Bonus-Malus Systems in Automobile Insurance

Belgium is a heavily regulated country, where third-party automobile insurance was made compulsory in 1956. Traditionally, insurers have used very few classification variables in the rating of automobile third-party liability policies. Only some characteristics of the car model (engine displacement or power, sporting nature) were used to differentiate premiums, along with a very moderate deductible for young drivers. Territory was used by only a few companies, for selected remote areas of the country.

In this chapter four different probability models are developed to represent the distribution of the number of claims in an insurance portfolio. The Poisson distribution is shown to provide a good description of the behavior of individual policyholders. It will be used in chapters 4 to 9 to compare the different BMS from a policyholder’s perspective. The Poisson is, however, inadequate to describe the number of losses in an insurance portfolio. The negative binomial distribution provides much better fits and will be used in chapters 10 to 14 to build optimal BMS. Two other distributions that belong to the class of mixed Poisson processes are also introduced.

Part 2 develops tools for analyzing bonus-malus systems. Extensive simulations are used to compare and rank thirty different systems, from twenty-two countries: six countries from East Asia (Hong Kong, Japan, South Korea, Malaysia-Singapore, Taiwan, Thailand), fourteen European countries (Belgium, Denmark, Finland, France, Germany, Italy, Luxembourg, the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, and the United Kingdom), as well as Kenya and Brazil. Several of these countries have recently modified their system. In these cases both the old and the new BMS are studied in order to investigate the impact of the recent modifications. In some countries many different BMS coexist; only one or two typical systems are then analyzed. All BMS are described in appendix B.

Insurance consists of a transfer of risk from the policyholder to the carrier. Without experience rating, the transfer is total (perfect solidarity): the variability of insureds’ payments is zero. With experience rating, personalized premiums will vary from year to year, according to claims history; cooperation between drivers is weakened. Solidarity between insureds can be evaluated by a measure of the variability of annual premiums. The coefficient of variation (standard deviation divided by mean) was selected, as it is a dimensionless parameter. There is thus no need for currency conversions.

The main goal of the establishment of a bonus-malus system is to reduce the premium for good drivers and to increase it for bad drivers. It is often assumed that the random variables “number of claims” and “claim amount” are independent. This assumption essentially states that the cost of an accident is for the most part beyond the control of a policyholder. If someone drives too fast and skids off the road, he may end up in a tree, or he may permanently disable a senior executive, father of five young children. The cost of the accident is largely independent of the mistake that caused it. The degree of care exercised by a driver will mostly influence his number of accidents, but in a much lesser way the cost of these accidents. Bad drivers will provoke more claims, but not necessarily more expensive claims.

All four measures defined in chapters 4 to 7 can be used to evaluate the “mildness” or “toughness” of a bonus-malus system. A system that penalizes claims heavily will exhibit high RSAL, high premium variability, high elasticity, and high average optimal retentions. These four measures, presented in tables 4–1, 5–1, 6–1, and 7–1 for a policyholder with claim frequency 0.10, are, however, highly positively correlated, as shown in table 8–1.

As detailed in chapter 2, the old Belgian system, in force since 1971, exemplified the major problem faced by insurers using a mild bonus-malus system: a strong clustering of policies in the high-discount classes. With only a two-class penalty for the first claim, the system was designed for an average claim frequency close to 1/3. The much lower claim frequencies observed since the 1974 oil shock created an increasing lack of financial balance, with over 80 percent of the policyholders in one of the three lowest classes in 1992, and less than 0.5 percent of insureds in the malus zone. This led the Professional Union of Insurance Companies to set up a study group and suggest a new system to the regulatory authorities (see Lemaire, 1988a). The new system was implemented late in 1992. It penalizes the first claim by four classes.

In Part 3, Bayesian analysis is applied to design optimal bonus-malus systems using the different models for claim number distributions presented in chapter 3. The decision problem is formulated in a game-theoretic framework. Several loss functions and premium calculation principles are applied to calculate optimal BMS.

In chapter 10, it was shown that a bonus-malus system based on the expected value principle obtains when a quadratic loss function is used. Although this type of loss function possesses important theoretical properties, more often than not the main reason for using it lies in the simplicity of computations.

The following approach is an adaptation to the determination of a BMS of a suggestion by Ferreira (1977) for the Control Authorities of the Commonwealth of Massachusetts. It is based on the use of utility functions, which enable us to break the symmetry between overcharges and undercharges.

With the exception of Korea, all bonus-malus systems in force throughout the world penalize the number of reported claims, without taking the costs of such claims into account. A mere scratch causes the same premium increase as a serious bodily injury accident. This procedure is unfair to town dwellers, among others, since traffic density creates more, but less severe, accidents. The negative binomial model has been generalized by Picard (1976) to take into account the subdivision of claims into two categories, small and large losses. Picard’s model is applied here to the Belgian data used throughout part 3. Two options have been considered to separate large and small claims:

One of the important trends in actuarial research in the 1980s has been the development of several premium calculation principles, and the study of their properties (see, for instance, Gerber, 1974a, 1974b, and Goovaerts, De Vijlder and Haezendonck, 1984). While this line of research may have been instrumental in inducing insurers to reevaluate their rating principles, it mostly focused on risk premiums (net premiums and safety loadings), while casting aside the determination of the loading for expenses, commissions, taxes, and profits. This chapter attempts to show that this neglect may have some severe consequences. It is futile to try to assess the risk premium with great accuracy if expense loadings are only be roughly calculated. Risk premiums with desirable properties can be distorted through the loading process. This should be obvious since in many cases the expense loading exceeds the risk premium. Note that the same remark was made by Jewell (1980): The next step in premium setting is to determine the additional 50–200% increase which determines the commercial premium by adding expense and profit loadings. Except in life insurance where there are specific cost models for sales commissions (in many cases of regulated form), there seems to be no further modelling principles used, except [multiplying the risk premium by a factor 1+α]. This lacuna in the literature is all the more surprising, as it is in sharp contrast to the fields of engineering and business management, where extensive and sophisticated cost allocation and modelling are the order of the day. Are these activities outside the realm of the actuary?

The two main reasons for the introduction of a bonus-maius system are (1) to reward the accident-free drivers with bonuses, while penalizing bad drivers with maluses and (2) to induce policyholders to drive more carefully. Other policy incentives have been suggested to achieve the same goals. In this part, we will investigate the effect of the introduction of a deductible. With a deductible, policyholders participate in the financial consequences of an accident. They are thus encouraged to drive carefully. Moreover, bad drivers are penalized, since over a lifetime they are expected to pay more deductibles than good drivers. So it may be argued that a deductible achieves the same goals as a BMS. Vandebroek (1993), using a dynamic programming model, even argues that partial coverage is superior to BMS in inducing a proper level of care by policyholders.

The preceding chapter recommends the use of a very high deductible as an alternative to BMS. The suggested deductible is much higher than values commonly used in automobile insurance. In this chapter, we use utility theory to determine the optimal value of a deductible. As deductibles are not commonly offered in third-party liability, the model uses automobile collision insurance data and terminology.