Insurance Mathematics

The economic importance of insurance continues to grow in developed countries. Insurers regularly offer their customers new types of contract to cover new risks, or to improve coverage for known risks. This calls for constantly updated insurance engineering that is capable of taking into account the various facets of this activity, which calls on several disciplines or techniques, such as mathematics, statistics, law, economics, accounting... In this article dealing with insurance mathematics, we will focus on quantitative aspects that are traditionally handled by actuaries, engineers specialized in insurance and financial techniques. The sheer breadth of the subject, as can be seen from the number of publications and references on the Web, means that we have to limit ourselves to a few key topics.

Every insurance transaction begins with the signing of a contract between an insurer and an insured. This contract provides for an immediate and certain payment, the insurance premium, in exchange for a future and uncertain payment, which will take the form of indemnities paid to the insured to compensate for claims incurred during the period of validity of the contract.

Risk is thus transferred from the insured to the insurer. The viability of this operation is only made possible by the pooling of many risks within the insurance company, which allows a certain statistical compensation, without however completely eliminating fluctuations in the total amount of claims, the object of future payments. As a result, the risk borne by the insurer must first be assessed in terms of the premium, which must be sufficient to cover the average cost of claims, as well as possible fluctuations in this cost. If these fluctuations are exceptional and unfavorable, they can have an impact on the balance of the insurer's annual accounts, or even jeopardize its solvency. This is why the insurer not only evaluates the premium, but also endeavors to estimate the probability law of the cumulative amount of claims, of which the premium, despite its importance, is only one indicator among others.

This article examines the problem of premium assessment, which is based not only on the analysis of past claims experience, but also on the theoretical principles of premium calculation that the insurer is likely to choose according to his own criteria, in particular his attitude to risk. As far as the probability distribution of cumulative claims is concerned, given the large size of the risk portfolios managed by insurance companies, elementary valuation methods based on convolutions are of limited use. The actuary must therefore turn to more efficient approximation methods or specific algorithms, such as Panjer's recursive method or the Fast Fourier Transform. The Monte-Carlo...