR) and Tail Value at Risk (TVaR). [ABSTRACT FROM AUTHOR]

The current paper provides a general approach to construct distortion operators that can price financial and insurance risks. Our approach generalizes the (Wang 2000) transform and recovers multiple distortions proposed in the literature as particular cases. This approach enables designing distortions that are consistent with various pricing principles used in finance and insurance such as no-arbitrage models, equilibrium models and actuarial premium calculation principles. Such distortions allow for the incorporation of risk-aversion, distribution features (e.g. skewness and kurtosis) and other considerations that are relevant to price contingent claims. The pricing performance of multiple distortions obtained through our approach is assessed on CAT bonds data. The current paper is the first to provide evidence that jump-diffusion models are appropriate for CAT bonds pricing, and that natural disaster aversion impacts empirical prices. A simpler distortion based on a distribution mixture is finally proposed for CAT bonds pricing to facilitate the implementation. [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]

Motivated by Cui et al. (2013) and Zheng and Cui (2014), we study in this paper the optimal (from the reinsurer’s point of view) reinsurance problem where the risk is measured by distortion risk measures, the premiums are calculated under the distortion premium principle, and both the upper and lower premium constraints are involved. Our objective is to seek for the optimal reinsurance strategy which minimizes the reinsurer’s risk measure of its total loss. Suppose an reinsurer is exposed to the risk f ( X ) that is transferred from an insurer, who faces a total loss X and decides to buy from our reinsurer the reinsurance contract. The reinsurance contract specifies that the reinsurer covers f ( X ) and the insurer covers X − f ( X ) . In addition, the insurer is obligated to compensate our reinsurer for undertaking the risk by paying the reinsurance premium under the distortion premium principle. We present a direct method for discussing the optimization problem. Based on our method, the optimal (or, suboptimal) reinsurance strategy is sought out. [ABSTRACT FROM AUTHOR]

In this paper, we build simple extreme analogues of Wang distortion risk measures and we show how this makes it possible to consider many standard measures of extreme risk, including the usual extreme Value-at-Risk or Tail-Valueat- Risk, as well as the recently introduced extreme Conditional Tail Moment, in a unified framework. We then introduce adapted estimators when the random variable of interest has a heavy-tailed distribution and we prove their asymptotic normality. The finite sample performance of our estimators is assessed in a simulation study and we showcase our techniques on two sets of data. [ABSTRACT FROM AUTHOR]

Evaluating risk measures, premiums, and capital allocation based on dependent multi-losses is a notoriously difficult task. In this paper, we demonstrate how this can be successfully accomplished when losses follow the multivariate Pareto distribution of the second kind, which is an attractive model for multi-losses whose dependence and tail heaviness are influenced by a heavy-tailed background risk. A particular attention is given to the distortion and weighted risk measures and allocations, as well as their special cases such as the conditional layer expectation, tail value at risk, and the truncated tail value at risk. We derive formulas that are either of closed form or follow well-defined recursive procedures. In either case, their computational use is straightforward.

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]

It is well known that a random vector with given marginals is comonotonic if and only if it has the largest convex sum, and that a random vector with given marginals (under an additional condition) is mutually exclusive if and only if it has the minimal convex sum. This paper provides an alternative proof of these two results using the theories of distortion risk measure and expected utility.

By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy-Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles). [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]

Abstract: The theme of this paper relates to solving portfolio selection problems using linear programming. We extend the well-known linear optimization framework for Conditional Value-at-Risk (CVaR)-based portfolio selection problems to optimization over a more general class of risk measure known as the class of Coherent Distortion Risk Measure (CDRM). CDRM encompasses many well-known risk measures including CVaR, Wang Transform measure, Proportional Hazard measure, and lookback measure. A case study is conducted to illustrate the flexibility of the linear optimization scheme, explore the efficiency of the 1/n-portfolio strategy, as well as compare and contrast optimal portfolios with respect to different CDRMs. [Copyright &y& Elsevier]

Catastrophic risk and insurance risk have required the use of specific risk measures for reinsurance companies to survive over the centuries. The goal in this paper is to apply the distortion risk measures introduced in actuarial sciences, as described by Wang (2000) in the Journal of Risk and Insurance, Vol. 67, No. 1, pp. 15-36, to the assessment of hedge funds risk. An empirical analysis of the Hedge Funds Research daily database over the period 2003-09 exhibits that these measures outperform the value-at-risk (VaR) or even extreme value-at-risk (EVaR) approaches in the capture of tail risks. [ABSTRACT FROM AUTHOR]

Abstract: The classical functional delta method (FDM) provides a convenient tool for deriving the asymptotic distribution of statistical functionals from the weak convergence of the respective empirical processes. However, for many interesting functionals depending on the tails of the underlying distribution this FDM cannot be applied since the method typically relies on Hadamard differentiability w.r.t. the uniform sup-norm. In this article, we present a version of the FDM which is suitable also for nonuniform sup-norms, with the outcome that the range of application of the FDM enlarges essentially. On one hand, our FDM, which we shall call the modified FDM, works for functionals that are “differentiable” in a weaker sense than Hadamard differentiability. On the other hand, it requires weak convergence of the empirical process w.r.t. a nonuniform sup-norm. The latter is not problematic since there exist strong respective results on weighted empirical processes obtained by Shorack and Wellner (1986) , Shao and Yu (1996) , Wu (2008) , and others. We illustrate the gain of the modified FDM by deriving the asymptotic distribution of plug-in estimates of popular risk measures that cannot be treated with the classical FDM. [ABSTRACT FROM AUTHOR]