Loss Models: From Data to Decisions, 4th Edition

9.8 The individual risk model

9.8.1 The model

The individual risk model represents the aggregate loss as a fixed sum of independent (but not necessarily identically distributed) random variables:

This formula is usually thought of as the sum of the losses from n insurance contracts, for example, n persons covered under a group insurance policy.

The individual risk model was originally developed for life insurance in which the probability of death within a year is q_j and the fixed benefit paid for the death of the jth person is b_j. In this case, the distribution of the loss to the insurer for the jth policy is

The mean and variance of aggregate losses are

and

because the X_js are assumed to be independent. Then, the pgf of aggregate losses is

(9.33)

In the special case where all the risks are identical with q_j = q and b_j = 1, the pgf reduces to

and in this case S has a binomial distribution. ...