Loss Models: From Data to Decisions, 4th Edition

APPENDIX C

FREQUENCY AND SEVERITY RELATIONSHIPS

Let N^L 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 Y^L, the amount paid per loss:

In this case, the appropriate frequency distribution continues to be N^L.

An alternative approach is to create Y^P, the amount paid per payment:

In this case, the frequency random variable must be altered to reflect the number of payments. Let this variable be N^P. Assume that for each loss the probability is v = 1 − F_X(d) that a payment will result. Further assume that the incidence of making a payment is independent of the number of losses. Then N^P = L₁ + L₂ + ··· + L_N, where L_j is 0 with probability 1 − v and is 1 with probability v. Probability generating functions yield the relationships in Table C.1.

Table C.1 Parameter adjustments.

The geometric distribution is not presented as it is a special case of the negative binomial with r = 1. For zero-truncated distributions, the same formulas are still used as the distribution for N^P will now be zero modified. For compound distributions, modify only the secondary distribution. For ...