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:

equation

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

equation

The mean and variance of aggregate losses are

equation

and

equation

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

(9.33) equation

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

equation

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

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