The individual risk model: a compound distribution

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The two approaches to modelling aggregate claims—the individual and the collective models—have been regarded as arising by considering a portfolio of policies in different ways. The individual risk model (IRM) is derived by considering the claims on individual policies and summing over all policies in the portfolio, while the collective risk model (CRM) is derived from the portfolio as a whole. This is sometimes held to be the main difference between the IRM and the CRM. In fact the IRM can be derived in exactly the same way as the CRM and can be regarded as a compound binomial distribution. This makes a unified treatment of risk models possible, simplifies the calculation of the mean and variance of the IRM, and facilitates the calculation of higher moments. The treatment of the IRM as a compound distribution has proved useful and effective in teaching risk theory and one of the purposes of this paper is to set out an alternative approach to that given in Bowers et al.(1) Using this approach, it becomes clearer that the distinction between the IRM and CRM is not the treatment of the policics in the portfolio (individually or collectively), as the names imply, but that in the IRM there can be at most one claim per policy, while in the CRM there is no such restriction. In addition, the use of a compound Poisson distribution as the limit of a compound binomial distribution to approximate the IRM is clearer.