Definitions and Notations

Abstract

Conventional actuarial techniques are largely based on frequencies and the average amounts of claims. For example, if an insurer has a portfolio of N policies at risk and if the expected mean value of the claim frequency for these policies during a specified period is q and the expected average size of the claim is m, then the expected total amount of claims is Nqm. However the actual amounts arising from several successive periods will differ from this expected figure and will fluctuate around it. In probabilistic terms, the actual amount of claims is a random variable. Conventional actuarial techniques are in fact based on a simplified model of an insurance portfolio in which random variables are replaced by this mean value, i.e. the fluctuation phenomenon is disregarded. Whilst for many purposes this simplified model is sufficient in the hands of experts, it is undeniably an over-simplification of the facts and it is both useful and interesting to develop the principles of insurance mathematics on a more general basis, in which both the number and size of claims are considered as random variables. Studies of the different kinds of fluctuation appearing in an insurance portfolio which start from this point of view constitute the branch of actuarial mathematics termed the theory of risk.