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 qj and the fixed benefit paid for the death of the jth person is bj. In this case, the distribution of the loss to the insurer for the jth policy is
The mean and variance of aggregate losses are
because the Xjs are assumed to be independent. Then, the pgf of aggregate losses is
In the special case where all the risks are identical with qj = q and bj = 1, the pgf reduces to
and in this case S has a binomial distribution. ...