Appendix C
Frequency and Severity Relationships

Let img be the number of losses random variable and let X be the severity random variable. If there is a deductible of d imposed, there are two ways to modify X. One is to create img, the amount paid per loss:

equation

In this case, the appropriate frequency distribution continues to be img.

An alternative approach is to create img, the amount paid per payment:

equation

In this case, the frequency random variable must be altered to reflect the number of payments. Let this variable be img. Assume that, for each loss, the probability is img that a payment will result. Further assume that the incidence of making a payment is independent of the number of losses. Then, img, where img is 0 with probability img and is 1 with probability v. Probability generating functions yield the relationships in Table C.1.

Table C.1 Parameter adjustments.

img Parameters for img
Poisson img
ZM Poisson img
Binomial img
ZM binomial img
img
Negative binomial img
ZM negative binomial img
img
ZM logarithmic img
img

The geometric distribution is not presented as it is a special case of the negative binomial with img. For zero-truncated distributions, the same formulas are still used as the distribution for img will now be zero modified. For compound distributions, modify only the secondary distribution. For ETNB secondary distributions, the parameter for the primary distribution is multiplied by img as obtained in Table C.1, while the secondary distribution remains zero truncated (however, img.

There are occasions on which frequency data are collected that provide a model for img. There would have to have been a deductible d in place and therefore v is available. It is possible to recover the distribution for img, although there is no guarantee that reversing the process will produce a legitimate probability distribution. The solutions are the same as in Table C.1, only now img.

Now suppose that the current frequency model is img, which is appropriate for a deductible of d. Also suppose that the deductible is to be changed to img. The new frequency for payments is img and is of the same type. Then use Table C.1 with img.

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