Credibility Premium Research Articles

The credibility models with equal correlation risks

In almost all credibility estimators considered previously, the claims are assumed to be independent over risks. However, from practical point of view, this paper investigates the credibility premium when risks are dependent on each other. The authors build the Buhlmann’s credibility models to account for an equal correlation structure over risks, and parameter estimations are also considered. Some desired results are derived.

Some Bayesian credibility premiums obtained by using posterior regret {$\Gamma$}-minimax methodology

In credibility theory, the premium charged to a policyholder is com- puted on the basis of his/her own past claims and the accumulated past claims of the corresponding portfolio of policyholders. In order to obtain an appropriate formula for this, various methodologies have been proposed in actuarial literature, most of them in the fleld of Bayesian decision methodology. In this paper, following the robust Bayesian paradigm, a procedure based on the posterior regret i-minimax principle is applied to derive, in a straightforward way, new credibility formula, making use of simple classes of distributions. This methodology is applied to the most commonly used premium calculation principles in insurance, namely the net, Esscher and variance principles.

The credibility premiums under generalized weighted loss functions

In the classical credibility theory, almost all the credibilitypremium models are built on the basis of pure premium. However, theinsurance practice demands that the premium must have a positivesafety loading. In this paper, we consider the premium principleinduced by a generalized loss function that can provide the premiumprinciple with\ a positive safety loading. Under this generalizedloss function, we derive its Bayes premium and two types ofcredibility premiums. Both credibility premiums are approximatelyconvex combinations of the collective premium and some functions ofthe historical claims; while in a first case the function is linearin the historical claims and the corresponding credibility premiumis not consistency, in the other one the function is taken as anempirical version of the individual premium and the correspondingcredibility premium converges to the individual premium.

The credibility premiums for models with dependence induced by common effects

In classical Bühlmann credibility models, claims are assumed to be independent between different risks. In many practical situations, however, this assumption may be violated because there are situations that could drive possible relationship among the insured individuals. This paper aims to extend the Bühlmann and Bühlmann–Straub credibility models to account for a special type of dependence between risks induced by common stochastic effects. By means of the projection method, the corresponding credibility premiums are obtained, which generalize some well known existing results in credibility theory.

A generalization of the credibility theory obtained by using the weighted balanced loss function

In this paper an alternative to the usual credibility premium that arises for weighted balanced loss function is considered. This is a generalized loss function which includes as a particular case the weighted quadratic loss function traditionally used in actuarial science. From this function credibility premiums under appropriate likelihood and priors can be derived. By using weighted balanced loss function we obtain, first, generalized credibility premiums that contain as particular cases other credibility premiums in the literature and second, a generalization of the well-known distribution free approach in [Bühlmann, H., 1967. Experience rating and credibility. Astin Bull. 4 (3), 199–207].

Credibility premiums for the zero-inflated Poisson model and new hunger for bonus interpretation

The purpose of this paper is to explore and compare the credibility premiums in generalized zero-inflated count models for panel data. Predictive premiums based on quadratic loss and exponential loss are derived. It is shown that the credibility premiums of the zero-inflated model allow for more flexibility in the prediction. Indeed, the future premiums not only depend on the number of past claims, but also on the number of insured periods with at least one claim. The model also offers another way of analysing the hunger for bonus phenomenon. The accident distribution is obtained from the zero-inflated distribution used to model the claims distribution, which can in turn be used to evaluate the impact of various credibility premiums on the reported accident distribution. This way of analysing the claims data gives another point of view on the research conducted on the development of statistical models for predicting accidents. A numerical illustration supports this discussion.

Robust and Efficient Methods for Credibility When Claims Are Approximately Gamma-Distributed

As is well known in actuarial practice, excess claims (outliers) have a disturbing effect on the ratemaking process. To obtain better estimators of premiums, which are based on credibility theory, Künsch and Gisler and Reinhard suggested using robust methods. The estimators proposed by these authors are indeed resistant to outliers and serve as an excellent example of how useful robust models can be for insurance pricing. In this article we further refine these procedures by reducing the degree of heuristic arguments they involve. Specifically we develop a class of robust estimators for the credibility premium when claims are approximately gamma-distributed and thoroughly study their robustness-efficiency trade-offs in large and small samples. Under specific datagenerating scenarios, this approach yields quantitative indices of estimators’ strength and weakness, and it allows the actuary (who is typically equipped with information beyond the statistical model) to choose a procedure from a full menu of possibilities. Practical performance of our methods is illustrated under several simulated scenarios and by employing expert judgment.

Robust regression credibility: The influence function approach

In classical credibility theory we assume that the vector of claims X j conditionally on Θ j has independent components with identical means. However, this assumption is sometimes unrealistic. To relax this condition Hachemeister (Hachemeister, C.A., 1975. Credibility for regression models with application to trend. In: Kahn, P. (Ed.), Credibility, Theory and Applications. Academic Press, New York) introduced regressors. The presence of large claims can perturb the credibility premium estimation. The lack of robustness of regression credibility estimators, as well as the fairness of tariff evaluation, led to the development of this paper. Our proposal is to apply robust statistics to the regression credibility estimation by using the robust influence function approach of M-estimators.

On the consistency of credibility premiums regarding Esscher principle

In this paper, we investigate the problems of convergence of experience-based ratemakings regarding the Esscher principle. In addition to the Bayes and the classical credibility premiums, we suggest a new credibility formula for the Esscher premium. Then we show the convergence of the Bayes and the newly defined credibility premiums towards the individual premium and point out that the classical credibility premium does not generally converge to the individual premium by presenting a sufficient and necessary condition under which the classical credibility Esscher premium converges to the individual premium. A simulation study is carried out to illustrate the theoretical conclusions.

On Bayesian Mixture Credibility

We introduce a class of Bayesian infinite mixture models first introduced by Lo (1984) to determine the credibility premium for a non-homogeneous insurance portfolio. The Bayesian infinite mixture models provide us with much flexibility in the specification of the claim distribution. We employ the sampling scheme based on a weighted Chinese restaurant process introduced in Lo et al. (1996) to estimate a Bayesian infinite mixture model from the claim data. The Bayesian sampling scheme also provides a systematic way to cluster the claim data. This can provide some insights into the risk characteristics of the policyholders. The estimated credibility premium from the Bayesian infinite mixture model can be written as a linear combination of the prior estimate and the sample mean of the claim data. Estimation results for the Bayesian mixture credibility premiums will be presented.

On Bayesian Mixture Credibility

We introduce a class of Bayesian infinite mixture models first introduced by Lo (1984) to determine the credibility premium for a non-homogeneous insurance portfolio. The Bayesian infinite mixture models provide us with much flexibility in the specification of the claim distribution. We employ the sampling scheme based on a weighted Chinese restaurant process introduced in Lo et al. (1996) to estimate a Bayesian infinite mixture model from the claim data. The Bayesian sampling scheme also provides a systematic way to cluster the claim data. This can provide some insights into the risk characteristics of the policyholders. The estimated credibility premium from the Bayesian infinite mixture model can be written as a linear combination of the prior estimate and the sample mean of the claim data. Estimation results for the Bayesian mixture credibility premiums will be presented.

Claim dependence with common effects in credibility models

Several credibility models found in published literature have largely been single dimensional in the sense that the observable claims are derived from a single individual risk or a single group of homogeneous risks over a period of time. In the case where the additional dimension of observing different individual risks or different groups of risks are allowed for, the assumption of independence across the observable claims is often made. This is a matter of convenience and mathematical tractability, though in general, everyone agrees this may seem unrealistic. As such, dependence must be taken into account when modelling risks for assessing credibility premiums. In this paper, we introduce the notion of modelling claim dependence across individuals and simultaneously across time within individuals using common effects. The resulting model is then used to predict expected claims given the history of all observable claims. It is well-known that this conditional expectation actually gives the best predictor of the next period claim for a single individual in the mean-squared error sense. We express this conditional expectation in the form of a credibility premium. We further give illustrative examples to demonstrate the ideas.

An Empirical Assessment of the Credibility Premium Associated with Meeting or Beating Both Time-Series Earnings Expectations and Analysts' Forecasts

Recent research results indicate a market premium for firms that met or beat analysts' forecasts. We find evidence consistent with these results. More important, however, we find a market premium for firms that met or beat time-series forecasts, and also the highest market premium for firms that met or beat both analysts' and time-series forecasts, relative to firms that met or beat one or neither forecast. In fact, there is no premium for firms that met or beat only analysts' or only time-series forecasts. Investors seem to consider both analysts' and time-series forecasts jointly, with the act of meeting or beating both forecasts providing the most credible signal of superior future financial performance. This 'credibility' premium to firms that met or beat time-series forecasts (as proxied for by the Foster model, 1977) is in addition to the premium for meeting or beating analysts' forecasts. The premium is supported by assessments of future financial performance over the two subsequent years and by tests of the predictability of earnings for the following quarter. Finally, we find that abnormal trading was greatest when both forecasts were met or beaten, compared to when only one or the other or neither forecast was met or beaten. This is further evidence that investors view the meeting or beating of both forecasts as the strongest signal of enhanced credibility in reported earnings.

Ethics, Earnings, and Equity Valuation

Business ethics has been much in the news since the bursting of the bubble in 2000—but that ethics can have a direct bearing on corporate earnings and on valuation receives too little attention. Many observers draw comfort from the crackdown on fraudulent accounting, although dubious corporate accounting is but the tip of the iceberg. More benign (or rather more legal) forms of aggressive accounting probably boost U.S. corporate earnings by at least 20% even now. This means that valuation levels are higher than they seem by a like margin. A more subtle but pernicious impact of aggressive accounting affects the market indirectly—if people don9t trust the numbers, they may price assets to reflect both a risk premium, reflecting objective risks in the markets, and a credibility premium, reflecting lingering investor concerns about the merits of the data that form the basis for valuation.

On the stochastic increasingness of future claims in the Bühlmann linear credibility premium

Since Nelder & Verrall (1997), the connection between Generalized Linear Models (GLM’s) and credibility theory has been recognized in actuarial science. Specifically, the credibility construction amounts to add a random effect on the same scale as the fixed effects to model unexplained heterogeneity. The present paper aims to examine the dependence existing between future claims (severity or frequency components) and the Buhlmann linear credibility premium. As expected, future claims are shown to increase with the amount of Buhlmann premium.

An application of nonparametric regression estimation in credibility theory

In this paper, we use the nonparametric regression method to establish estimators for credibility premiums under some principles of premium calculation. The asymptotic properties of the estimators are studied.

Equity and Exact Credibility

AbstractWe consider an alternative to the usual credibility premium that arises from squared-error loss, namely, a so-called equitable credibility premium (Promislow and Young, 1999). We derive formulas for the credibility weight in certain cases and give sufficient conditions for exact credibility.

233010 (M31, M10) Risk-adjusted credibility premiums using distorted probabilities: Wang S.S., Young V.R.,Scandinavian Actuarial Journal, 1998, No.2

233010 (M31, M10) Risk-adjusted credibility premiums using distorted probabilities: Wang S.S., Young V.R.,Scandinavian Actuarial Journal, 1998, No.2

Risk-adjusted credibility premiums using distorted probabilities

Denneberg (1990) and Wang (1996a) propose that one calculate risk-adjusted insurance premiums as the expectation with respect to a distorted probability measure, a non-additive set function. This premium principle is supported by the theories of decision making of Yaari (1987) and of Schmeidler (1989). Denneberg (1994a) presents three conditioning rules for updating non-additive set functions in light of available information. In this work, we show how to apply these three update rules to calculate a risk-adjusted credibility premium and, thereby, combine credibility theory with this relatively new premium principle. Our main result is that, for some pairs of distortion function and update rule, one gets the same risk-adjusted credibility premium by distorting the predictive probability distribution, as required by the theory of Yaari, or by updating the distorted probability, as required by the theory of Schmeidler.

Experience rating under weighted squared error loss

In the classical model of experience rating, optimum premiums are obtained by minimizing squared error loss over suitable classes of premiums containing the constants. As a result of this particular choice of the loss function, the expected values of the Bayes and credibility premiums agree with the net premium.