Distortion Risk Measures Research Articles

Risk measures induced by efficient insurance contracts

The Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics, which has recently been characterized via sets of axioms from perspectives of portfolio risk management and statistics. Meanwhile, there is large literature on insurance design with ES as an objective or a constraint. A visible gap is to justify the special role of ES in insurance and actuarial science. To fill this gap, we study the characterization of risk measures induced by efficient insurance contracts, i.e., those that are Pareto optimal for the insured and the insurer. One of our major results is that we characterize a mixture of the mean and ES as the risk measure of the insured and the insurer, when contracts with deductibles are efficient. Characterization results of other risk measures, including the mean and distortion risk measures, are also presented by linking them to different sets of contracts.

Computing and Estimating Distortion Risk Measures: How to Handle Analytically Intractable Cases?

Computing and Estimating Distortion Risk Measures: How to Handle Analytically Intractable Cases?

New risk measures for variance distortion and catastrophic financial risk measures

In recent years, expectation distortion risk measures have been widely used in financial and insurance applications due to their attractive properties. The author introduced two new classes of financial risk measures “VaR raised to the power of t” and “ES raised to the power of t” in his works and also investigated the issue of the belonging of these risk measures to the class of risk measures of expectation distortion, and described the corresponding distortion functions. The aim of this study is to introduce a new concept of variance distortion risk measures, which opens up a significant area for investigating the properties of these risk measures that may be useful in applications. The paper proposes a method of finding new variance distortion risk measures that can be used to acquire risk measures with special properties. As a result of the study, it was found that the class of risk measures of variance distortion includes risk measures that are in a certain way related to “VaR raised to the power of t” and “ES raised to the power of t” measures. The article describes the composite method for constructing new variance distortion functions and corresponding distortion risk measures. This method is used to build a large set of examples of variance distortion risk measures that can be used in assessing certain financial risks of a catastrophic nature. The author concludes that the study of the variance distortion risk measures introduced in this paper can be used both for the development of theoretical risk management methods and in the practice of business risk management in assessing unlikely risks of high catastrophe.

Systemic risk: Conditional distortion risk measures

In this paper, we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution (ΔCoD) measures as measures of systemic risk and analyze their properties and representations. The classes unify, and significantly extend, existing systemic risk measures such as the conditional Value-at-Risk, conditional Expected Shortfall, and risk contribution measures in terms of the VaR and ES. We provide sufficient conditions for two random vectors to be ordered by the proposed CoD-risk measures and ΔCoD-measures. These conditions are expressed using the conventional stochastic dominance, increasing convex/concave, dispersive, and excess wealth orders for the marginals and canonical positive/negative stochastic dependence notions.

Portfolio risk analysis of excess of loss reinsurance

Consider a catastrophe insurance market in which primary insurers purchase excess of loss reinsurance to transfer their higher-layer losses to a reinsurer. We conduct a portfolio risk analysis for the reinsurer. In doing so, we model the losses to the primary insurers by a mixture structure, which effectively integrates three risk factors: common shock, systematic risk, and idiosyncratic risk. Assume that the reinsurer holds an initial capital Cn that is in accordance with its market size n. When expanding its business, the reinsurer needs to comply with a certain VaR-based solvency capital requirement, which determines an infimal retention level rn according to the initial capital Cn. As our main results, we find the limit of rn as n→∞ and then establish a weak convergence for the reinsurance portfolio loss. The latter result is applied to approximate the distortion risk measures of the reinsurance portfolio loss. In our numerical studies, we examine the accuracy of the obtained approximations and conduct various sensitivity tests against some risk parameters.

Bowley reinsurance with asymmetric information: a first-best solution

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 Journal 2021, 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.

Optimal reinsurance policy under a new distortion risk measure

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.

Optimal reinsurance with multiple reinsurers: Competitive pricing and coalition stability

This paper studies economic pricing of reinsurance contracts via competition of an insurer with multiple reinsurers. All firms are assumed to be endowed with distortion risk measures or expected exponential utilities. Reinsurance contracts are required to be Pareto optimal, individually rational, and satisfy a competition constraint that we call coalition stability. As shown in the literature, it holds that Pareto optimality is equivalent to a structure on the indemnities. This paper characterizes the corresponding premiums by a competition argument. The competition among reinsurers imposes constraints on the premiums that the reinsurers are able to charge and this may lead to a strictly positive profit for the insurer. When the firms use distortion risk measures, this constraint yields stability for subcoalitions, which is a condition akin to the core in cooperative game theory. The premiums and the profit of the insurer are derived in closed-form. This paper illustrates this premium function with the Mean Conditional Value-at-Risk and the GlueVaR. If the firms use expected exponential utilities, the premium is represented by an exponential premium.

Меры «VaR в степени t» и «ES в степени t» как новые подклассы мер катастрофических финансовых рисков во множестве мер риска искажения (Measures “VaR to the Extent of T” and “ES to the Extent of t” As New Subclasses of Measures of Catastrophic Financial Risks in a Variety of Measures of Risk Distortion)

Russian Abstract: Меры риска искажения в последние годы широко используются в финансовых и страховых приложениях благодаря своим привлекательным свойствам. Целью работы является исследование вопроса принадлежности мер риска «VaR в степени t», введенных в научный оборот автором к классу мер риска искажения, а также описание соответствующих функций искажения. В данной работе вводится новый класс мер риск «ES в степени t» и также исследуется вопрос о принадлежности этих мер риска, к классу мер риска искажения, а также описания соответствующих функций искажения. В работе описывается композитный метод, как метод построения новых функций искажения и соответствующих мер риска искажения. Этот метод используется для доказательства принадлежности мер риска «VaR в степени t» и «ES в степени t» к классу мер риска искажения. Представлены также различные примеры для иллюстрации соответствующих понятий и результатов, проявляющих важность мер риска «VaR в степени t» и «ES в степени t», как подмножеств мер риска искажения, позволяющих выявлять финансовые риски различной степени катастрофичности. Автор делает вывод, что меры риска «VaR в степени t» и «ES в степени t» может быть полезно в практике риск-менеджмента компаний при оценке маловероятных рисков высокой катастрофичности. English Abstract: Distortion risk measures have been widely used in financial and insurance applications in recent years due to their attractive properties. The aim of the work is to study the issue of belonging to the VaR to degree t risk measures introduced into scientific circulation by the author to the class of distortion risk measures, as well as to describe the corresponding distortion functions. This work introduces a new class of risk measures ES to degree t and also examines the issue of belonging to these risk measures, to the class of distortion risk measures, as well as descriptions of the corresponding distortion functions. The paper describes the composite method as a method for constructing new distortion functions and corresponding distortion risk measures. This method is used to prove that the risk measures VaR to degree t and ES to degree t belong to the class of distortion risk measures. Various examples are also presented to illustrate relevant concepts and outcomes that demonstrate the importance of VaR to degree t and ES to degree t risk measures as subsets of distortion risk measures that identify financial risks of varying degrees of catastrophicity. The author concludes that the risk measures VaR to the extent of t and ES to the extent of t can be useful in the practice of risk management of companies in assessing unlikely risks of high catastrophe.

Tail Distortion Risk Measure for Portfolio with Multivariate Regularly Variation

For the multiplicative background risk model, a distortion-type risk measure is used to measure the tail risk of the portfolio under a scenario probability measure with multivariate regular variation. In this paper, we investigate the tail asymptotics of the portfolio loss $$\sum _{i=1}^{d}R_iS$$ , where the stand-alone risk vector $${\mathbf {R}}=(R_1,\ldots ,R_d)$$ follows a multivariate regular variation and is independent of the background risk factor S. An explicit asymptotic formula is established for the tail distortion risk measure, and an example is given to illustrate our obtained results.

OPTIMAL REINSURANCE DESIGN WITH DISTORTION RISK MEASURES AND ASYMMETRIC INFORMATION

ABSTRACTThis paper studies a problem of optimal reinsurance design under asymmetric information. The insurer adopts distortion risk measures to quantify his/her risk position, and the reinsurer does not know the functional form of this distortion risk measure. The risk-neutral reinsurer maximizes his/her net profit subject to individual rationality and incentive compatibility constraints. The optimal reinsurance menu is succinctly derived under the assumption that one type of insurer has a larger willingness to pay than the other type of insurer for every risk. Some comparative analyses are given as illustrations when the insurer adopts the value at risk or the tail value at risk as preferences.

Computing Sensitivities for Distortion Risk Measures

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.

Extremile Regression

Regression extremiles define a least squares analogue of regression quantiles. They are determined by weighted expectations rather than tail probabilities. Of special interest is their intuitive meaning in terms of expected minima and maxima. Their use appears naturally in risk management where, in contrast to quantiles, they fulfill the coherency axiom and take the severity of tail losses into account. In addition, they are comonotonically additive and belong to both the families of spectral risk measures and concave distortion risk measures. This article provides the first detailed study exploring implications of the extremile terminology in a general setting of presence of covariates. We rely on local linear (least squares) check function minimization for estimating conditional extremiles and deriving the asymptotic normality of their estimators. We also extend extremile regression far into the tails of heavy-tailed distributions. Extrapolated estimators are constructed and their asymptotic theory is developed. Some applications to real data are provided.

A Theory for Measures of Tail Risk

The notion of “tail risk” has been a crucial consideration in modern risk management and financial regulation, as very well documented in the recent regulatory documents. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures that quantify the tail risk, that is, the behaviour 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, value at risk (VaR) and expected shortfall, 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 other than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.

Bowley reinsurance with asymmetric information on the insurer's risk preferences

ABSTRACT 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.

Preference Robust Distortion Risk Measure and Its Application

Distortion risk measure (DRM) plays a crucial role in risk measuring and managing, especially in insurance pricing. Various DRMs have been introduced but little is discussed about which DRM at hand should be chosen to address a decision maker (DM)'s risk attitude. This paper aims to fill out the gap. Specifically we consider a situation where the true distortion function is unknown either because it is difficult to identify/elicit and/or because the DM's risk attitude is ambiguous. We introduce a preference robust distortion risk measure (PRDRM) which is based on the worst-case distortion function from an ambiguity set of distortion functions to mitigate the impact arising from the ambiguity. The ambiguity set is constructed under well-known general principals such as concavity and inverse S-shapedness of distortion functions (over-weighting on events from impossible to possible or possible to certainty and under-weighting on those from possible to more possible) as well as new user-specific information such as sensitivity to tail losses, confidence intervals to some lotteries, and preferences to certain lotteries over others. To calculate the proposed PRDRM, we use the convex and/or concave envelop of a set of points to characterise the curvature of the distortion function and derive a tractable reformulation of the PRDRM when the underlying random loss is discretely distributed. Moreover, we show that the worst-case distortion function is a non-decreasing piece-wise linear function and can be determined by solving a linear programming problem. Finally, we apply the proposed PRDRM to a risk capital allocation problem and carry out some numerical tests to examine the efficiency of the PRDRM model.

An examination of the tail contribution to distortion risk measures

Extreme losses are specially relevant in the finance and insurance sectors. Here, we analyze the tail behavior of risks and its influence on risk measures. Specifically, we examine the part of the risk value of a distortion risk measure (DRM) that is attributable to extreme losses. We analyze the additive properties of tail contributions to risk values when several risks are aggregated. We show that the partial contributions are subadditive if the distortion function is concave in the tail. We examine the tail behavior for quantile-based DRMs, including value-at-risk and tail value-at-risk. We conclude that such an evaluation will allow decision makers to obtain relevant information about the contribution of extreme losses to risk values and about the fraction of the diversification benefit attributable to the tails. An example is used to illustrate our results.

Risk Measures Induced by Efficient Insurance Contracts

The Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics, which has recently been characterized via sets of axioms from perspectives of portfolio risk management and statistics. Meanwhile, there is large literature on insurance design with ES as an objective or a constraint. A visible gap is to justify the special role of ES in insurance and actuarial science. To fill this gap, we study characterization of risk measures induced by efficient insurance contracts, i.e., those that are Pareto optimal for the insured and the insurer. One of our major results is that we characterize a mixture of the mean and ES as the risk measure of the insured and the insurer, when contracts with deductibles are efficient. Characterization results of other risk measures, including the mean and distortion risk measures, are also presented by linking them to different sets of contracts.

Reverse Sensitivity Analysis for Risk Modelling

We consider the problem where a modeller conducts sensitivity analysis of a model consisting of random input factors, a corresponding random output of interest, and a baseline probability measure. The modeller seeks to understand how the model (the distribution of the input factors as well as the output) changes under a stress on the output's distribution. Specifically, for a stress on the output random variable, we derive the unique stressed distribution of the output that is closest in the Wasserstein distance to the baseline output's distribution and satisfies the stress. We further derive the stressed model, including the stressed distribution of the inputs, which can be calculated in a numerically efficient way from a set of baseline Monte Carlo samples. The proposed reverse sensitivity analysis framework is model-free and allows for stresses on the output such as (a) the mean and variance, (b) any distortion risk measure including the Value-at-Risk and Expected-Shortfall, and (c) expected utility type constraints, thus making the reverse sensitivity analysis framework suitable for risk models.

New Risk Measures “<i>VaR</i> to the Power of <i>t</i>” and “<i>ES</i> to the Power of <i>t</i>” and Distortion Risk Measures

Distortion risk measures have been popular in financial and insurance applications in recent years due to their attractive properties. The aim of the article is to investigate whether risk measures “VaR in the power of t”, introduced by the author, belong to the class of distortion risk measures, as well as to describe the corresponding distortion functions. The author introduces a new class of risk measures “ES to the power of t” and investigates whether it belongs to distortion risk measures, and also describes the corresponding distortion functions. The author used the composite method to design new distortion functions and corresponding distortion risk measures, to prove that risk measures “VaR to the power of t” and “ES to the power of t” belong to the class of distortion risk measures. The paper presents examples to illustrate the relevant concepts and results that show the importance of risk measures “VaR to the power of t” and “ES to the power of t” as subsets of distortion risk measures that allow identifying various financial catastrophic risks. The author concludes that risk measures “VaR to the power of t” and “ES to the power of t” can be used in risk management of companies when assessing remote, highly catastrophic risks.