Introduction

Conventional actuarial and insurance techniques are based on a simplified model of an insurance portfolio in which random variables are replaced by their mean values. De Moivre, a French mathematician, proved as early as 1700 that an insurance business would eventually be ruined if it failed to include a margin in its favour in the price it charged for its contingent payments: in other words, the insurer must include a safety loading in his premium to guard against losses due to random fluctuations.

Studies of the random fluctuations (stochastic variations) which occur in accumulated claim amounts constitute the branch of actuarial mathematics termed the theory of risk. The theory is useful as a guide to the relationship between reserves, retentions and the level of risk, and the general order of magnitude of these quantities.

Example 11.1.1. Consider three insurance contracts. Under the first, the only possible claim amount is $10; only one claim is possible during the year the policy is in force, and the probability of a claim is 1.0. The other two contracts are of the same form, but with claim sizes $100 and $1000 respectively, and claim probabilities 0.1 and 0.01 respectively

The risk premium is the same for all three contracts, namely, $10 (section 7.1).