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Multi-Event Bonus-Malus Scales
Abstract
This article is devoted to the design of bonus-malus scales involving different types of claims. Typically, claims with or without bodily injuries, or claims with full or partial liability of the insured driver, are distinguished and entail different penalties. Under mild assumptions, claim severities can also be taken into account in this way. Numerical illustrations enhance the interest of the approach. Copyright The Journal of Risk and Insurance, 2006.
... To eliminate the first drawback, a few other approaches have been proposed. Bonus-malus systems involving different claim types are designed in [18]. Each claim type induces a specific penalty for the policyholder. ...
... We try to eliminate both drawbacks mentioned above. To take into account claim amounts, we consider different claim types and use the multi-event bonus-malus systems introduced in [18]. To eliminate the second drawback, we introduce varying deductibles for the policyholders who are in the malus zone. ...
... The rest of the paper is organized as follows. In Section 2, we describe the multievent bonus-malus systems introduced in [18]. In Section 3, we introduce varying deductibles in such bonus-malus systems and study basic properties of such systems. ...
The paper deals with bonus-malus systems with different claim types and varying deductibles. The premium relativities are softened for the policyholders who are in the malus zone and these policyholders are subject to per claim deductibles depending on their levels in the bonus-malus scale and the types of the reported claims. We introduce such bonus-malus systems and study their basic properties. In particular, we investigate when it is possible to introduce varying deductibles, what restrictions we have and how we can do this. Moreover, we deal with the special case where varying deductibles are applied to the claims reported by policyholders occupying the highest level in the bonus-malus scale and consider two allocation principles for the deductibles. Finally, numerical illustrations are presented.
... Por lo tanto, el sistema de BM "afina" o ajusta la clasificación tarifaria efectuada a priori. Es decir, el sistema de BM redefine a posteriori la tarificación efectuada a priori (Dionne y Ghali, 2005;Pitrebois et al., 2006). Esta asignación es esencial desde el punto de vista financiero, porque si los asegurados con más riesgo están mal clasificados en la escala del BM, la compañía podría incurrir en costes y riesgos muy altos. ...
... Sin embargo, hay otra serie de factores "no observables" que podrían ajustar dicha tarificación y que las compañías de seguros no miden directamente. Teóricamente se supone que dichas variables deberían estar reflejadas en el BM, ya que como hemos dicho anteriormente los "factores escondidos" son revelados, en parte, por el número de siniestros acreditados por los asegurados (Pitrebois et al., 2006) y la prima puede reajustarse de acuerdo con el número de siniestros de los que han informado los asegurados. Esto se hace normalmente integrando el historial relativo a los siniestros pasados en el denominado "sistema bonus-malus". ...
La gestión de riesgos, asociada al seguro del automóvil, es una cuestión crucial a la que se enfrentan en la actualidad tanto actuarios como profesionales del sector. Es clave seleccionar
adecuadamente los factores de riesgos para asignar las tarifas a los asegurados en función del riesgo asociado. Por tanto, el objetivo de este trabajo es comprobar empíricamente la validez de la utilización de los niveles de “bonus-malus” para clasificar adecuadamente a los asegurados a través de dos modelos de ecuaciones estructurales. Los análisis sobre una muestra de 4.365 pólizas automovilísticas españolas descritas a través de 11 factores de riesgo muestran que la variable BM contribuye a mejorar la capacidad explicativa del modelo pero no de manera significativa.
In this paper we consider a discrete-time risk model, which allows the premium to be adjusted according to claims experience. This model is inspired by the well-known bonus-malus system in the non-life insurance industry. Two strategies of adjusting periodic premiums are considered: aggregate claims or claim frequency. Recursive formulae are derived to compute the finite-time ruin probabilities, and Lundberg-type upper bounds are also derived to evaluate the ultimate-time ruin probabilities. In addition, we extend the risk model by considering an external Markovian environment in which the claims distributions are governed by an external Markov process so that the periodic premium adjustments vary when the external environment state changes. We then study the joint distribution of premium level and environment state at ruin given ruin occurs. Two numerical examples are provided at the end of this paper to illustrate the impact of the initial external environment state, the initial premium level and the initial surplus on the ruin probability.
Using a unique data set that rates a cohort of drivers on two distinct experience-rating mechanisms, we examine the impact of these experience-rating mechanism on premiums charged to low-risk, high-risk, and novice drivers. The first mechanism is a pure no-claims discount that classifies drivers by 0 to 6 or more years of at-fault claims-free driving. The second is a bonus-malus system with 32 driving record (DR) classes. Using standard stochastic modeling techniques, we find that having a greater number of DR classes does not result in lower prices for the new drivers nor does it create significant savings to the lowest risk drivers. And, not surprisingly, with an increasing number of malus DR classes, the highest risk insureds pay a substantially higher premium that may not be sustainable over time. Due to its mandatory nature in most countries, monitoring the fairness of the classification mechanism and the affordability and availability of auto insurance are common regulatory goals. Charging novice drivers excessively high premiums may conflict with affordability and fairness objectives. Thus, we conclude with an examination of other pricing mechanisms that can be used for novice drivers.
Objective:
The objective of this paper is to test the validity of using ‘bonus-malus’ (BM) levels to classify policyholders satisfactorily.
Design/methodology/approach:
In order to achieve the proposed objective and to show empirical evidence, an artificial intelligence method, Rough Set theory, has been employed.
Findings:
The empirical evidence shows that common risk factors employed by insurance companies are good explanatory variables for classifying car policyholders’ policies. In addition, the BM level variable slightly increases the explanatory power of the a priori risks factors.
Practical implications:
To increase the prediction capacity of BM level, psychological questionnaires could be used to measure policyholders’ hidden characteristics.
Contributions:
The main contribution is that the methodology used to carry out research, the Rough Set Theory, has not been applied to this problem.
There are a wide range of variables for actuaries to consider when calculating a motorist's insurance premium, such as age, gender and type of vehicle. Further to these factors, motorists' rates are subject to experience rating systems, including credibility mechanisms and Bonus Malus systems (BMSs). Actuarial Modelling of Claim Counts presents a comprehensive treatment of the various experience rating systems and their relationships with risk classification. The authors summarize the most recent developments in the field, presenting ratemaking systems, whilst taking into account exogenous information. The text: Offers the first self-contained, practical approach to a priori and a posteriori ratemaking in motor insurance. Discusses the issues of claim frequency and claim severity, multi-event systems, and the combinations of deductibles and BMSs. Introduces recent developments in actuarial science and exploits the generalised linear model and generalised linear mixed model to achieve risk classification. Presents credibility mechanisms as refinements of commercial BMSs. Provides practical applications with real data sets processed with SAS software. Actuarial Modelling of Claim Counts is essential reading for students in actuarial science, as well as practicing and academic actuaries. It is also ideally suited for professionals involved in the insurance industry, applied mathematicians, quantitative economists, financial engineers and statisticians.
One of the pricing strategies for Bonus-Malus (BM) systems relies on the decomposition of the claims' randomness, namely, one part accounting for claims' frequency and the other part for claims' severity. For a Negative Binomial number of claims, we focused on the modelling of claim severities using a hybrid structure. While Weibull distribution is suggested for constructing smaller-sized claims, Pareto severities employed by (1) were applied to model large ones. A generalised function for the BM premium rates is then derived. Results are illustrated by a numerical example.
A bonus-malus system calculates the premiums for car insurance based on the previous claim experience (class). In this paper, we propose a model that allows dependence between the claim frequency and the class occupied by the insured using a copula function. It also takes into account zero-excess phenomenon. The maximum likelihood method is employed to estimate the model parameters. A small simulation is performed to illustrate the proposed model and method.
Purpose – The purpose of this paper is to transfer the concept of market-consistent embedded value (MCEV) from life to non-life insurance. This is an important undertaking since differences in management techniques between life and non-life insurance make management at the group level very difficult. The purpose of this paper is to offer a solution to this problem. Design/methodology/approach – After explaining MCEV, the authors derive differences between life and non-life insurance and develop a MCEV model for non-life business. The model framework is applied to a German non-life insurance company to illustrate its usefulness in different applications. Findings – The authors show an MCEV calculation based on empirical data and set up an economic balance sheet. The value implications of varying loss ratios, cancellation rates, and costs within a sensitivity analysis are analyzed. The usefulness of the model is illustrated within a value-added analysis. The authors also embed the MCEV concept in a simplified model for an insurance group, to derive group MCEV and outline differences between local GAAP, IFRS and MCEV. Practical implications – The analysis provides new and relevant information to the stakeholders of an insurance company. The model provides information comparable to that provided by embedded value models currently used in the life insurance industry and fills a gap in the literature. The authors reveal significant valuation difference between MCEV and IFRS and argue that there is a need for a consistent MCEV approach at the insurance-group level. Originality/value – The paper presents a new valuation technique for non-life insurance that is easy to use, simple to interpret, and directly comparable to life insurance. Despite the growing policy interest in embedded value, not much academic attention has been given to this methodology. The authors hope that this work will encourage further discussion on this topic in academia and practice.
This paper presents statistical models which lead to experience rating in insurance. Serial correlation for risk variables
can receive endogeneous or exogeneous explanations. The paper recalls that the main interpretation for automobile insurance
is exogeneous, since positive contagion is always observed for the number of claims reported and since true contagion should
be negative. This positive contagion can be explained by the revelation throughout time of a hidden features in the risk distributions.
These features are represented by heterogeneity components in a heterogeneous model. Prediction on longitudinal data can be
performed through the heterogeneous model, and the paper provides consistent estimators for models related to number and cost
of claims. Examples are given for count data models with a constant or time-varying heterogeneity components, one or several
equations, and for a cost-number model on events. Empirical results are presented, which are drawn from the analysis of a
French data base of automobile insurance contracts.
KeywordsObserved and real contagion–overdispersion–fixed and random effects models–Poisson and linear models with heterogeneity–experience rating
This paper focuses on techniques for constructing Bonus-Malus systems in third party liability automobile insurance. Specifically,
the article presents a practical method for constructing optimal Bonus-Malus scales with reasonable penalties that can be
commercially implemented. For this purpose, the symmetry between the overcharges and the undercharges reflected in the usual
quadratic loss function is broken through the introduction of parametric asymmetric loss functions of exponential type. The
resulting system possesses the desirable financial stability property.
Diese Arbeit konzentriert sich auf die Konstruktion von Bonus-Malus-Systemen in der Kfz-Haftpflichtversicherung. Insbesondere
wird eine praktische Methode vorgestellt, urn optimale Bonus-Malus-Tarife mit vemünftigen Buβen zu konstruieren, welche kommerziell
implementiert werden konnen. Dazu wird die Symmetric zwische Uber-und Unterbelastung, die sich in der üblichen quadratischen
Verlustfunktion ausdrückt, durch die Einführung parametrischer asymmetrischer Verlustfunktionen vom Exponentialtyp aufgebrochen.
Das resultierende System besitzt die erwiinschte finanzielle Stabilitätseigenschaft.
The objective of this paper is to provide an extension of well-known models of tarification in automobile insurance. The analysis begins by introducing a regression component in the Poisson model in order to use all available information in the estimation of the distribution. In a second step, a random variable is included in the regression component of the Poisson model and a negative binomial model with a regression component is derived. We then present our main contribution by proposing a bonus-malus system which integrates a priori and a posteriori information on an individual basis. We show how net premium tables can be derived from the model. Examples of tables are presented.
Automobile insurance is an example of a market where multi-period contracts are observed. This form of contract can be justified by asymmetrical information between the insurer and the insured. Insurers use risk classification together with bonus-malus systems. In this paper the authors show that the actual methodology for the integration of these two approaches can lead to inconsistencies. They develop a statistical model that adequately integrates risk classification and experience rating. For this purpose they present Poisson and negative binomial models with regression component in order to use all available information in the estimation of accident distribution. A bonus-malus system which integrates a priori and a posteriori information on an individual basis is proposed, and insurance premium tables are derived as a function of time, past accidents, and the significant variables in the regression. Statistical results were obtained from a sample of 19,013 drivers. Copyright 1992 by John Wiley & Sons, Ltd.
In this paper, we propose an analytic analogue to the simulation procedure described in Taylor (1997). We apply the formulas to a Belgian data set and discuss the interaction between a priori and a posteriori ratemakings.
In this paper, we study bonus systems in an open portfolio, i.e. we consider that a policyholder can transfer his policy to a different insurance company at any time. We make use of inhomogeneous Markov chains to model the system and show, under reasonable assumptions, that the stationary distribution is independent of the market shares, and is easily calculated.
Time-Dependent Risk ModelsPoisson Arrival ProcessesRuin Probabilities: The Compound Poisson ModelBounds. Asymptotics and ApproximationsNumerical Evaluation of Ruin FunctionsFinite-Horizon Ruin Probabilities
A new criterion for the evaluation of automobile bonus systems is proposed. It states that a bonus system should be constructed such as to minimize a weighted average of the expected squared rating errors in various insurance periods. The criterion generalizes an asymptotic criterion given earlier by Norberg in 1976. In addition, the new nonasymptotic criterion makes it possible to discuss various short term aspects such as the optimal choice of starting class and the time heterogeneity of risks. Our treatment is illustrated by examples with numerical results.
In this paper, we propose an analytic analogue to the simulation procedure described in Taylor (1997). We apply the formulas to a Belgian data set and discuss the interaction between a priori and a posteriori ratemakings.
The operation of a bonus-malus system, superimposed on a premium system involving a number of other rating variables, is considered. To the extent that good risks are rewarded in their base premiums, through the other rating variables, the size of the bonus they require for equity is reduced. This issue is discussed quantitatively, and a numerical example given.
The objective of this paper is to provide an extension of well-known models of tarification in automobile insurance. The analysis begins by introducing a regression component in the Poisson model in order to use all available information in the estimation of the distribution. In a second step, a random variable is included in the regression component of the Poisson model and a negative binomial model with a regression component is derived. We then present our main contribution by proposing a bonus-malus system which integrates a priori and a posteriori information on an individual basis . We show how net premium tables can be derived from the model. Examples of tables are presented.
This paper deals with the problem of designing experience rating systems of the bonus type, commonly used in automobile insurance. On the basis of a simple model the mean squared deviation between a policy's expected claim amount and its premium in the nth insurance period as n→∞, is taken as a measure of the efficiency of a bonus system. It is shown that to any set of bonus rules (which determines the bonus class transitions of the policies), there is an optimal premium scale, which coincides with the one proposed by Pesonen in 1963. Thus the problem of choosing an efficient bonus system reduces to choosing efficient bonus rules. Examples are given of comparison between different bonus rules. In one example the present Norwegian bonus system is compared to alternative systems. Comments are made on earlier papers on bonus systems. The credibility theoretic foundation is laid in a separate section.
In the present paper we study bonus systems with linear premium scales in a set-up presented in a paper by Borgan, Hoem & Norberg. Some numerical examples are given.
The Storebrand Insurance Company has introduced a new bonus—malus system for automobile insurance (passenger cars) into the Norwegian market. The new system differs from traditional bonus—malus systems mainly due to the fact that bonus reduction after a claim is expressed by a fixed monetary amount. This paper describes the practical implementation of the bonus—malus system.
In this paper, we study bonus systems in an open portfolio, i.e. we consider that a policyholder can transfer his policy to a different insurance company at any time. We make use of inhomogeneous Markov chains to model the system and show, under reasonable assumptions, that the stationary distribution is independent of the market shares, and is easily calculated.
- R Kaas
- M J Goovaerts
- J Dhaene
- M Denuit
Kaas, R., M. J. Goovaerts, J. Dhaene, and M. Denuit, 2001, Modern Actuarial Risk Theory
(Dordrecht: Kluwer Academic Publishers).
Setting a BMS in the Presence of Other Rating Factors: Taylor's
- S Pitrebois
- M Denuit
- J.-F Walhin
Pitrebois, S., M. Denuit, and J.-F. Walhin, 2003, Setting a BMS in the Presence of Other
Rating Factors: Taylor's Work Revisited, ASTIN Bulletin, 33: 419-436.