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.

The aggregate claims process is modelled by a process with independent, stationary and nonnegative increments. Such a process is either compound Poisson or else a process with an infinite number of claims in each time interval, for example a gamma process. It is shown how classical risk theory, and in particular ruin theory, can be adapted to this model. A detailed analysis is given for the gamma process, for which tabulated values of the probability of ruin are provided.

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.

In this paper we give a recursive scheme, involving Panjer's recursion, to compute the distribution of a compound sum of integer claims, when the number of summands follows a Generalized Poisson distribution. Also, an elegant derivation is given for some basic properties of this counting distribution.

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.

In this paper we present an algorithm for the approximate calculation of finite time survival probabilities for the classical risk model. We also show how this algorithm can be applied to the calculation of infinite time survival probabilities. Numerical examples are given and the stability of the algorithms is discussed.

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.

Constraints imposed on premium calculation principles are studied under one aspect of competitive market theory: the impossibility of systematic arbitrage. Principles based on second moments or utility theory are shown to lead to arbitrage possibilities, while some other principles do not.

The model introduced may be treated as a mixed two-way analysis of variance with fixed company effects and random time effects. Further, the risk volumes are integrated into the model in such a way that the unexplained variance is inversely proportional to the risk volume of each company. The proposed model is used to analyze loss ratio data from the general insurance market in Kuwait. The maximum likelihood estimates of the structural parameters are obtained. These estimates are then used to compute the loss ratios and solvency margins for the four domestic insurance companies.

Instead of determining for a fire insurance portfolio the loss distribution purely based on the claims experience, we try to determine it based on the sums insured.

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.