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Content available remote FAIR BONUS-MALUS SYSTEMS
In spite of over 50 years of existence of a bonus-malus system (BMS) many crucial problems concerning its modelling, analysis and optimisation remain unsolved. Definitions of BMS proposed in literature are so general that they include systems which could not exist and perform well in the competitive automobile insurance market, and therefore they are not used in practice. The objective of this article is to present two definitions of fair bonus-malus systems, which differ in the criterion of distinguishing BMS, however both allow for eliminating systems with non-realistic structures. The first proposal is the definition of so called bonus-malus system fair by premium (BMSF(PR)). The concept of this system consists in excluding from considerations such systems, in which policyholders are penalized with the greater premium after reporting fewer losses or after claiming a given number of losses if they were in better class. The criterion for distinguishing BMS in the second definition is based on the transition rules i.e. rules governing the transition of the insured, having reported a given number of claims, from one class to another. Therefore, these systems are named fair bonus-malus systems by transition rules (BMSF(TR)). In this paper it is also proved that each BMSF(TR) is also BMSF(PR)
After describing theoretical basis and properties of the Markov set-chains, their application to the analysis of an automobile insurance system is presented. The bonus-malus system is a system of assigning premiums on the basis of the premium paid in the preceding period and the number of claims reported by a policyholder at that time. In the literature this system is commonly modelled with the use of homogeneous Markov chains, which requires often unrealistic assumption of constant transition matrix and consequently unchanged loss number distribution. The basic parameter of the loss number distribution is its mean called an average claim frequency. Its value may fluctuate from time to time due to insurance companies' actions, changes in the behaviour of a policyholder as well as external factors such as weather conditions or state of roads. A model of a bonus-malus system is constructed in the framework of the Markov set-chain theory. It enables to examine consequences of average claim frequency changes. It is shown how the fluctuation of the average claim frequency may influence both a stationary probability that a policyholder belongs to the class of a distinct premium and expected time that is needed by an insured from a particular class to reach another or once again the same class. The results of the study are crucial to insurance companies having interest not only in system evaluation but also in predicting changes in its performance.
The subject of the paper are basic properties of bonus-malus system fair by the transition rules between classes (BMSF(TR)), of which definition excludes unrealistic bonus-malus systems. The paper presents an ergodic Markov chain which is a BMSF(TR) model and which allows to analyze the properties of expected value of insurance premium according to the features characterizing an insured and a system i.e. claims frequency, class in the initial year, insurance duration and maximum number of claims acknowledged in the system.
In the article four extreme variants of BMSF(TR) in which extreme transition rules are valid, i.e. rules of maximum/minimum advancement and maximum/minimum fall, are presented. These four systems allow us to determine the lowest and highest expected premium in any insurance year in any BMSF(TR) and the intervals of values of expected premium in the systems of BMSF(TR) type which are modifications of these four extreme systems.
The goal of this article is to apply panel data approach to the analysis of claim frequency in automobile insurance. The model which is constructed estimates the influence of particular characteristics of the insured on their insurance loss number, but it also enables identification of the hunger for bonus effect. Panel data approach allows for identification of drivers' individual effects that influence their driving quality, but cannot be quantified directly, such as for example tendency to drive fast. This is done thanks to repetitive observation of the same individuals. Having information on their number of losses claimed in different bonus-malus system classes, it is possible to separate their individual skills from the hunger for bonus phenomenon, as well as identify the scale of the latter, which differs in particular classes. Chapter one is an introduction. In chapter two main benefits from the use of panel data have been described. Recent publications considering the topic are mentioned as well, with emphasis on the differences between other authors' approaches and this one. Chapter three contains a brief description of the methods applied, which are Poisson regression mixed models. In chapter four the basic model is adjusted to the conditions of hunger for bonus and it is shown, how this phenomenon is identified. In chapter five empirical analysis based on the real market data of approximately 21 thousand observations is done. The model is estimated and the conclusions are discussed with a short simulation study of the insurance company financial state.
The aim of the article is to discuss a model used to determine the number of losses in automobile insurance. The model is based on panel data. Although the aim is to model the number of losses, due to hunger for bonus not all the losses are revealed. Thus the data on the number and also the value of claims are used. Common use of these two types of data enables estimation of the true number of losses that occur (not just those that are claimed). This is done with the use of true data from the Polish market. The discussion of particular factors that influence the severity of losses (moral hazard, hunger for bonus, observed and unobserved characteristics of the insured) is included.
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