Bonus Made Easy

Abstract

The paper introduces an alternative approach to the traditional experience rating theory in automobile insurance. The approach is based on a simple theory of how high deductibles financed by loans maintain the risk differentiation in an automobile insurance arrangement. Thus the approach differs totally from the usual bonus-malus classes as well as from the credibility based experience rating ideas. The paper is of a theoretical nature and leads up to a mathematical description of how the approach may be optimalized within the framework of a risk model. KEYWORDSBonus-malus systems; optimal deductibles financed by loans. BACKGROUNDFrom a practical point of view it is well-known that the existing automobile bonus-malus systems possess several considerable disadvantages which are difficult, or even impossible, to handle within the traditional theory of experience rating. The aim of this paper is to introduce an alternative bonus-malus approach which, at least theoretically, eliminates the most important ones of these disadvantages. CRITICISM OF EXISTING BONUS SYSTEMSTo motivate the new bonus-malus (B-M) approach it is appropriate to stress the usual criticism of the existing B-M systems. In particular, the existing systems are, among other things, based on two general characteristics:(i) The claim amounts are omitted as a posterior tariff criterion, (ii) At any time the policyholders may leave an insurance company without any further financial commitments to the company.These characteristics lead to three of the most considerable disadvantages:(2.1) Regarding an occurred claim, the future loss of bonus will in many cases exceed the claim amount.1 An earlier version of this work has been presented at the ASTIN Colloquium, Stockholm 1991. ASTIN BULLETIN, Vol. 24. No. 1, 1994 Because only the number of claims (and of course the discount rate) in an insurance period determines the premium in the following period, it follows that (2.1) is an immediate consequence of (i). In many cases (2.1) gives the policyholder a feeling of unfairness, especially if the loss of bonus is much higher than the occurred claim amount. A consequence of this is the wellknown bonus hunger behaviour of the policyholders.Disadvantage (2.2) is of course a consequence of (ii). Malus evaders let the remaining policyholders pay the bill for their (the evaders') claim costs. This has, at least in Norway, been a serious problem in the insurance industry, mainly because of an unsatisfactory exchange of bonus information between the insurance companies.Because all insurance arrangements attached to existing B-M systems are exposed to bonus hunger as well as malus evasion, it follows that (2.3) is a secondary consequence of (2.1) and (2.2). A higher average rate of discount is contrary to risk differentiation, which is the objective of all B-M systems. In an extreme situation the result might be that the great majority of the policyholders are at, or close to, the maximum rate of discount.A number of authors have focused on the disadvantages...

Optimal Insurance Coverage under Bonus-Malus Contracts

“…As a concluding remark to the above discussion, we may point out that ordinary deductibles are usually used in the insurance market as the main instrument to reduce the claim probability (moral hazard) and to reduce the costs generated by claims handling. Therefore, the main intention of a bonus-malus system is to handle the problem of adverse selection generated by individual asymmetric information (even if Holtan (1994) outlines a model with high deductibles financed over a period of time as an adverse selection alternative to bonus-malus systems). Hence, as a general rule bonus-malus systems should only be used if individual loss experience is a significant risk parameter within the insurance market.…”

Note on the Papers by J. Holtan and By J. Lemaire & H. Zi

“…In this note I want to give some general comments on the papers by LEMAIRE & Zi (1994) and HOLTAN (1994).…”

“…In our strive for maximizing BM advantages and minimizing BM disadvantages, actuarial BM research should instead simultaneously focus on both components (a) and (b). The construction of the High-Deductible System (HDS) in HOLTAN (1994) is an example of this strategy. However, as pointed out in LEMAIRE & Zi (1994) (see Section 1 and 4) and HOLTAN (1994) (see Section 3, 5 and 6), a HDS compared with existing BM systems both eliminates and generates important disadvantages which are linked to component (a).…”

Fair Actuarial Values for Deductible Insurance Policies in the Presence of Parameter Uncertainty

“…They proved that an insured would file a claim only if it is larger than some critical value that is higher than the positive deductible amount. Holtan [8,9] investigated the characteristics of the deductible insurance policy in the presence of the bonus-malus adjustments and showed that the bonus-malus policies cannot be mutually optimal (in sense of being Pareto optimal) to both the insured and the insurer.…”