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FAIR BONUS-MALUS SYSTEMS

100%
Podgorska M., Kryszen B., Niemiec M.
Roczniki Kolegium Analiz Ekonomicznych
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2008
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vol. 18
11-24
EN
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)
2
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EXPECTED PREMIUM IN THE MODEL OF BONUS-MALUS SYSTEM FAIR BY TRANSITION RULES

80%
Podgorska M.
Roczniki Kolegium Analiz Ekonomicznych
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2008
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vol. 18
25-41
EN
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.
3
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STATIONARY PROPERTIES OF EXTREME BONUS-MALUS SYSTEMS FAIR BY TRANSITION RULES

80%
Niemiec M.
Roczniki Kolegium Analiz Ekonomicznych
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2008
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vol. 18
69-81
EN
The purpose of this paper is to present stationary properties of four 'extreme' cases of bonus-malus systems fair by transition rules, characterized by rules of maximum/minimum advancement and maximum/minimum fall in each class. General formulae for the stationary distributions and mean stationary premiums for these systems are derived. Also the first attempt to define the impact of a change in the number of classes on the values of mean stationary premium has been made.
4
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A SET OF BONUS-MALUS SYSTEMS FAIR BY TRANSITION RULES - ELEMENTS OF ANALYSIS

80%
Niemiec M., Topolewski M.
Roczniki Kolegium Analiz Ekonomicznych
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2008
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vol. 18
113-145
EN
The paper considers a set of bonus-malus systems fair by transition rules (BMSF(TR)). Although the definition of such system reduces a set of all BM systems, the number of systems of BMSF(TR) type still remains huge. First the authors use a special algorithm to generate all possible systems of BMSF(TR) type with a given number of classes. Then they propose to implement a premium criterion that fulfils the definition of BMSF(TR) and, which is very important, makes systems financially balanced. Next they choose appropriate measures which allow to analyse and compare all possible systems of a given size in respect of their performance in different portfolios i.e. portfolios which differ by claim frequency and claim variance. The results of the analysis are general conclusions on how transition rules and portfolio's characteristics influence system performance.
5

Markov Set-Chain - Application to Analysis of Bonus-Malus System

80%
Niemiec M.
Przegląd Statystyczny
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2005
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vol. 52
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issue 1
87-102
EN
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.
6
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EXTREME TYPES OF BONUS-MALUS SYSTEM FAIR BY TRANSITION RULES

80%
Podgorska M.
Roczniki Kolegium Analiz Ekonomicznych
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2008
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vol. 18
43-68
EN
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.
7
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Expected premium in bonus-malus system as a non-increasing claims frequency function

80%
Kryszen B.
Roczniki Kolegium Analiz Ekonomicznych
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2005
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vol. 14
45-57
EN
Bonus-malus system is most commonly used in automobile insurance to set the premium. In the system the premium depends on the number of claims reported by the driver in the previous period. That's why claims frequency model determines a posteriori classification. Generally there is assumed, that in the automobile insurance, the mixed Poisson distribution describes number of claims. Taking into a consideration above, in the article the system is analyzed for the individual client whose claims model is characterized by the Poisson distribution.To evaluate expected premium the non-realistic systems were eliminated. The bonus-malus system classification presented in the article allows distinguishing those, present on the competitive market, later called 'fair'. In the research it was proved that in the fair systems, the expected premium is non-increasing claim frequency function, in the stationary as well as in the non-stationary periods. The basic condition for a correct a posteriori risk assessment is non-positive change of the expected premium level while claim frequency is increasing. The aim of the research was to describe conditions determining the above theorem and to present formal prove. The fair systems defined in the article are commonly used in practice. Moreover, the actuarial literature assumes that Poisson distribution describes number of claims for the individual insured. Based on that, proved theorem is related to wide category of the system used on the market. .
8
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Two-element perturbation of Markov chain in analysis of bonus-malus system

80%
Niemiec M.
Roczniki Kolegium Analiz Ekonomicznych
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2005
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vol. 14
59-74
EN
The purpose of the article is to apply the two-element perturbation of a Markov chain to the analysis of a bonus-malus system commonly employed in automobile insurance in order to classify policyholders. In the literature the bonus-malus system is modelled in the framework of the finite irreducible ergodic discrete Markov chain theory, which requires the assumption of a constant transition matrix and thus restricts the analysis of consequences of changes in the system's structure. In the article the application of the perturbed Markov chain is investigated. The two-element perturbation consists in an increase in one element of the transition matrix at the expense of an equal decrease in one other element in the same raw. In spite of its simplicity the perturbation allows for analyzing structure modification in the bonus-malus system due to specific transition matrix of its model. The perturbation proves to be an adequate tool in the study of consequences of changes in the transition rules i.e. rules determining the transfer of a policyholder from one class to another. It enables to examine the influence of their changes on stationary probabilities, mean first passage and return times and consequently on the system evaluation and performance. Hereby, it provides insurance companies with valuable information, indispensable for constructing a new system or modifying the old one.
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