Stochastic loss reserving: A new perspective from a Dirichlet model

Forecasting the outstanding claim liabilities to set adequate reserves is critical for a nonlife insurer's solvency. Chain–Ladder and Bornhuetter–Ferguson are two prominent actuarial approaches used for this task. The selection between the two approaches is often ad hoc due to different underlying assumptions. We introduce a Dirichlet model that provides a common statistical framework for the two approaches, with some appealing properties. Depending on the type of information available, the model inference naturally leads to either Chain–Ladder or Bornhuetter–Ferguson prediction. Using claims data on Worker's compensation insurance from several U.S. insurers, we discuss both frequentist and Bayesian inference.     

Stochastic Payments per Claim Incurred

We propose a Bayesian model to quantify the uncertainty associated with the payments per claim incurred (PPCI) algorithm. Based on the PPCI algorithm, two submodels are proposed for the number of reported claims run-off triangle and the PPCI run-off triangle, respectively. The model for the claims amount is then derived from the two submodels under the assumption of independence between the number of incurred claims and the PPCI. The joint likelihood of the number of reported claims and claims amount is derived. The posterior distribution of parameters is estimated via the Hamiltonian Monte Carlo (HMC) sampling approach. The Bayesian estimator, the process variance, the estimation variance, and the predictive distribution of unpaid claims are also studied. The proposed model and the HMC inference engine are applied to to an empirical claims dataset of the WorkSafe Victoria to estimate the unpaid claims of the doctor benefit. The Bayesian modeling procedure is further refined by including a preliminary generalized linear model analysis. The results are compared with those in a PwC report. An alternative model is compared with the proposed model based on various information criteria.     

Spatial modelling of claim frequency and claim size in non-life insurance

In this paper, models for claim frequency and average claim size in non-life insurance are considered. Both covariates and spatial random effects are included allowing the modelling of a spatial dependency pattern. We assume a Poisson model for the number of claims, while claim size is modelled using a Gamma distribution. However, in contrast to the usual compound Poisson model, we allow for dependencies between claim size and claim frequency. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC). The issue of model comparison is thoroughly addressed. Besides the deviance information criterion and the predictive model choice criterion, we suggest the use of proper scoring rules based on the posterior predictive distribution for comparing models. We give an application to a comprehensive data set from a German car insurance company. The inclusion of spatial effects significantly improves the models for both claim frequency and claim size, and also leads to more accurate predictions of the total claim sizes. Further, we detect significant dependencies between the number of claims and claim size. Both spatial and number of claims effects are interpreted and quantified from an actuarial point of view.     

Die Haftpflichtversicherung nach der VVG-Reform

The reform of the German Insurance Contract Act (Versicherungsvertragsgesetz, ?VVG“) also targets key aspects of third-party liability insurance. The changes go beyond the findings made by both the courts and legal authorities to date.Compulsory insurance aside, the law still provides that an injured third party has no standing to assert a claim directly against the tortfeasor’s liability insurer. The tortfeasor may assign its indemnity claim against the insurer solely to the injured third party and may no longer be precluded from doing so under the General Insurance Conditions (AVB). Consequently, the tortfeasor’s indemnity claim against the insurer effectively becomes a pecuniary claim. This is criticised by the insurance industry particularly with regard to eliminating the prohibition against acknowledgment and satisfaction of claims.In the future, third parties will be able to assert claims directly against the tortfeasor’s insurer and this will be the case for compulsory insurance across the board. Provisions currently in effect in the motor vehicle liability insurance industry will be carried over to the entire compulsory insurance sector. Compulsory insurance does permit agreements involving self-deductibles. However, such agreements are generally effective only as between the insurer and the tortfeasor inter se, i.e. they are not effective as against third parties — in contrast to valid disclaimers of risk.Another change in compulsory insurance is the hierarchy of claims for compensatory damages and relief in the event the insured amount is inadequate. Specifically, the hierarchy gives preference to individual claims of injured parties which are not otherwise covered, such as claims for pain and suffering.The prohibition against the retroactive loss of provisional coverage for failure to pay the first premium, which had been criticised primarily by motor vehicle liability insurers, has been omitted in the Government bill.     

An application of two-stage quantile regression to insurance ratemaking

Two-part models based on generalized linear models are widely used in insurance rate-making for predicting the expected loss. This paper explores an alternative method based on quantile regression which provides more information about the loss distribution and can be also used for insurance underwriting. Quantile regression allows estimating the aggregate claim cost quantiles of a policy given a number of covariates. To do so, a first stage is required, which involves fitting a logistic regression to estimate, for every policy, the probability of submitting at least one claim. The proposed methodology is illustrated using a portfolio of car insurance policies. This application shows that the results of the quantile regression are highly dependent on the claim probability estimates. The paper also examines an application of quantile regression to premium safety loading calculation, the so-called Quantile Premium Principle (QPP). We propose a premium calculation based on quantile regression which inherits the good properties of the quantiles. Using the same insurance portfolio data-set, we find that the QPP captures the riskiness of the policies better than the expected value premium principle.     

Longitudinal modeling of insurance claim counts using jitters

Modeling insurance claim counts is a critical component in the ratemaking process for property and casualty insurance. This article explores the usefulness of copulas to model the number of insurance claims for an individual policyholder within a longitudinal context. To address the limitations of copulas commonly attributed to multivariate discrete data, we adopt a ‘jittering’ method to the claim counts which has the effect of continuitizing the data. Elliptical copulas are proposed to accommodate the intertemporal nature of the ‘jittered’ claim counts and the unobservable subject-specific heterogeneity on the frequency of claims. Observable subject-specific effects are accounted in the model by using available covariate information through a regression model. The predictive distribution together with the corresponding credibility of claim frequency can be derived from the model for ratemaking and risk classification purposes. For empirical illustration, we analyze an unbalanced longitudinal dataset of claim counts observed from a portfolio of automobile insurance policies of a general insurer in Singapore. We further establish the validity of the calibrated copula model, and demonstrate that the copula with ‘jittering’ method outperforms standard count regression models.     

索赔特征对保险理赔(诉讼)额影响的实证分析——基于交强险人身伤害判决数据

面对日益增多的保险理赔(诉讼),亟需加强索赔原因及索赔特征对法院判决结果的影响研究.本文通过对北京市大兴区人民法院2007年1月至2010年8月涉及人身伤害的交强险判决案例进行回归分析,发现:索赔原因与索赔人损失大小、就业状况、法律规定的赔偿上限有关;索赔人在交通事故中承担的过错责任与其性别、医疗费支出状况、是否死亡有...     

Fat-Tailed Regression Modeling with Spliced Distributions

Insurance claims data usually contain a large number of zeros and exhibits fat-tail behavior. Misestimation of one end of the tail impacts the other end of the tail of the claims distribution and can affect both the adequacy of premiums and needed reserves to hold. In addition, insured policyholders in a portfolio are naturally non-homogeneous. It is an ongoing challenge for actuaries to be able to build a predictive model that will simultaneously capture these peculiar characteristics of claims data and policyholder heterogeneity. Such models can help make improved predictions and thereby ease the decision-making process. This article proposes the use of spliced regression models for fitting insurance loss data. A primary advantage of spliced distributions is their flexibility to accommodate modeling different segments of the claims distribution with different parametric models. The threshold that breaks the segments is assumed to be a parameter, and this presents an additional challenge in the estimation. Our simulation study demonstrates the effectiveness of using multistage optimization for likelihood inference and at the same time the repercussions of model misspecification. For purposes of illustration, we consider three-component spliced regression models: the first component contains zeros, the second component models the middle segment of the loss data, and the third component models the tail segment of the loss data. We calibrate these proposed models and evaluate their performance using a Singapore auto insurance claims dataset. The estimation results show that the spliced regression model performs better than the Tweedie regression model in terms of tail fitting and prediction accuracy.     

Bayesian Modelling of Outstanding Liabilities Incorporating Claim Count Uncertainty

Abstract

This paper deals with the prediction of the amount of outstanding automobile claims that an insurance company will pay in the near future. We consider various competing models using Bayesian theory and Markov chain Monte Carlo methods. Claim counts are used to add a further hierarchical stage in the model with log-normally distributed claim amounts and its corresponding state space version. This way, we incorporate information from both the outstanding claim amounts and counts data resulting in new model formulations. Implementation details and illustrations with real insurance data are provided.     

Semiparametric estimation in the optimal dividend barrier for the classical risk model

In the context of an insurance portfolio which provides dividend income for the insurance company’s shareholders, an important problem in risk theory is how the premium income will be paid to the shareholders as dividends according to a barrier strategy until the next claim occurs whenever the surplus attains the level of ‘barrier’. In this paper, we are concerned with the estimation of optimal dividend barrier, defined as the level of the barrier that maximizes the expected discounted dividends until ruin, under the widely used compound Poisson model as the aggregate claims process. We propose a semi-parametric statistical procedure for estimation of the optimal dividend barrier, which is critically needed in applications. We first construct a consistent estimator of the objective function that is complexly related to the expected discounted dividends and then the estimated optimal dividend barrier as the minimizer of the estimated objective function. In theory, we show that the constructed estimator of the optimal dividend barrier is statistically consistent. Numerical experiments by both simulated and real data analyses demonstrate that the proposed estimators work reasonably well with an appropriate size of samples.     

An Empirical Analysis of the Effects of Increasing Deductibles on Moral Hazard

Using information on timing and number of claims in a unique data set pertaining to comprehensive automobile insurance with the increasing deductible provision in Taiwan, the authors provide new evidence for moral hazard. Time-varying correlations between the choice of the insurance coverage and claim occurrence are significantly positive and exhibit a smirk pattern across policy months. This empirical finding supports the existence of asymmetric information. A subsample estimation depicts insured drivers' significant responses to increasing deductibles, which implies the existence of moral hazard. According to the probit regression results, the increasing deductible makes policyholders who have ever filed claims less likely to file additional claims later in the policy year. The empirical findings strongly support the notion that the increasing deductible provision helps control moral hazard.     

Micro-level stochastic loss reserving for general insurance

The vast literature on stochastic loss reserving concentrates on data aggregated in run-off triangles. However, a triangle is a summary of an underlying data-set with the development of individual claims. We refer to this data-set as ‘micro-level’ data. Using the framework of Position Dependent Marked Poisson Processes) and statistical tools for recurrent events, a data-set is analyzed with liability claims from a European insurance company. We use detailed information of the time of occurrence of the claim, the delay between occurrence and reporting to the insurance company, the occurrences of payments and their sizes, and the final settlement. Our specifications are (semi)parametric and our approach is likelihood based. We calibrate our model to historical data and use it to project the future development of open claims. An out-of-sample prediction exercise shows that we obtain detailed and valuable reserve calculations. For the case study developed in this paper, the micro-level model outperforms the results obtained with traditional loss reserving methods for aggregate data.     

Wird der vom Regierungsentwurf eines VVG geplante Direktanspruch des Geschädigten gegen den Pflichthaftpflichtversicherer ein stumpfes Schwert? Zur Notwendigkeit eines Informationsanspruchs des Geschädigten in der neuen Pflichthaftpflichtversicherung

The amendment of the German Act on Insurance Contracts comprises a number of substantial modifications regarding liability insurance. The most important modification is the introduction of a direct claim against the insurer with regard to all compulsory insurances. Such a direct claim is up to now only known from the motor vehicle liability insurance. Yet, a direct claim will be advantageous for the aggrieved claimant only if he manages to identify the proper insurer. As a result of the structural differences to the motor vehicle liability insurance the act of identifying the insurer is almost utterly impossible for the claimant without any help by the insured. Thus the claimant must also be given a claim against the insured to be furnished with all necessary information regarding the insurer. This right originates from sec. 242 of the German Civil Code.     

On semiparametric estimation of ruin probabilities in the classical risk model

The ruin probability of an insurance company is a central topic in risk theory. We consider the classical Poisson risk model when the claim size distribution and the Poisson arrival rate are unknown. Given a sample of inter-arrival times and corresponding claims, we propose a semiparametric estimator of the ruin probability. We establish properties of strong consistency and asymptotic normality of the estimator and study bootstrap confidence bands. Further, we present a simulation example in order to investigate the finite sample properties of the proposed estimator.     

Ex Post Moral Hazard and Bayesian Learning in Insurance

We study a dynamic insurance market with asymmetric information and ex post moral hazard. In our model, the insurance buyer's risk type is unknown to the insurer; moreover, the buyer has the option of not reporting losses. The insurer sets premia according to the buyer's experience rating, computed via Bayesian estimation based on buyer's history of reported claims. Accordingly, the buyer has strategic incentive to withhold information about losses. We construct an insurance market information equilibrium model and show that a variety of reporting strategies are possible. The results are illustrated with explicit computations in a two‐period risk‐neutral case study.     

Lognormal Mixed Models for Reported Claims Reserves

Abstract

Traditional claims-reserving techniques are based on so-called run-off triangles containing aggregate claim figures. Such a triangle provides a summary of an underlying data set with individual claim figures. This contribution explores the interpretation of the available individual data in the framework of longitudinal data analysis. Making use of the theory of linear mixed models, a flexible model for loss reserving is built. Whereas traditional claims-reserving techniques don’t lead directly to predictions for individual claims, the mixed model enables such predictions on a sound statistical basis with, for example, confidence regions. Both a likelihood-based as well as a Bayesian approach are considered. In the frequentist approach, expressions for the mean squared error of prediction of an individual claim reserve, origin year reserves, and the total reserve are derived. Using MCMC techniques, the Bayesian approach allows simulation from the complete predictive distribution of the reserves and the calculation of various risk measures. The paper ends with an illustration of the suggested techniques on a data set from practice, consisting of Belgian automotive third-party liability claims. The results for the mixed-model analysis are compared with those obtained from traditional claims-reserving techniques for run-off triangles. For the data under consideration, the lognormal mixed model fits the observed individual data well. It leads to individual predictions comparable to those obtained by applying chain-ladder development factors to individual data. Concerning the predictive power on the aggregate level, the mixed model leads to reasonable predictions and performs comparable to and often better than the stochastic chain ladder for aggregate data.     

Medical claims review in an emerging market

Traditionally, insurance medicine has made significant contributions to sound underwriting and to the rise of insurer's margin of risk ratio. However, it is may be less effective in emerging markets, because emerging markets have few clinical follow-up studies, and there is a lack of medicoactuarial research. Making a medical expert's opinion to insurance claim administration is called a medical claims review. It is subdivided into medical verification and advice for claims staff. In addition to medical risk selection, it is a new field of insurance medicine in emerging insurance markets, which are often characterized by assertive insurance consumers. Medical contributions at the stage of claims adjudication compare the coverage provided in the product, with the information provided in the claims, based on medical records and the agreement between them. This is called medical verification. The insurance doctors can also use their medical knowledge to help the claim staff with informing claimants about the medical basis of claim decisions.     

Securitization with recourse: An instrument that offers uninsured bank depositors sequential claims

This paper expands on the observation that securitization with recourse offers a lender the protections of an uninsured bank deposit with an additional senior claim if the bank fails. The senior claim derives from the commitment of the revenues from a group of securitized assets that pay off the securitized lender first. Thus, securitization with recourse provides sequential claims for bank liability holders, which improves the allocation of risk sharing among them. Securitization with recourse may also improve the selection of loans granted by partially offsetting the moral hazard incentives toward risk-taking created by fixed-rate deposit insurance. Empirical results are given which are consistent with the theoretical model.     

我国机动车辆保险代位追偿清算模型研究

近期,我国机动车辆保险理赔中的"无责免赔"问题引起了社会广泛关注,随之而来的代位追偿案件也将引发保险公司之间的账务清算支付,但我国目前尚未建立行业统一的机动车辆保险清算系统。本文说明了我国车险业务快速发展过程中消费者理赔问题,分析了代位追偿及建设我国行业清算体系的必要性和重要意义,借鉴国内外金融业清算体系的成熟案例,提出了构建我国机动车辆保险代位追偿清算模型,为提高行业效率、优化保险公司之间的清算流程提供有力保障。     

Claims‐Made and Reported Policies and Insurer Profitability in Medical Malpractice

The liability crisis of the 1970s led to the introduction of a new type of insurance policy designed to reduce the undiversifiable uncertainty associated with writing long‐tail liability lines. These new claims‐made and reported policies gained favor in place of the traditional occurrence coverage in the early 1980s not only in medical malpractice but also in the general liability arena. The main question we want to address in this article is why two types of contracts that cover the same risk exposure exist in the medical malpractice insurance industry whereas only one exists primarily in other insurance lines.