The Theory of Risk and Some Applications

Abstract

Some of the classical assumptions and formulae of the collective risk theory are presented first and a model utilizing dynamic programming is constructed. Some classical applications of risk theory such as solvency tests, risk reserves, reinsurance, and rate making are briefly reviewed. The scope of the theory is extended to the field of dynamic system design. Finally, problems concerning the teaching of risk theory to actuaries and non-actuaries is discussed. 1. Theory or practice? Conventional insurance mathematics operates mainly by deterministic quantities, such as mortalities, loss ratios, and their expected or capitalized values, and so on. However, the deterministic models give a truncated picture of the insurance business, because random and other fluctuations are omitted. Attempts to complete the deterministic model by including the probabilistic variations have been made for a long time. The so-called individual risk theory was developed mainly as a supplementary field to life insurance mathematics. Subsequently, following an idea originally introduced by Filip Lundberg, the risk business of an insurer (or any other collective) as a whole has been interpreted as a stochastic process and collective risk theory is built mainly on that basis. The simultaneous development of the general theory of stochastic processes and its numerous applications has influenced fruitfully the development of risk theory, too. A well-developed theory already exists, even if many questions are still open. When speaking of risk theory we limit the consideration, as the European risk theorists conventionally do, to concern the risk process of an insurer as a whole. Theories dealing with individual risks, behaviour of policyowners or individual risk contracts are not dealt with even if many of the formulae are also applicable to them. By some simplifying one observes that research has had two main directions: the scientific school, which has successfully developed the theory on the basis of rigorous discipline, and the school, which has attempted to modify the existing theories and formulae to make them applicable in the environments of insurers. Often the starting point has been a practical problem, e.g. the making of premium rates (including the safety margin), defining the minimum need for reserve funds, choosing the level of reassurance, and so on. In practice, many rules of thumb have been applied and to turn for help to risk theory, which originally was