The paper presents an extension of the classical Cramér–Lundberg ruin theory: the famous upper bound for the ruin probability with an infinite time horizon can be extended in a certain sense to the short and middle term case. Furthermore, a relation between the average values of lifetime and ruin amount is given.

We give an extension of the Economic Premium Principle treated in Astin Bulletin, Volume 11 where only exponential utility functions were admitted. The case of arbitrary risk averse utility functions leads to similar quantitative results. The role of risk aversion in the treatment is essential. It also permits an easy proof for the existence of equilibrium.

We consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

Much of the risk theory literature deals with the total claims distribution where pk = the probability of k claims and S(x) is the distribution function of severity. Both pk and S(x) are univariate probability distributions. Thus, F(x) can be interpreted as a model of claims from one class of policies or as an aggregate model where pk and S(x) represent mixed probability distributions from a heterogeneous portfolio of policies. An alternative approach to modelling total claims in the latter case would be to recognize explicitly that total claims are the result of the interaction of multivariate processes. In the most general case, total claims arise from a multivariate accident process where each accident triggers multivariate claims frequency and severity processes.

The purpose of this article is to present a multivariate model of total claims and to develop the cumulant generating function of this distribution. Such a model is superior to the traditional model in two respects: (1) It permits explicit recognition of shifts in the overall portfolio composition. Applications of the traditional model, in contrast, rely on the assumption that the portfolio composition is relatively constant over time. (2) It facilitates the evaluation of the effects of reinsurance on the total claims distribution when the reinsurance arrangements are not the same in different segments of the portfolio.

Probabilities of ruin are solutions of differential or integrodifferential equations. Solving such equations numerically can be performed by means of approximate quadrature formulae for the convolution part of the equation. In this contribution it is shown how applicable recursion formulae, giving results within a prescribed tolerance level, can be obtained. Some numerical results are displayed.

The allocation of operating costs among the lines of an insurance company is one'of the toughest problems of accounting; it is first shown that most of the methods used by the accountants fail to satisfy some natural requirements. Next it is proved that a cost allocation problem is identical to the determination of the value of a cooperative game with transferable utilities, and 4 new accounting methods that originate from game theory are proposed. One of those methods, the proportional nucleus, is recommended, due to its properties. Several practical examples are discussed throughout the paper.