## Bonus Malus Auto Explication Essay

Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

Received 3 June 2014; Revised 13 August 2014; Accepted 1 September 2014; Published 29 September 2014

Copyright © 2014 Yu Chen and Long Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A bonus-malus system plays a very important role in actuarial mathematics through determining its relativity premium, which is extensively used in automobile insurance. There are many ways including Bayesian estimator and ordinary linear estimator to calculate the relativity premium. There is no doubt that Bayesian estimator is the most accurate estimator; however, it is undesirable for commercial purposes for its rather irregular pattern. This paper aims to introduce an optimal linear estimator for relativity premium, which has a simple pattern and is obtained under the quadratic loss function such that the result is close to Bayesian method. The Loimaranta efficiency of such an optimal linear estimator has been studied and compared with the two methods mentioned above.

#### 1. Introduction

Bonus-malus system (BMS) is an important research field in modern actuary. In insurance, BMS is a system that adjusts the premium paid by a policyholder according to his individual claim history. Bonus usually is a discount in the premium which is given on the renewal of the policy if no claim is made in the previous year. Malus is an increase in the premium if there is a claim in the previous year. Bonus-malus systems are very common in vehicle insurance. And it is a topic of interest of how to compute the relativity premium of a bonus-malus system.

Consider a bonus-malus system containing levels, labeled 1 to , and the next level is determined by the current level and the number of claims reported during the current period. If the number of claims in different periods is independent, then the trajectory of a given policyholder will be a Markov chain. To deal with the heterogeneity of BMS, it is natural to consider that the claim numbers caused by the policyholder are assumed to be mixed Poisson distributed. More precisely, has conditional discrete probability function of the following form: where and represent the mean frequency and random risk effect, respectively. The random effect represents the risk proneness of the policyholder, that is, unknown risk characteristics having a significant impact on the occurrence of claims. is the distribution function of . According to Norberg [1], the probability mass function associated with level for a given policyholder is and the Bayesian relativity premium under the quadratic loss function is where is the distribution function of and is the stationary probability for a policyholder with level . The idea of Bayesian way is to attach a relativity premium to the randomly picked policyholder according to his relative risk parameter and the closeness or relativity is measured by Bayesian methods. Obviously the Bayesian estimator has a rather complex and irregular pattern and one cannot see that the relativity premium is regularly increasing according to the level .

Many authors have discussed the problem of how to design an optimal bonus-malus system. For example, Lemaire and Zi [2] compared the validity of 30 bonus-malus systems by four different tools, such as the relative stationary average premium level, the coefficient of variation of the insured’s premiums, the efficiency of the bonus-malus system, and the average optimal retention. Lemaire [3] suggested that the claim frequency as negative binomial distributed and used the quadratic loss function to study the bonus-malus system. Walhin and Paris [4] discussed the problem mainly using a finite Poisson mixture distribution as the claim frequency distribution. All those authors took the claim frequency as the most important factor and used the Bayesian estimator. The Bayesian estimator not only presents a rather irregular pattern but also may result unfairly without taking the severity of each claim into account. Frangos and Vrontos [5] designed an optimal bonus-malus system by taking both the frequency and the severity of the claim into consideration, and Mahmoudvand and Hassani [6] developed the system to a generalized form with a frequency and a severity component based both on the a priori and on the a posteriori classification criteria. And this makes sense for the system allowing for both the frequency and severity to distinguish different premiums with equal claim times and it is fair from the point of insured.

Obviously, the Bayesian estimator has a rather complex and irregular pattern. In consideration of the drawbacks of the Bayesian relativity premium, Gilde and Sundt [7] suggested to smooth by a linear pattern; that is, , , and obtained the parameter values of and under the condition of minimizing . The solution of can be easily gotten as follows: where is the covariance of level and risk parameter and and are the expectation and variance of level . With this estimator derived from the quadratic loss function of and , we can compute the relativity premium easily and the linear pattern has the increasing property associate with the level . But it still does not guarantee fairness from the insurers’ side because the linear form imposes the same degree when the bonus-malus system penalizes or rewards a certain policyholder.

Though assuming that the frequency or severity is differently distributed, successively, researchers used Bayesian estimator or linear estimator to calculate the relative premium when designing or developing bonus-malus system. No matter how well the heterogeneity of auto insurance is depicted from the insured’s side through the frequency or severity distribution, however, the behavior from the insurers’ side is not represented that well. Thus the relativity premiums calculated by Bayesian estimator as equation (3) shows or by the Ordinary Linear estimator as equation (4) shows, miss the insurers’ point. On the other hand, the Bayesian estimator has a rather complex pattern for commercial use and an irregular pattern for merit rating mechanism; that is, the relationship between the relativity premium and level is implicit, though it is most accurate. And the Ordinary Linear estimator smoothed from the Bayesian estimator cannot be as accurate as the Bayesian method and is not that fair due to the same degree or effect imposed on a policyholder when insurers do penalizing and rewarding, though it is easy to compute. Given for this, this paper introduces a new estimator, namely, the Optimal Linear estimator which we expect to be close to Bayesian estimator to calculate the relativity premium. The Optimal Linear estimator, derived from the surplus capital equation from the point of insurers and with the linear pattern by smoothing the Bayesian estimator under the quadratic loss function of surplus capital, will not only be easily used for computing but also be much closer to the Bayesian estimator than the Ordinary Linear estimator. And the Optimal Linear estimator is regularly increasing according to the level too. Then, we will show how the new method performs better than the Ordinary Linear estimator by introducing the Loimaranta efficiency as a tool for comparing. The comparison includes the demonstration of the Hong Kong auto insurance market and the simulation of a bonus-malus system allowing for claim amounts.

The organization of this paper is as follows. Section 2 introduces a simple method, namely, the Optimal Linear estimator, which can be used to compute the relativity premium in a steady state bonus-malus system. A comparison for the three estimators’ efficiency based on Loimaranta efficiency is discussed in Section 3. And some simulations and demonstrations allowing for severity of the claims of each policyholder are also presented in Section 3. Then, the conclusion remarks come.

#### 2. The Optimal Linear Relativity

As mentioned above, the BMS discussed in this paper has levels, labeled 1 to , with as the relativity premium associated with level . The problem addressed here is to determine the solution of Optimal Linear relativity premium . In order to go further, suppose that , , represents the number of policyholders in level at time and is the initial capital and is the benchmark of the premium. Then, the surplus capital of the insurer at time can easily be obtained as where denotes the whole premium incomes and is the amount of all the claims at time . Similarly, we can naturally get the whole premium incomes associated to the Bayesian relativity and associated to the Optimal Linear relativity. The question of questions is to smooth the Bayesian relativity and achieve the Optimal Linear relativity which has an explicit linear pattern, denoted as We embark on this issue from the point of the surplus capital of the insurer and achieve the parameter values of and by minimizing the quadratic loss function

Actuaries in insurance companies try to design a tariff structure that will fairly distribute the burden of claims among policyholders. Therefore they try to find the best model for an estimation of the insurance premium. The paper deals with an estimate of *a priori* annual claim frequency and application of bonus-malus system in the vehicle insurance.In this paper, analysis of the portfolio of vehicle insurance data using generalized linear model (GLM) is performed. Based on large real-world sample of data from 67 857 vehicles, the present study proposes a classification analysis approach addressing the selection of predictor variables. The models with different predictor variables are compared by the analysis of deviance. Based on this comparison, the model for the best estimate of annual claim frequency is chosen. Then the bonus-malus (BM) system is used for each class of drivers and Bayesian relative premium is calculated. Finally a fairer premium for different groups of drivers is proposed.

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