This survey paper presents the basic concepts of cooperative game theory, at an elementary level. Five examples, including three insurance applications, are progressively developed throughout the paper. The characteristic function, the core, the stable sets, the Shapley value, the Nash and Kalai-Smorodinsky solutions are defined and computed for the different examples.

In this paper, we seek to find the optimal retentions for an insurance company which intends to reinsure each of n risks belonging to its portfolio, by means of a pure quota-share treaty, a pure excess of loss treaty or any combination of the two. The criterion chosen to the selection of the optimal programme is the maximization of the adjustment coefficient, attending to the relationship existing between this coefficient and Lundberg's upper bound of the ruin probability.

This paper discusses distribution of surplus in life insurance within a general Markov chain framework. A conservative interest rate and a conservative set of transition intensities are used for reserving purposes whereas more realistic assumptions are used for the purpose of distributing surplus. The paper examines various actuarial aspects of distributing surplus through either cash bonuses, terminal bonuses or increased benefits. The results are illustrated by some examples.

Traditional credibility models have treated the process generating the losses as stable over time, perhaps with a deterministic trend imposed. However, there is ample evidence that these processes are not stable over time. What is required is a method that allows for time-varying parameters in the process, yet still provides the shrinkage needed for sound ratemaking. In this paper we use an automobile insurance example to illustrate how this can be accomplished.

It is shown that there is a connection between rating in automobile insurance and the estimation of IBNR claims amounts because automobile insurance tariffs are mostly cross-classified by at least two variables (e.g. territory and driver class) and IBNR claims run-off triangles are always cross-classified by the two variables accident year and development year. Therefore, by translating the most well-known automobile rating methods into the claims reserving situation, some known and some unknown claims reserving methods are obtained. For instance, the automobile rating method of Bailey and Simon produces a new claims reserving method, whereas the model leading to the rating method called “marginal totals” produces the well-known IBNR claims estimation method called “chain ladder”. A drawback of this model is the fact that it is designed for the number of claims and not for the total claims amount for which it is usually applied.

As an alternative for both, rating and claims reserving, we describe a simple but realistic parametric model for the total claims amount which is based on the Gamma distribution and has the advantage of providing the possibility of assessing the goodness-of-fit and calculating the estimation error. This method is not very well known in automobile insurance—although a satisfactory application is reported—and seems to be completely unknown in the field of claims reserving, although its execution is nearly as simple as that of the chain ladder method.

A simple model for IBNR claims is presented. Estimates for the loss reserves and for the ultimate claims rate are derived. Approximations to the mean square error of the estimators are produced. A more specific parametric model is suggested for the case that we deal with claim numbers instead of claim amounts. The general method is illustrated by a practical application to the pricing of a casualty excess of loss cover.

At the 1990 ASTIN-colloquium, Schmitter posed the problem of finding the extreme values of the ultimate ruin probability ψ(u) in a risk process with initial capital u, fixed safety margin θ, and mean μ and variance σ2 of the individual claims. This note aims to give some more insight into this problem. Schmitter's conjecture that the maximizing individual claims distribution is always diatomic is disproved by a counterexample. It is shown that if one uses the distribution maximizing the upper bound e−Ru to find a ‘large’ ruin probability among risks with range [0, b], incorrect results are found if b is large or u small.

The related problem of finding extreme values of stop-loss premiums for a compound Poisson (λ) distribution with identical restrictions on the individual claims is analyzed by the same methods. The results obtained are very similar.