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This article presents an optional bonus-malus contract based on a priori risk classification of the underlying insurance contract. By inducing self-selection, the purchase of the bonus-malus contract can be used as a screening device. This gives an even better pricing performance than both an experience rating scheme and a classical no-claims bonus system. An application to the Danish automobile insurance market is considered. INTRODUCTION If one tells the truth, one is sure, sooner or later, to be found out. Oscar Wilde In a world with two levels of risk and asymmetric information, where the insurance company cannot distinguish between customers from the two risk groups, there is no pooling equilibrium and there may not be an equilibrium at all (Rothschild and Stiglitz, 1976). In a pooling equilibrium, the lower risk policyholders subsidize the higher risk policyholders. If the policyholders are aware of their risks and the difference is too large between risk and price, the lower risk individuals will not buy insurance. Thus, either the market for insurance breaks down or each type of policyholder buys an insurance contract with a payoff that caters to the specific riskiness of the policyholder. This idea of sufficiently tailor-made (i.e., differentiated) contracts may not fit exactly with what we see in current insurance markets. Insurance contracts offered by insurance companies are only to some extent differentiated, depending on the sophistication of the models, the available data, and regulations. However, Allard, Cresta, and Rochet (1997) show that pooling equilibria may exist if even the slightest distributional cost exists. Differences in risk aversion can also make policyholders with different risks accept pooling (at least to some extent), according to Rothschild and Stiglitz (1976). Covariate-based regression is generally used to differentiate pricing and the most widely used models for this are generalized linear models (GLMs); see Pinquet (2001) for applications in actuarial science and McCullagh and Nelder (1989) for a general discussion of the statistical theory of GLMs. One way to improve the a priori pricing is to make use of a posteriori corrections based on experience rating, such as credibility theory and bonus-malus systems (BMS). Simple time-independent credibility models with extensions to both time-dependence and multivariate experience rating are found in Pinquet, Guillen, and Bolance (2001) and Englund et al. (2009). Brouhns et al. (2003) provide a quick survey of different types of BMS. Unfortunately, these experience-based methods have a limited impact when individual claim information is rare, as for new policyholders, and it may take several years of observation before the precision is reasonable. However, Donnelly, Englund, and Nielsen (2013) find evidence of adverse selection within the Danish automobile insurance market, meaning that the degree of coverage chosen by the policyholder is based on the ex ante assessment that a policyholder makes of his riskiness and wealth. Therefore, we seek new ways to differentiate the policyholders by risk at the very outset, that is, at the time of purchase of the insurance product. This article gives examples of an insurance contract that may induce self-selection within the existing risk classes of the rating scheme. The new type of insurance contract has some features that are not found in contracts of standard BMS. Similar to BMS, the payoff that is offered to policyholders depends on the actual number of experienced claims during each insurance period, but the contract has a more general form of payoff than a standard contract with bonus. The payoff can be tailor-made to individual policyholders' needs or preferences. Implicit in this is the idea that insurance companies should offer a menu of contracts in such a way that self-selection is induced among the policyholders. … |
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