Stop-loss and limited loss random variables are two important transforms of a loss random variable and appear in many modeling problems in insurance, finance, and other fields. Risk levels of a loss variable and its transforms are often measured by risk measures. When only partial information on a loss variable is available, risk measures of the loss variable and its transforms cannot be evaluated effectively. To deal with the situation of distribution uncertainty, the worst-case values of risk measures of a loss variable over an uncertainty set, describing all the possible distributions of the loss variable, have been extensively used in robust risk management for many fields. However, most of these existing results on the worst-case values of risk measures of a loss variable cannot be applied directly to the worst-case values of risk measures of its transforms. In this paper, we derive the expressions of the worst-case values of distortion risk measures of stop-loss and limited loss random variables over an uncertainty set introduced in Bernard et al. (2023). This set represents a decision maker’s belief in the distribution of a loss variable. We find the distributions under which the worst-case values are attainable. These results have potential applications in a variety of fields. To illustrate their applications, we discuss how to model optimal stop-loss reinsurance problems and how to determine optimal stop-loss retentions under distribution uncertainty. Explicit and closed-form expressions for the worst-case TVaRs of stop-loss and limited loss random variables and optimal stop-loss retentions are given under special forms of the uncertainty set. Numerical results are presented under more general forms of the uncertainty set.
%B European Journal of Operational Research %V 318 %P 310-326 %G eng %U https://doi.org/10.1016/j.ejor.2024.03.016 %N 1 %0 Journal Article %J Mathematics of Operations Research %D 2022 %T Inf-Convolution, Optimal Allocations, and Model Uncertainty for Tail Risk Measures %A Fangda Liu %A Tiantian Mao %A Ruodu Wang %A Linxiao Wei %XInspired by the recent developments in risk sharing problems for the value at risk (VaR), the expected shortfall (ES), and the range value at risk (RVaR), we study the optimization of risk sharing for general tail risk measures. Explicit formulas of the infconvolution and Pareto-optimal allocations are obtained in the case of a mixed collection of left and right VaRs, and in that of a VaR and another tail risk measure. The inf-convolution of tail risk measures is shown to be a tail risk measure with an aggregated tail parameter, a phenomenon very similar to the cases of VaR, ES, and RVaR. Optimal allocations are obtained in the settings of elliptical models and model uncertainty. In particular, several results are established for tail riskmeasures in the presence ofmodel uncertainty, which may be of independent interest outside the framework of risk sharing. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Our analysis generalizes in several directions the recent literature on quantile-based risk sharing.
%B Mathematics of Operations Research %V 47 %P 2494-2519 %8 12 Sep 2021 %G eng %U https://doi.org/10.1287/moor.2021.1217 %N 3 %0 Journal Article %J Journal of Mathematical Economics %D 2022 %T Insurance With Heterogeneous Preferences %A Tim J. Boonen %A Fangda Liu %X This paper studies an optimal insurance problem with finitely many potential policyholders. A monopolistic, risk-neutral insurer offers an insurance contract, and exponential utility maximizing individuals accept the offer or not. We allow for heterogeneity in the preferences of the individuals, while the insurer cannot discriminate in the insurance premium. We show that it is optimal for the insurer to offer only a full insurance contract, and the price optimization problem is reduced to a discrete problem, where the premium is an indifference premium for one individual in the market. Moreover, if individuals can self-select their insurance coverage given the market premium rate, then we find that partial insurance is generally optimal. Since the risk aversion parameters of individuals is generally unobserved, we also present a simulation-based framework. We show its convergence, and provide numerical examples. %B Journal of Mathematical Economics %V 102 %P Article 102742 %G eng %U https://doi.org/10.1016/j.jmateco.2022.102742 %0 Journal Article %J Economic Theory %D 2021 %T Competitive Equilibria in a Comonotone Market %A Tim J. Boonen %A Fangda Liu %A Ruodu Wang %X We investigate competitive equilibria in a special type of incomplete markets, referred to as a comonotone market, where agents can only trade such that their risk allocation is comonotonic. The comonotone market is motivated by the no-sabotage condition. For instance, in a standard insurance market, the allocation of risk among the insured, the insurer and the reinsurers is assumed to be comonotonic a priori to the risk-exchange. Two popular classes of preferences in risk management and behavioral economics, dual utilities (DU) and rank-dependent expected utilities (RDU), are used to formulate agents’ objectives. We present various results on properties and characterization of competitive equilibria in this framework, and in particular their relation to complete markets. For DU-comonotone markets, we find the equilibrium in closed form and for RDU-comonotone markets, we find the equilibrium in closed form in special cases. The fundamental theorems of welfare economics are established in both the DU and RDU markets. We further propose an algorithm to numerically obtain competitive equilibria based on discretization, which works for both the DU-comonotone market and the RDU-comonotone market. Although the comonotone and complete markets are closely related, many of our findings are intriguing and in sharp contrast to results in the literature on complete markets in terms of existence, uniqueness, and closed-form solutions of the equilibria, and comonotonicity of the pricing kernel. %B Economic Theory %V 72 %P 1217–1255 %G eng %U https://doi.org/10.1007/s00199-020-01319-4 %0 Journal Article %J Insurance: Mathematics & Economics %D 2021 %T Enhancing an insurer's expected value by reinsurance and external financing %A Chi, Yichun %A Liu, Fangda %X In this paper, we analyze a decision-making problem for an insurer with limited liability, who is subject to a solvency constraint and wants to maximize the expected value through reinsurance purchase and external financing. We impose mild conditions on the reinsurance premium principle, which are the axioms of law invariance, risk loading and preserving the convex order. These three axioms are satisfied by all the widely used premium principles, except the Esscher principle, listed in . If value at risk (VaR) or conditional value at risk (CVaR) is adopted to calculate the insurer's regulatory capital, we show that the optimal reinsurance can be in the form of two layers. The optimal reinsurance form is reduced to one layer once the premium principle further satisfies a weak condition. In particular, under Wang's, expected value, variance and Dutch premium principles, we derive the insurer's optimal strategies of reinsurance and financing, which are obtained explicitly under the VaR risk measure but have to be solved numerically for the CVaR risk measure. Results indicate that the insurer has three types of optimal strategies: the reinsurance only strategy, external financing only strategy, and mixed strategy. Moreover, the insurer's attitude toward ceding the risk through reinsurance and increasing capital through external financing is often greatly affected by the regulatory regime, the financing cost and the reinsurance price. %B Insurance: Mathematics & Economics %V 101 %P 466-484 %G eng %U https://doi.org/10.1016/j.insmatheco.2021.08.010 %N Part B %0 Journal Article %J Insurance: Mathematics & Economics %D 2021 %T The Fourier-cosine Method for Finite-time Ruin Probabilities %A Wing Yan Lee %A Xiaolong Li %A Fangda Liu %A Yifan Shi %A S. C. Phillip Yam %X In this paper, we study the finite-time ruin probability in the risk model driven by a Lévy subordinator, by incorporating the popular Fourier-cosine method. Our interest is to propose a general approximation for any specified precision provided that the characteristic function of the Lévy Process is known. To achieve this, we derive an explicit integral expression for the finite-time ruin probability, which is expressed in terms of the density function and the survival function of Lt. Moreover, we apply the rearrangement inequality to further improve our approximations. In addition, with only mild and practically relevant assumptions, we prove that the approximation error can be made arbitrarily small (actually an algebraic convergence rate up to 3, which is the fastest possible approximant known upon all in the literature), and has a linear computation complexity in a number of terms of the Fourier-cosine expansion. The effectiveness of our results is demonstrated in various numerical studies; through these examples, the supreme power of the Fourier-cosine method is once demonstrated. %B Insurance: Mathematics & Economics %V 99 %P 256-267 %G eng %U https://doi.org/10.1016/j.insmatheco.2021.03.001 %N July 2021 %0 Journal Article %J Mathematics of Operations Research %D 2021 %T A theory for measures of tail risk, %A Fangda Liu %A Ruodu Wang %XThe notion of "tail risk" has been a crucial consideration in modern risk management. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures which quantify the tail risk, that is, the behavior of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, the Value-at-Risk (VaR) and the Expected Shortfall (ES), are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk measures, risk aggregation, elicitability, and dual representations. In particular, there is no elicitable tail convex risk measure rather than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.
SSRN link: https://papers.ssrn.com/abstract_id=2841909
%B Mathematics of Operations Research %V 46 %P 835-1234, C2 %G eng %U https://doi.org/10.1287/moor.2020.1072 %N 3 %0 Journal Article %J Insurance: Mathematics & Economics %D 2020 %T Convex risk functionals: representation and applications. %A Fangda Liu %A Jun Cai %A Christiane Lemieux %A Ruodu Wang %X We introduce the family of law-invariant convex risk functionals, which includes a wide majority of practically used convex risk measures and deviation measures. We obtain a unified representation theorem for this family of functionals. Two related optimization problems are studied. In the first application, we determine worst-case values of a law-invariant convex risk functional when the mean and a higher moment such as the variance of a risk are known. Second, we consider its application in optimal reinsurance design for an insurer. With the help of the representation theorem, we can show the existence and the form of optimal solutions. %B Insurance: Mathematics & Economics %V 90 %P 66-79 %G eng %U https://doi.org/10.1016/j.insmatheco.2019.10.007 %0 Journal Article %J Journal of Economic Behavior & Organization %D 2020 %T Optimal insurance in the presence of multiple policyholders. %A Carole Bernard %A Fangda Liu %A Steven Vanduffel %X The literature on optimal insurance typically considers optimal risk sharing between one insurer (or reinsurer) and one insurance prospect. However, the insurance business is based on diversification benefits that arise when pooling many insurance policies. In this paper, we first show that the classical results on optimal insurance in the case of a single insurance prospect remain valid when there are multiple prospects, provided their insurance claims are independent. Specifically, all prospects receive coverage. However, due to phenomena such as medical progress, longevity risk, and natural or man-made disasters, insurance claims tend to be correlated. We show that in the case of interdependent insurance policies, it may become optimal for the insurer to refuse to sell insurance to some prospects, and this decision is driven by the prospects’ attitudes towards risk and their risk exposure characteristics. This finding calls for government policies to ensure that insurance remains available and affordable to everyone. %B Journal of Economic Behavior & Organization %V 18 %P 638-656 %G eng %U https://doi.org/10.1016/j.jebo.2020.02.012 %0 Journal Article %J Journal of Industrial and Management Optimization %D 2018 %T Analysis of a dynamic premium strategy: from theoretical and marketingperspectives. %A Wing Yan Lee %A Fangda Liu %X Premium rate for an insurance policy is often reviewed and updated periodically according to past claim experience in real-life. In this paper, a dynamic premium strategy that depends on the past claim experience is proposed under the discrete-time risk model. The Gerber-Shiu function is analyzed under this model. The marketing implications of the dynamic premium strategy will also be discussed. %B Journal of Industrial and Management Optimization %V 14 %P 1545-1577 %G eng %U https://www.aimsciences.org/article/doi/10.3934/jimo.2018020 %N 4 %0 Journal Article %J Insurance: Mathematics & Economics %D 2017 %T Optimal insurance design in the presence of exclusion clauses %A Chi, Yichun %A Liu, Fangda %XThe present work studies the design of an optimal insurance policy from the perspective of an insured, where the insurable loss is mutually exclusive from another loss that is denied in the insurance coverage. To reduce ex post moral hazard, we assume that both the insured and the insurer would pay more for a larger realization of the insurable loss. When the insurance premium principle preserves the convex order, we show that any admissible insurance contract is suboptimal to a two-layer insurance policy under the criterion of minimizing the insured’s total risk exposure quantified by value at risk, tail value at risk or an expectile. The form of optimal insurance can be further simplified to be one-layer by imposing an additional weak condition on the premium principle. Finally, we use Wang’s premium principle and the expected value premium principle to illustrate the applicability of our results, and find that optimal insurance solutions are affected not only by the size of the excluded loss but also by the risk measure chosen to quantify the insured’s risk exposure.
%B Insurance: Mathematics & Economics %V 76 %P 185-195 %G eng %U https://doi.org/10.1016/j.insmatheco.2017.07.003 %0 Journal Article %J ASTIN Bulletin %D 2016 %T Optimal reinsurance from the perspectives of both an insurer and areinsurer. %A Cai, Jun %A Lemieux, Christiane %A Liu, Fangda %X Optimal reinsurance from an insurer's point of view or from a reinsurer's point of view has been studied extensively in the literature. However, as two parties of a reinsurance contract, an insurer and a reinsurer have conflicting interests. An optimal form of reinsurance from one party's point of view may be not acceptable to the other party. In this paper, we study optimal reinsurance designs from the perspectives of both an insurer and a reinsurer and take into account both an insurer's aims and a reinsurer's goals in reinsurance contract designs. We develop optimal reinsurance contracts that minimize the convex combination of the Value-at-Risk (VaR) risk measures of the insurer's loss and the reinsurer's loss under two types of constraints, respectively. The constraints describe the interests of both the insurer and the reinsurer. With the first type of constraints, the insurer and the reinsurer each have their limit on the VaR of their own loss. With the second type of constraints, the insurer has a limit on the VaR of his loss while the reinsurer has a target on his profit from selling a reinsurance contract. For both types of constraints, we derive the optimal reinsurance forms in a wide class of reinsurance policies and under the expected value reinsurance premium principle. These optimal reinsurance forms are more complicated than the optimal reinsurance contracts from the perspective of one party only. The proposed models can also be reduced to the problems of minimizing the VaR of one party's loss under the constraints on the interests of both the insurer and the reinsurer. %B ASTIN Bulletin %V 46 %P 815-849 %G eng %U https://www.cambridge.org/core/journals/astin-bulletin-journal-of-the-iaa/article/optimal-reinsurance-from-the-perspectives-of-both-an-insurer-and-a-reinsurer/D52E0DC74B5ECE527C39EC10A1DC8D4F %N 3 %0 Journal Article %J Insurance: Mathematics & Economics %D 2014 %T Optimal reinsurance with regulatory initial capital and default risk. %A Jun Cai %A Christiane Lemieux %A Fangda Liu %X In a reinsurance contract, a reinsurer promises to pay the part of the loss faced by an insurer in exchange for receiving a reinsurance premium from the insurer. However, the reinsurer may fail to pay the promised amount when the promised amount exceeds the reinsurer’s solvency. As a seller of a reinsurance contract, the initial capital or reserve of a reinsurer should meet some regulatory requirements. We assume that the initial capital or reserve of a reinsurer is regulated by the value-at-risk (VaR) of its promised indemnity. When the promised indemnity exceeds the total of the reinsurer’s initial capital and the reinsurance premium, the reinsurer may fail to pay the promised amount or default may occur. In the presence of the regulatory initial capital and the counterparty default risk, we investigate optimal reinsurance designs from an insurer’s point of view and derive optimal reinsurance strategies that maximize the expected utility of an insurer’s terminal wealth or minimize the VaR of an insurer’s total retained risk. It turns out that optimal reinsurance strategies in the presence of the regulatory initial capital and the counterparty default risk are different both from optimal reinsurance strategies in the absence of the counterparty default risk and from optimal reinsurance strategies in the presence of the counterparty default risk but without the regulatory initial capital. %B Insurance: Mathematics & Economics %V 57 %P 13-24 %G eng %U https://doi.org/10.1016/j.insmatheco.2014.04.006 %0 Journal Article %J ASTIN Bulletin %D 2012 %T Average Value-at-Risk minimizing reinsurance under Wang’spremium principle with constraints %A Cheung, K. C. %A Liu, Fangda %A Yam, S. C. Phillip %X In the present work, we study the optimal reinsurance decision problem in which the Average Value-at-Risk of the retained loss is minimized under Wang's premium principle and is also subject to either (1) a budget constraint on reinsurance premium, or (2) a reinsurer's probabilistic benchmark constraint of his potential loss. We show that the optimal reinsurance is a single-insurance layer under Constraint (1), and a cap insurance or a double-insurance layer under Constraint (2); moreover, under Constraint (2), we further establish that under most common circumstances (see Remark after Theorem 3), a cap insurance will suffice to be optimal. Finally, some numerical illustrations will be provided. %B ASTIN Bulletin %V 42 %P 575-600 %G eng %U https://www.cambridge.org/core/journals/astin-bulletin-journal-of-the-iaa/article/average-valueatrisk-minimizing-reinsurance-under-wangs-premium-principle-with-constraints/DCE7FD3CC893538369FA555A3582C11A %N 2