Loss Models: From Data to Decisions, 4th Edition

8.6 The impact of deductibles on claim frequency

An important component in analyzing the effect of policy modifications pertains to the change in the frequency distribution of payments when the deductible (ordinary or franchise) is imposed or changed. When a deductible is imposed or increased, there will be fewer payments per period, while if a deductible is lowered, there will be more payments.

We can quantify this process if it can be assumed that the imposition of coverage modifications does not affect the process that produces losses or the type of individual who will purchase insurance. For example, those who buy a 250 deductible on an automobile property damage coverage may (correctly) view themselves as less likely to be involved in an accident than those who buy full coverage. Similarly, an employer may find that the rate of permanent disability declines when reduced benefits are provided to employees in the first few years of employment.

To begin, suppose X_j, the severity, represents the ground-up loss on the jth such loss and there are no coverage modifications. Let N^L denote the number of losses. Now consider a coverage modification such that v is the probability that a loss will result in a payment. For example, if there is a deductible of d, v = Pr(X > d). Next, define the indicator random variable I_j by I_j = 1 if the jth loss results in a payment and I_j = 0 otherwise. Then I_j has a Bernoulli distribution with parameter v and the pgf of I_j is P_Ij (z) = 1 − v + vz