Commercial automobile insurance: should fleet policies differ from single-vehicle plans


A “fleet” insurance policy covers a number of motor vehicles, usually five or more, owned by a business firm. The issue addressed in this paper is whether the design of such policies should differ from that of single-vehicle insurance plans. Specifically, I inquire whether the size of a fleet matter for the loss reimbursement schedules. The intuition is that even over a single contract period (say a year), the loss experience for a large fleet may be expected to provide relatively precise information with respect to a firm’s risk class or risk management policies. Presumably, this should make it possible to provide better insurance coverage, while maintaining the screening of bad risks and the incentives to reduce accident frequencies. Single-vehicle policies involve a per-occurrence deductible for own-fault damages to the insured vehicle (as well as for theft, etc.).

One reason for this feature is the elimination of small claims that would be too costly to process. Another explanation is that deductibles are useful for loss prevention and for the screening of bad risks. A deductible can be seen as a penalty contingent on ex-post information about the policy holder’s risk characteristics.

The prospect of a loss-contingent penalty induces the insured to take care; it can also lead higher-risk individuals to reveal their type by self-selecting policies with smaller deductibles. In either case, deductibles are actually only part of the penalty structure facing the insured. Typically, insurers use elaborate forms of experience rating to adjust premiums on the basis of the policy holder’s loss experience.

The model

A firm owns a fleet of N identical vehicles. Without insurance, the firm’s end-of-period wealth is Y = WN – I7=1 Xi’ where WN is the firm’s wealth if no loss occurs and Xi is the loss on the ith vehicle over the period considered. WN may vary with the size of the fleet of vehicles and with the firm’s self-protection expenditures as explained below.

The X/s are i.i.d. variables with support {O,x} where x is the amount of loss. Depending on whether the firm is a high or low risk, the probability of loss per vehicle is PH or PL’ with 0 < PL < PH” Thus, for a firm with fleet size N, the end-of-period wealth without insurance can be rewritten as Y = WN – XkN where the random variable NE {O, …,N} counts the number of loss occurrences. This variable has the binomial distribution with parameters PH (or P L) and N..

Adverse Selection

The insurance market is perfectly competitive, insurers are risk-neutral and there are no transaction costs (i.e., there is zero loading). For the MH model, it is assumed that the second-best equilibrium contract is such as to induce the firm to incur self-protection expenditures; this implies that the contract provides only partial coverage.

For the AS model, I assume the existence of a separating equilibrium in the manner of Rothschild and Stiglitz (1976). In such an equilibrium, the high-risk firm purchases complete coverage at the high-risk fair price. The low-risk firm purchases coverage at the low-risk fair price, but coverage is partial so as not to attract the high-risk type4.

Adverse Selection

Interpret WN in (1) as the low-risk firm’s wealth when eN has been expended on self-protection. If no self-protection cost is incurred, the firm is high risk and this term is replaced by WN + eN. The second-best contract under moral hazard then solves the same problem as above, with the self-selection condition (4) replaced

Losses are completely covered except for a penalty D(k) that is strictly increasing in the number of loss occurrences, as long as the firm’s wealth is not driven down to the liability limit. The contract involves zero profit and expected losses are paid by the insurance premium and the expected penalty.

Last word

The proposition states that the best way to provide coverage for L , while preventing H from purchasing the contract, is to penalize more heavily when there is a large number of losses. The intuition for this result is that a penalty contingent on ex post information (i.e., the loss experience) is relatively more costly for H than for L, the more unfavorable the information.

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