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 Xj, the severity, represents the ground-up loss on the jth such loss and there are no coverage modifications. Let NL 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 Ij by Ij = 1 if the jth loss results in a payment and Ij = 0 otherwise. Then Ij has a Bernoulli distribution with parameter v and the pgf of Ij is PIj (z) = 1 − v + vz