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.     

The Cost of Pension Insurance

This article estimates the cost of the federal pension insurance program. Pension insurance claims have an important market‐risk component, which means that the cost of the exposure cannot be estimated by discounting future claims by the risk‐free rate. Moreover, owing to the complexity of the insurance contract, its price cannot be estimated with known options formulas without introducing an error of nonquantifiable magnitude. To circumvent these problems, we model the insurance program in its full complexity and use a Monte Carlo method. By hedging the exposure with a dynamic premium policy that offloads the market risk to the insureds, one can calculate the risk‐free, or actuarial, cost of that policy. One can also characterize the nature of the subsidy and its structure across insured plans. Finally, we provide an estimate of the implicit cost of the hedge function that taxpayers currently are providing for zero remuneration. The model shows that simple contingent claims models of pension insurance result in a price that is about triple the true market cost of the insurance, and that pension insurance models that ignore market risk understate the cost by half. The solution demonstrates the broad characteristics that might characterize a credible private‐sector version of pension insurance.     

Dynamic Spanning in the Consumption-Based Capital Asset Pricing Model

Under the assumptions of the Consumption-based Capital Asset Pricing Model (CCAPM), Pareto optimal consumption allocations are characterized by each agent's consumption process being adapted to the filtration generated by the aggregate consumption process of the economy. The wealth processes of the agents, however, are adapted to the finer filtration generated by aggregate consumption and the conditional distribution of future aggregate consumption. Therefore, in order to achieve Pareto optimal consumption allocations, a sufficiently varied set of assets must exist such that any wealth process adapted to this finer filtration can be implemented by dynamically trading in that set of assets. We provide sufficient conditions for the existence of such a set of assets based on dynamically trading contingent claims on aggregate consumption. In addition, we give sufficient conditions for the existence of equilibria in a dynamically effectively complete market in which agents are only able to trade in contingent claims on aggregate consumption, the market portfolio of firms, and a (numeraire) zero-coupon bond. We demonstrate the role of short- and long-term contingent claims on aggregate consumption for the implementation of Pareto optimal allocations in the presence of short- andlong-term risks. In addition, in the presence of personal risks, we demonstrate the role of insurance contracts.     

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.     

Rechtsnatur und Zuordnung der Alterungsrückstellungen in der PKV

Discussions about the inclusion of the previously privately insured into the statutory government health insurance abound. Specifically, questions regarding the transfer of specific rights under the private contract persist. In particular this article addresses the ageing reserve into which the insured contributed as part of their original policies. Since contractual claims are protected by fundamental property rights, it was first investigated whether the ageing reserve would allow the insured to claim a reimbursement. However, this disbursement cannot be supported as long as the ageing reserve is still being accrued and has not yet been dissolved since the monies paid into the reserve cannot be traced to their individual sources. The legal principles of trust (Treuhand) and reversion (Anwartschaftsrecht) are also not applicable to this situation as they are based on different factual premises. While the expectation of a constant premium throughout old age is not protected by German Basic Constitutional Law, the insured’s right to expect a continuance of the contract is protected; in addition, required services and remuneration in case of a claim are guaranteed by article 14.     

Grenzen der Auskunfts- und Belegpflicht in der D&O Versicherung

A large number of claims brought under German D&O insurance regard Insured vs. Insured cases, i.e. claims brought by the company against its own directors and officers (Executive Directors, Supervisory Board Members etc.). After notification of and examination by the insurer of such an insured event, the insurer will in most cases opt to grant the insured defence cover in order to fight off the claim. The insurer hereby expresses that it regards the claim of the company (= its own policyholder) against the board member (= the insured) to be without merit. This situation—where the policyholder is at the same time the damaged party—though typical under (German) D&O-policies is uncommon for liability insurance cover in general. It, thus, raises the issue as to the limits of the policyholder’s duty to disclose information. The scope of said obligation is not unlimited. It rather has to be ascertained pursuant to Sect. 31 VVG (German Insurance Contract Act), by taking into account the policyholder’s interests in commercial and industrial confidentiality and the burden of poof as provided by Sect. 93 para. 2 AktG (German Stock Companies Act). In case legal proceedings ensue between the company and the insured, and, as a consequence, the insurer exercises its obligation to conduct the case for the insured or the insurer joins the lawsuit on the side of the insured (by declaring a Third Party Notice [Streiverkündung]), the insurer clearly becomes an adversary to the company. Under such circumstances, the company is irrevocably released from its duty to disclose information.     

Comparisons of aggregate claim numbers and amounts: a study of heterogeneity

In this paper, we investigate how the heterogeneity among occurrence probabilities and claim severities affects the aggregate claim numbers and aggregate claim amount for an insurance portfolio. We show that higher heterogeneity (and dependence) among occurrence probabilities results in both smaller aggregate claim numbers and aggregate claim amount in the sense of the mean residual lifetime order. We also prove that as the heterogeneity among the claims increases, the aggregate claim amount increases in the sense of the usual stochastic order when the vector of occurrence probabilities is left tail weakly stochastic arrangement increasing. These theoretical findings are applied to (i) study ordering properties of convolutions of binomial random variables, (ii) provide upper bounds for the mean residual lifetime functions of the aggregate claim numbers and amount, and (iii) compare stop-loss premiums and risk capital of different insurance portfolios.     

Expiration Dates in Automobile Insurance Contracts: The Curious Case of Last Policy Month Claims in Taiwan

This paper investigates a widespread trend in the Taiwanese automobile insurance market in which the loss claims of vehicle damage insurance contracts have a high propensity to occur just before the end of the policy year (as opposed to calendar year). We show that certain uncommon characteristics of claim data are consistently observed in the last policy month. We indirectly show that there is a severe time-varying excess claim problem in this market. The major sources of excess claims can be explained by the bonus-malus system problem and the auto-dealer incentive issue.     

Peer-to-Peer-Absicherung privater Risiken: Eine Exploration am Beispiel von Wildschaden

This paper examines the extent to which business model development is an effective method of surmounting market failure in the insurance industry. In particular, it endeavours to demonstrate that replacing the traditional relationship between the insured and the insurer with a bilateral agreement opens up new markets, thus enabling hitherto non-insurable risks to be covered by new peer-to-peer business models. The insurance against risks caused by wild animals was chosen as the empirical field, since it is known to be an area which exhibits market failure. Based on 16 episodic interviews with representatives of the hunting community, demand structures and relevant contextual factors are revealed and analysed in terms of risk coverage and claims management among community members. This paper thus proposes an alternative position to that of the traditional insurance business, in which cover is based on a large and diversified risk group.     

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.     

Spatial statistical modelling of insurance risk: a spatial epidemiological approach to car insurance

Spatial models, such as the Besag, York and Mollie (BYM) model, have long been used in epidemiology and disease mapping. A common research question in these subjects is modelling the number of disease events per region; here the BYM models provides a holistic framework for both covariates and dependencies between regions. We use these tools to assess the relative insurance risk associated with the policyholders geographical location. A Bayesian modelling approach is presented and an elastic net is used to reduce the large number of possible geographic covariates. The final inference is performed using Integrated Nested Laplace Approximation. The model is applied to car insurance data from If P&C Insurance together with spatially referenced covariate data of high resolution, provided by Insightone. The entire analysis is performed using freely available R-packages. Including spatial dependence when modelling the number of claims significantly improves on the result obtained using ordinary generalised linear models. However, the support for adding a spatial component to the model for claims cost is weaker.     

Credibility Using Copulas

Abstract

Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a “cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes.

This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross section of risk classes (towns) with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time; specifically, this paper is the first to propose using a t-copula in the context of generalized linear models. The t-copula is the copula associated with the multivariate t-distribution; like the univariate tdistributions, it seems especially suitable for empirical work. Moreover, we show that the t-copula gives rise to easily computable predictive distributions that we use to generate credibility predictors. Like Bayesian methods, our copula credibility prediction methods allow us to provide an entire distribution of predicted claims, not just a point prediction.

We present an illustrative example of Massachusetts automobile claims, and compare our new credibility estimates with those currently existing in the literature.     

Assessing Consumer Fraud Risk in Insurance Claims

Abstract

We present an unsupervised learning method for classifying consumer insurance claims according to their suspiciousness of fraud versus nonfraud. The predictor variables contained within a claim file that are used in this analysis can be binary, ordinal categorical, or continuous variates. They are constructed such that the ordinal position of the response to the predictor variable bears a monotonic relationship with the fraud suspicion of the claim. Thus, although no individual variable is of itself assumed to be determinative of fraud, each of the individual variables gives a “hint” or indication as to the suspiciousness of fraud for the overall claim file. The presented method statistically concatenates the totality of these “hints” to make an overall assessment of the ranking of fraud risk for the claim files without using any a priori fraud-classified or -labeled subset of data. We first present a scoring method for the predictor variables that puts all the variables (whether binary “red flag indicators,” ordinal categorical variables with different categories of possible response values, or continuous variables) onto a common –1 to 1 scale for comparison and further use. This allows us to aggregate variables with disparate numbers of potential values. We next show how to concatenate the individual variables and obtain a measure of variable worth for fraud detection, and then how to obtain an overall holistic claim file suspicion value capable of being used to rank the claim files for determining which claims to pay and the order in which to investigate claims further for fraud. The proposed method provides three useful outputs not usually available with other unsupervised methods: (1) an ordinal measure of overall claim file fraud suspicion level, (2) a measure of the importance of each individual predictor variable in determining the overall suspicion levels of claims, and (3) a classification function capable of being applied to existing claims as well as new incoming claims. The overall claim file score is also available to be correlated with exogenous variables such as claimant demographics or highvolume physician or lawyer involvement. We illustrate that the incorporation of continuous variables in their continuous form helps classification and that the method has internal and external validity via empirical analysis of real data sets. A detailed application to automobile bodily injury fraud detection is presented.     

Die Ermittlung von Risikozuschlägen bei der Zeitwertbewertung von Verpflichtungen aus Schadenversicherungsgeschäften

The IASB proposes fair value accounting of insurance liabilities in the new IFRS on insurance contracts. These liabilities are not systematically traded in markets. Therefore the estimation of a fair value is only possible by simulating a market transaction. This simulation can be carried out by using financial models like the Capital Asset Pricing Model and the Economic Capital Model. In order to determine the fair value it has to be tested if those models can realistically calculate the insurance risk of the liabilities. This includes analysing the nature and extent of risk measurement as well as the assumptions the models are based on. The particular problem of the Capital Asset Pricing Model consists in measuring the risk by betas. An insurance beta can only be determined by relating it to other directly measurable betas. Those relationships can only be developed by putting forward special assumptions which increases the likelihood of a subjective valuation. The Economic Capital Model on the opposite is able to measure the insurance risk. The analysis of the models is carried out under simplified assumptions. Therefore it remains to be proven that the Economic Capital Model can also handle a more specific view of the insurance risk.     

The Incentive Effects of Increasing Per-Claim Deductible Contracts in Automobile Insurance

A new rating system of automobile insurance for vehicle damage in Taiwan was launched in 1996, introducing a deductible that increases with the number of claims. In this article, we provide a theoretical rationale for the existence of an increasing per‐claim deductible system and show that the new system is most likely an optimal choice for those insured who tend to have lower claims probability when incentives are present. Using a unique dynamic data set, we are able to conduct a natural experiment to examine the incentive effects (both positive and negative) by looking at the change in claim tendency before and after switching between two deductible plans: an increasing per‐claim deductible and a zero deductible. Our results provide direct evidence of the effects of deductible structures on claim behavior.     

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.     

新保险法热点问题之中港比较

比较大陆和香港两地保险实践,会发现许多热点问题在大陆地区和香港地区之法律规制仍存在较大差异.伴随着大陆地区保险市场的逐步成熟,大陆保险法律体系的完善亦可借鉴香港地区相关理念和做法,充分发挥行业自律的重要作用,并逐步推进以市场行为为中心的监管模式向以偿付能力为中心的监管模式转变.     

Lebensversicherung — stille Reserven — Überschussbeteiligung

Hidden reserves play an important role in life insurance policies; the values of the bonus as well as the value of the final bonus are dependent on them. The article points out that insured people in the German system do not have a claim in respect of hidden reserves. The authors of the article also suggest a resolve may be sought by reflecting on the ?fair market value“ idea; an idea originating from the latest suggestions in Brussels (Regulation on the application of IAS)     

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.