Distortion Premium Principle Research Articles

Pareto-optimal insurance with an upper limit on the insurer's exposure

We examine the problem of determining Pareto-optimal (PO) insurance contracts when the insurer imposes an ex ante upper limit on disbursement. The problem is similar in spirit to that of Cummins & Mahul (2004), but it extends it in two directions: first, we use the more general and more flexible class of distortion premium principles; and second, we allow for heterogeneity in beliefs between the insurer and the insured. We unify the settings of Ghossoub (2019a, 2019b), and we adapt the approaches therein to encompass the case of a policy limit. First, we show that PO contracts are those that result from a budget-constrained optimization problem for the DM. We then provide a closed-form characterization of optimal contracts. Our result is similar in spirit to that of Cummins & Mahul (2004), who show that when policy limits are introduced to Arrow's model, PO contracts are limited deductible contracts. While Ghossoub (2019a, 2019b) shows that variable deductible contracts are optimal, the results of the present paper indicate that limited variable deductible contracts are optimal when policy limits are present. We illustrate our results via numerical examples.

VaR and CTE Based Optimal Reinsurance from a Reinsurer’s Perspective

In this article, we study optimal reinsurance design. By employing the increasing convex functions as the admissible ceded loss functions and the distortion premium principle, we study and obtain the optimal reinsurance treaty by minimizing the VaR (value at risk) of the reinsurer’s total risk exposure. When the distortion premium principle is specified to be the expectation premium principle, we also obtain the optimal reinsurance treaty by minimizing the CTE (conditional tail expectation) of the reinsurer’s total risk exposure. The present study can be considered as a complement of that of Cai et al. [5].

Optimal reinsurance with multiple reinsurers: Distortion risk measures, distortion premium principles, and heterogeneous beliefs

This paper unifies the work on multiple reinsurers, distortion risk measures, premium budgets, and heterogeneous beliefs. An insurer minimizes a distortion risk measure, while seeking reinsurance with finitely many reinsurers. The reinsurers use distortion premium principles, and they are allowed to have heterogeneous beliefs regarding the underlying probability distribution. We provide a characterization of optimal reinsurance indemnities, and we show that they are of a layer-insurance type. This is done both with and without a budget constraint, i.e., an upper bound constraint on the aggregate premium. Moreover, the optimal reinsurance indemnities enable us to identify a representative reinsurer in both situations. Finally, two examples with the Conditional Value-at-Risk illustrate our results.

A METHOD FOR CONSTRUCTING AND INTERPRETING SOME WEIGHTED PREMIUM PRINCIPLES

AbstractWe present a method for constructing and interpreting weighted premium principles. The method is based on modifying the underlying risk distribution in such a way that the risk-adjusted expected value (or premium) is greater than the expected value of some conveniently chosen function of claims, which defines the insurer’s perception of the risk. Under some assumptions on the function of claims, the method produces distortion premium principles. We provide several examples under different assumptions on the claim arrival process and different functions of claims, including record claims andkth record claims.

Optimal insurance under rank-dependent expected utility

We re-visit the problem of optimal insurance design under Rank-Dependent Expected Utility (RDEU) examined by Bernard et al. (2015), Xu (2018), and Xu et al. (2018). Unlike the latter, we do not impose the no-sabotage condition on admissible indemnities, that is, that indemnity and retention functions be nondecreasing functions of the loss. Rather, in a departure from the aforementioned work, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, hence automatically ruling out any ex post moral hazard that could otherwise arise from possible misreporting of the loss by the insured. We fully characterize the optimal indemnity schedule and discuss how our results relate to those of Bernard et al. (2015) and Xu et al. (2018). We then extend the setting by allowing for a distortion premium principle, with a distortion function that differs from that of the insured, and we provide a characterization of the optimal retention in that case.

A unifying approach to constrained and unconstrained optimal reinsurance

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).

Optimal Insurance Under Rank-Dependent Expected Utility

We re-visit the problem of optimal insurance design under Rank-Dependent Expected Utility (RDEU) examined by Bernard et al. (2015), Xu (2018), and Xu et al. (2015). Unlike the latter, we do not impose the no sabotage condition on admissible indemnities, that is, the comonotonicity of indemnity functions and retention functions with the loss. Rather, in a departure from the aforementioned work, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, hence automatically ruling out any ex post moral hazard that could otherwise arise from possible misreporting of the loss by the insured. Hence, monotonicity properties of indemnification schedules become of second-order concern. We fully characterize the optimal indemnity schedule and discuss how our results relate to those of Bernard et al. (2015) and Xu et al. (2015). We then extend the setting by allowing for a distortion premium principle, with a distortion function that differs from that of the insured, and we provide a characterization of the optimal retention in that case.

Optimal Reinsurance with Multiple Reinsurers: Distortion Risk Measures, Distortion Premium Principles, and Heterogeneous Beliefs

This paper unifies the work on multiple reinsurers, distortion risk measures, premium budgets, and heterogeneous beliefs. An insurer minimizes a distortion risk measure, while seeking reinsurance with finitely many reinsurers. The reinsurers use distortion premium principles, and they are allowed to have heterogeneous beliefs regarding the underlying probability distribution. We provide a characterization of optimal reinsurance indemnities, and we show that they are of a layer-insurance type. This is done both with and without a budget constraint, i.e., an upper bound constraint on the aggregate premium. Moreover, the optimal reinsurance indemnities enable us to identify a representative reinsurer in both situations. Finally, two examples with the Conditional Value-at-Risk illustrate our results.

Optimal Reinsurance from the Perspectives of Both Insurers and Reinsurers Under General Distortion Risk Measures

In this article, we propose new methodologies to tackle and the problems studied in [Cai, J., Lemieux, C. and Liu, F., 2016, Optimal reinsurance from the perspectives of both an insurer and a reinsurer, ASTIN Bulletin, 46(3), 815-849]. Optimal reinsurance contracts that minimize the convex combination of the risk measurements of the insurer's loss and the reinsurer's loss are characterized under different types of constraints. The results in Cai et al. (2016) are considerably extended to the general framework of distortion risk measures and distortion premium principle.

边际赔偿函数和违约风险下的最优再保险

本文考虑了再保险人的违约风险，首先运用失真风险度量和失真保费原理建立了含有违约风险的总风险模型。其次通过边际索赔(MIF)函数与分出函数之间的关系建立了与总风险模型等价的MIF再保险优化模型。然后对MIF再保险优化模型的求解得到最优的边际索赔(MIF)函数，进而得到最优的分出函数。最后应用该方法研究了在VaR风险度量和Wang’s保费原理下的最优分出函数。 In this paper, we consider the default risk of reinsurer. Firstly, we use the distortion risk measure and distortion premium principle to establish the total risk model with default risk. Secondly, by the relationship between the Marginal Indemnification Function (MIF) and the ceded loss function, we build MIF reinsurance optimization model equivalent to total risk model. Then, the optimal MIF function is obtained by solving the MIF reinsurance optimization model. Furthermore, the optimal ceded loss function is obtained. Finally, we apply this method to study the optimal loss function by the VaR risk measure and Wang’s premium principle.

Reinsurer’s optimal reinsurance strategy with upper and lower premium constraints under distortion risk measures

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.

A family of premium principles based on mixtures of TVaRs

Risk-adjusted distributions are commonly used in actuarial science to define premium principles. In this paper, we claim that an appropriate risk-adjusted distribution, besides satisfying other desirable properties, should be well-behaved under conditioning with respect to the original risk distribution. Based on a sequence of such risk-adjusted distributions, we introduce a family of premium principles that gradually incorporate the degree of risk-aversion of the insurer in the risk loading. Members of this family are particular distortion premium principles that can be represented as mixtures of TVaRs, where the weights in the mixture reflect the attitude toward risk of the insurer. We make a systematic study of this family of premium principles.

The role of a representative reinsurer in optimal reinsurance

In this paper, we consider a one-period optimal reinsurance design model with n reinsurers and an insurer. For very general preferences of the insurer and that all reinsurers use a distortion premium principle, we establish the existence of a representative reinsurer and this in turn facilitates solving the optimal reinsurance problem with multiple reinsurers. The insurer determines its optimal risk that it wants to reinsure via this pricing formula. The risk to be reinsured is then shared by the reinsurers via tranching. The optimal ceded loss functions among multiple reinsurers are derived explicitly under the additional assumptions that the insurer’s preferences are given by an inverse-S shaped distortion risk measure and that the reinsurers’ premium principles are some functions of the Conditional Value-at-Risk. We also demonstrate that under some prescribed conditions, it is never optimal for the insurer to cede its risk to more than two reinsurers.

Optimal Reinsurance with One Insurer and Multiple Reinsurers

In this paper, we consider a one-period optimal reinsurance design model with n reinsurers and an insurer. For very general preferences of the insurer, we obtain that there exists a very intuitive pricing formula for all reinsurers that use a distortion premium principle. The insurer determines its optimal risk that it wants to reinsure via this pricing formula. This risk it wants to reinsure is then shared by the reinsurers via tranching. The optimal ceded loss functions among multiple reinsurers are derived explicitly under the additional assumptions that the insurer’s preferences are given by an inverse-S shaped distortion risk measure and that the reinsurer’s premium principles are some functions of the Conditional Value-at-Risk. We also demonstrate that under some prescribed conditions, it is never optimal for the insurer to cede its risk to more than two reinsurers.

The uncertain premium principle based on the distortion function

In this paper, we discuss the premium principle in uncertain environment. First, the net premium principle for uncertain risks is presented within the framework of uncertainty theory. With the help of distortion function, a new uncertain premium principle is derived from the uncertain net premium. Some properties of uncertain distortion premium principle are proved. Furthermore, a sufficient and necessary condition that an uncertain premium principle is an uncertain distortion premium principle has been characterized. Finally, some examples are given to illustrate the calculations of the uncertain distortion premium.

A new characterization of distortion premiums via countable additivity for comonotonic risks

For premium calculation principles or risk measures, all existing works only consider the additivity for a finite number of comonotonic risks. As we all know, a limiting status of finite additivity is the additivity for countable risks. In this paper we investigate the countable additivity and generate new and elegant characterizations for Choquet pricing and distortion premium principles. We also study the countable exchangeability, as an extension to additivity. It leads to generalized Choquet pricing and generalized distortion premium principles.

Risk Measures and Theories of Choice

ABSTRACTWe discuss classes of risk measures in terms both of their axiomatic definitions and of the economic theories of choice that they can be derived from. More specifically, expected utility theory gives rise to the exponential premium principle, proposed by Gerber (1974), Dhaene et al. (2003), whereas Yaari's (1987) dual theory of choice under risk can be viewed as the source of the distortion premium principle (Denneberg, 1990; Wang, 1996). We argue that the properties of the exponential and distortion premium principles are complementary, without either of the two performing completely satisfactorily as a risk measure. Using generalised expected utility theory (Quiggin, 1993), we derive a new risk measure, which we call the distortion-exponential principle. This risk measure satisfies the axioms of convex measures of risk, proposed by Föllmer & Shied (2002a,b), and its properties lie between those of the exponential and distortion principles, which can be obtained as special cases.

Risk capital allocation and cooperative pricing of insurance liabilities

The Aumann–Shapley [Values of Non-atomic Games, Princeton University Press, Princeton] value, originating in cooperative game theory, is used for the allocation of risk capital to portfolios of pooled liabilities, as proposed by Denault [Coherent allocation of risk capital, J. Risk 4 (1) (2001) 1]. We obtain an explicit formula for the Aumann–Shapley value, when the risk measure is given by a distortion premium principle [Axiomatic characterisation of insurance prices, Insur. Math. Econ. 21 (2) (1997) 173]. The capital allocated to each instrument or (sub)portfolio is given as its expected value under a change of probability measure. Motivated by Mirman and Tauman [Demand compatible equitable cost sharing prices, Math. Oper. Res. 7 (1) (1982) 40], we discuss the role of Aumann–Shapley prices in an equilibrium context and present a simple numerical example.

On Characterization of Distortion Premium Principle

In this paper, based on the additive measure integral representation of a non-additive measure integral, it is shown that any comonotonically additive premium principle can be represented as an integral of the distorted decumulative distribution function of the insurance risk. Furthermore, a sufficient and necessary condition that a premium principle is a distortion premium principle is given.

On Characterization of Distortion Premium Principle

In this paper, based on the additive measure integral representation of a non-additive measure integral, it is shown that any comonotonically additive premium principle can be represented as an integral of the distorted decumulative distribution function of the insurance risk. Furthermore, a sufficient and necessary condition that a premium principle is a distortion premium principle is given.