We investigate the implications of a dual approach to the graduation of the force of mortality based on the modelling of the exposures as gamma random variables, as opposed to the modelling of the numbers of deaths as Poisson random variables.

This paper deals with the bivariate generalized Poisson distribution. The distribution is fitted to the aggregate amount of claims for a compound class of policies submitted to claims of two kinds whose yearly frequencies are a priori dependent. A comparative study with the bivariate Poisson distribution and with two bivariate mixed Poisson distributions has been carried out, based on data concerning natural events insurance in the USA and third party liability automobile insurance in France.

The objective of this paper is to make allowance for cost of claims in experience rating. We design here a bonus-malus system for the pure premium of insurance contracts, from a rating based on their individual characteristics. Empirical results are presented, that are drawn from a French data base of automobile insurance contracts.

The upper bound provided by Lundberg's inequality can be improved for the probability of ruin in finite horizon, as Gerber (1979) has shown. This paper studies this upper bound as a function of the retention limit, for an excess of loss arrangement, and compares it with the probability of ruin.

This paper shows how credibility theory can be encompassed within the theory of Hierarchical Generalized Linear Models. It is shown that credibility estimates are obtained by including random effects in the model. The framework of Hierarchical Generalized Linear Models allows a more extensive range of models to be used than straightforward credibility theory. The model fitting and testing procedures can be carried out using a standard statistical package. Thus, the paper contributes a further range of models which may be useful in a wide range of actuarial applications, including premium rating and claims reserving.

Many authors have observed that Hachemeisters Regression Model for Credibility – if applied to simple linear regression – leads to unsatisfactory credibility matrices: they typically ‘mix up’ the regression parameters and in particular lead to regression lines that seem ‘out of range’ compared with both individual and collective regression lines. We propose to amend these shortcomings by an appropriate definition of the regression parameters:

–intercept

–slope

Contrary to standard practice the intercept should however not be defined as the value at time zero but as the value of the regression line at the barycenter of time. With these definitions regression parameters which are uncorrected in the collective can be estimated separately by standard one dimensional credibility techniques.

A similar convenient reparametrization can also be achieved in the general regression case. The good choice for the regression parameters is such as to turn the design matrix into an array with orthogonal columns.

A new two-parameter family of analytical functions will be introduced for the modelling of loss distributions and exposure curves. The curve family contains the Maxwell-Boltzmann, the Bose-Einstein and the Fermi-Dirac distributions, which are well known in statistical mechanics. The functions can be used for the modelling of loss distributions on the finite interval [0, 1] as well as on the interval [0, ∞]. The functions defined on the interval [0, 1] are discussed in detail and related to several Swiss Re exposure curves used in practice. The curves can be fitted to the first two moments μ and σ of a loss distribution or to the first moment μ and the total loss probability p.

We develop a semi-parametric predictor of the IBNR reserve in a macro-model when the claim amount for a certain accident and development year can be expressed in a loglinear form composed of a deterministic part and a random error. We need to make assumptions only on the first two moments of the error, without any specified parametric assumption on its distribution. We give its properties, present its advantages and compare the estimates obtained with various predictors of the IBNR reserve, parametric and non-parametric, using a data set.

Good estimates for the tails of loss severity distributions are essential for pricing or positioning high-excess loss layers in reinsurance. We describe parametric curve-fitting methods for modelling extreme historical losses. These methods revolve around the generalized Pareto distribution and are supported by extreme value theory. We summarize relevant theoretical results and provide an extensive example of their application to Danish data on large fire insurance losses.

Alexander McNeil's (1996) study of the Danish data on large fire insurance losses provides an excellent example of the use of extreme value theory in an important application context. We point out how several alternate statistical techniques and plotting devices can buttress McNeil's conclusions and provide flexible tools for other studies.