Academic journal article Journal of Risk and Insurance

Bonus-Malus Scales in Segmented Tariffs with Stochastic Migration between Segments

Academic journal article Journal of Risk and Insurance

Bonus-Malus Scales in Segmented Tariffs with Stochastic Migration between Segments

Article excerpt


This article proposes a computer-intensive methodology to build bonus-malus scales in automobile insurance. The claim frequency model is taken from Pinquet, Guillen, and Bolance (2001). It accounts for overdispersion, heteroskedasticity, and dependence among repeated observations. Explanatory variables are taken into account in the determination of the relativities, yielding an integrated automobile ratemaking scheme. In that respect, it complements the study of Taylor (1997).


A Priori Risk Classification

Nowadays, it has become extremely difficult for insurance companies to maintain cross-subsidies among different risk categories in a competitive setting. Therefore, actuaries have to design a tariff structure that will fairly distribute the burden of claims among policyholders. If the risks in the portfolio are not all identically distributed, it is fair to partition all policies into homogenous classes with all policyholders belonging to the same class paying the same premium.

The classification variables introduced to partition risks into cells are called a priori variables (as their values can be determined before the policyholder starts to drive). In automobile third-party liability insurance, they commonly include the age, gender, and occupation of the policyholders; the type and use of their car; the place where they reside; and sometimes even the number of cars in the household, marital status, smoking behavior, or the color of the vehicle. It is convenient to achieve a priori classification with the help of generalized linear models; see, e.g., Renshaw (1994) or Pinquet (2000) for applications in actuarial sciences, and Dobson (1990) for a general overview of the statistical theory.

A Posteriori Corrections

Many important factors cannot be taken into account a priori; think, for instance, of the swiftness of reflexes, aggressiveness behind the wheel, or knowledge of the highway code. Consequently, tariff cells are still quite heterogenous despite the use of many classification variables. Data involving claim counts exhibit variability exceeding that explained by Poisson models. This is due to residual heterogeneity and can be modeled by a random effect in a statistical model.

It is reasonable to believe that the hidden characteristics are partly revealed by the number of claims reported by the policyholders. Hence, the adjustment of the premium from the individual claims experience in order to restore fairness among policyholders. In that respect, the allowance of past claims in a rating model derives from an exogenous explanation of serial correlation for longitudinal data. In this case, correlation is only apparent and results from the revelation of hidden features in the risk characteristics.

It is worth mentioning that serial correlation for claim numbers can also receive an endogeneous explanation. In this framework, the history of individuals modifies the risk they represent; this mechanism is termed "true contagion," referring to epidemiology. For instance, a car accident may modify the perception of danger behind the wheel and lower the risk to report another claim in the future. Insurance rating schemes also provide incentives to careful driving and should induce negative contagion. Nevertheless, the main interpretation for automobile insurance is exogenous, since positive contagion (i.e., policyholders who reported claims in the past are more likely to produce claims in the future than those who did not) is always observed for numbers of claims, whereas true contagion should be negative.

Once estimated, the statistical model incorporating the portfolio heterogeneity can be used to perform prediction on longitudinal data and allows for experience rating in casualty insurance. In an empirical Bayesian setting, the prediction is derived from the expectation of a random effect with respect to a posterior distribution taking into account the history of the individual. …

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