A recursive expression is derived for computing exactly the distribution of aggregate claims of a portfolio of life insurance policies. The recursion generalizes a formula of White and Greville for the claim numbers distribution and improves Kornya's approximation method for the aggregate claims distribution. It can be seen as the counterpart in the individual model of Panjer's recursion formula for the collective model.

We explicitly calculate price equilibria for power and logarithmic utility functions which—together with the exponential utility functions—form the so-called HARA (Hyperbolic Absolute Risk Aversion) class.

A price equilibrium is economically admissible in the market which is a closed system. Furthermore it is on the one side individually optimal for each participant of the market (in the sense of maximal expected utility), on the other side it is a Pareto optimum and thus collectively optimal for the market as a whole.

A conventional practice in standard risk theory considerations has been to assume that claims are paid immediately as they have incurred (see BPP, item 3.1c, BPP is used as an abbreviation for the book “Risk Theory”, 1984 edition, by Beard, Pentikäinen, Pesonen). The delay of the claims settlement has been, of course, a central aspect in reserve calculation theory and practices, and numerous valuable works have been published on this topic in recent years. However, its regard in general model building and in risk theory considerations has gained little attention until recent years. The purpose of this paper is to contribute to this research work by discussing how the “run-off” risk, i.e., the variability due to the delay of the claims payment, could be incorporated into the standard risk theory models as a separate entry (see BPP, item 10.2e) and to find some evaluation of the order of magnitude of the “extra” (if any) fluctuation so rendered. We expect that the proposed technique can also be utilized in testing different reserve calculation methods and in comparing their effectiveness. The main ideas follow very much along the lines given by Rantala in his doctoral thesis (1984).

One should appreciate the fact that any risk theory model can never be more than an idealization of real-life processes. An intricate problem for practitioners is to evaluate the uncertainties ensuing from the fact that the model, more or less, ignores or only approximates the factors affecting the real events, and that the practical applications are often based on and their necessary parameters estimated from observed data that are subject to random fluctuations and to many other kinds of uncertainties. The problem complex of the run-off risk, when understood in a broad sense, is so wide that it requires a series of studies, and this paper should be regarded as a first step only. The posing of the problem follows the conventional risk theory approach by using the mixed compound Poisson process further allowing for long-term variations of risk exposure (“cycles”), and now extending the model to cope with the delayed settlement of the claims. At this stage of the on-going researchwork the impact of the parameter estimation is excluded from consideration. Therefore, our results and the numerical examples, as given in what follows, do not describe the total uncertainty of the claims or the reserves.

Under-insurance is one of the scourges of homeowner insurance: when it is established after a claim that the rebuilding cost of a dwelling has been underestimated by the policyholder, the proportional rule applies in the sense that the indemnity is reduced in proportion of the under-insurance. To avoid this dramatic problem, prejudicial to all parties concerned, a Belgian company has in 1983 marketed a model to evaluate rebuilding costs: if the policyholder fills in correctly a two-page form, the company forgoes the application of the proportional rule.

The construction of this model is explained; then, it is shown how a statistical study sponsored by the Professional Union of Insurance Companies allowed the model to improve.

The industrial risks insurance market has shown a deficit for a few years, in France as well as in other European countries. In order to change that situation, insurers, with the help of their professional organizations, have begun to re-build their tariffs. Further, the progressive opening of the services European market has led these countries towards a standardization of their actuarial methods to estimate premiums. Among these methods, the credibility approach has already been adopted by Germany, Switzerland and partly by Belgium.

We have applied this method to statistical data from the French market. The estimation of the pure premium is based on the Bühlmann-Straub model with a measure of volume for every category (sums insured in our case). Loadings for fluctuation have been evaluated by the N.P. approximation of the distribution function of the loss ratio.

Beyond the simple use of the mathematical formulas, we have tried to measure the reaction of the model under different scenarii: risks typology with one, two or three levels, rough disturbance of the loss ratio of a category of risk, stability of the tariff over a long period of time, comparison of this tariff with the tariff built with the help of the existing French method based on a correction for large claims.

The variance of statistical estimates of outstanding claim payments for long-tailed general insurance portfolios is examined. The variance's three components are discussed. As there is no accepted technique for measuring this variance three methods are investigated empirically for its measurement—a parametric method, the jackknife method, and the bootstrap method. No method stands out as superior to the others and it is recommended that all three be evaluated and used to gauge the possible errors in the estimation of outstanding claims.

Three methods for fitting multiplicative models to observed, cross-classified risk data are compared. They are the method of Bailey–Simon, the method of marginal totals and a maximum likelihood method. The methods are applied to a number of risk data sets and compared with respect to balance and goodness-of-fit.

The estimation of risk premium for individual car models is discussed. Cluster analysis is used to identify groups of car models with similar technical attributes. Credibility theory is used to combine estimates of risk premium from individual car model claim statistics, group claim statistics, and a technical assessment carried out by car experts. The procedure is applied to a small set of car models.