The majority of optimal Bonus-Malus Systems (BMS) presented up to now in the actuarial literature assign to each policyholder a premium based on the number of his accidents. In this way a policyholder who had an accident with a small size of loss is penalized unfairly in the same way with a policyholder who had an accident with a big size of loss. Motivated by this, we develop in this paper, the design of optimal BMS with both a frequency and a severity component. The optimal BMS designed are based both on the number of accidents of each policyholder and on the size of loss (severity) for each accident incurred. Optimality is obtained by minimizing the insurer's risk. Furthermore we incorporate in the above design of optimal BMS the important a priori information we have for each policyholder. Thus we propose a generalised BMS that takes into consideration simultaneously the individual's characteristics, the number of his accidents and the exact level of severity for each accident.

This paper examines an insurance or risk premium calculation method called the mean-value-distortion pricing principle in the general framework of anticipated utility theory. Then the relationship between comonotonicity and independence is explored. Two types of risk aversion and optimal reinsurance contracts are also discussed in the context of the pricing principle.

In this paper we consider the problem raised in the Astin Bulletin (1999) by Prof. Benktander at the occasion of his 80th birthday concerning the choice of an appropriate claim size distribution in connection with reinsurance rating problems. Appropriate models for large claim distributions play a central role in this matter. We review the literature on extreme value methodology and consider its use in reinsurance. Whereas the models in extreme-value methods are non-parametric or semi-parametric of nature, practitioners often need a fully parametric model for assessing a portfolio risk both in the tails and in more central portions of the claim distribution. To this end we propose a parametric model, termed the generalised Burr-gamma distribution, which possesses such flexibility. Throughout we consider a Norwegian fire insurance portfolio data set in order to illustrate the concepts. A small sample simulation study is performed to validate the different methods for estimating excess-of-loss reinsurance premiums.

A method of inverting the Laplace transform based on the integration between zeros technique and a simple acceleration algorithm is presented. This approach was designed to approximate ultimate ruin probabilities for Γ-convolutions claim sizes, but it can be also used with other distributions. The stable algorithm obtained yields interval approximations (lower and upper bounds) to any desired degree of accuracy even for very large values of u (1,000,000), initial reserves, without increasing the number of computations. This last fact can be considered an interesting property compared with other recursive methods previously used in actuarial literature or other methods of inverting Laplace transforms.

This paper proposes bonus-malus systems for fleets of vehicles, by using the individual characteristics of both the vehicles and the carriers. Bonus-malus coefficients are computed from the history of claims or from the history of safety offences of the carriers and the drivers. The empirical results are derived from a data set obtained from the Société de l'Assurance Automobile du Québec, the public insurer for bodily injuries and the regulator of road safety.

Based on the notions of value-at-risk and expected shortfall, we consider two functionals, abbreviated VaR and RaC, which represent the economic risk capital of a risky business over some time period required to cover losses with a high probability. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). Quantitatively, RaC is equal to VaR plus an additional stop-loss dependent term, which takes into account the average amount at loss. Furthermore, RaC is additive for comonotonic risks, which is an important extremal situation encountered in the modeling of dependencies in multivariate risk portfolios. Numerical illustrations for portfolios of gamma distributed risks follow. As a result of independent interest, new analytical expressions for the exact probability density of sums of independent gamma random variables are included, which are similar but different to previous expressions by Provost (1989) and Sim (1992).

In this paper we study a class of Mixed Bivariate Poisson Distributions by extending the Hofmann Distribution from the univariate case to the bivariate case.

We show how to evaluate the bivariate aggregate claims distribution and we fit some insurance portfolios given in the literature.

This study typically extends the use of the Bivariate Independent Poisson Distribution, the Mixed Bivariate Negative Binomial and the Mixed Bivariate Poisson Inverse Gaussian Distribution.

Recently, Wang's premium principle (Wang, 1995, 1996) has been discussed by many authors. Considerable attention has been given to the conditions under which Wang's premium principle can be reduced to the standard deviation premium principle. In this paper, we have got two results on this problem. One is that the natural set is a location-scale family if Wang's premium principle can be reduced to the SD premium principle for all surjective distortions. The other is that the natural set is a location-scale family for all power distortions.

Whittaker graduation is applied to the spatial smoothing of insurance data. Such data (e.g. claim frequency) form a surface over the 2-dimensional geographic domain to which they relate. Observations on this surface are subject to sampling error. They need to be smoothed spatially if a reliable estimate of the underlying surface is to be obtained.

A measure of smoothness of a surface has been defined. This has been incorporated in 2-dimensional Whittaker graduation to effect the necessary smoothing. The details of this are worked out in Section 4. The procedure is illustrated by numerical example in Section 5. The Bayesian interpretation of this form of spatial smoothing is discussed, and used to assist in the selection of the Whittaker relativity constant.

The paper analyses the question: Should an insurance customer carry an occurred loss himself, or should he make a claim to the insurance company? This question is important within bonus-malus contracts with individual experience adjustments of the premium. The analysis model includes a bonus hunger strategy where the customers prefer the most profitable financial alternative, that is, the alternative which represents the lowest rate of interest. Hence the loss of bonus after a claim is calculated as a rate of interest paid from the customer to the insurer. Within this model the paper outlines the existence of a true compensation function and a relative cost function for each customer. A set of properties for bonus-malus contracts are presented and discussed. A concrete example of a bonus-malus system and an insurance compensation function illustrates the theoretical framework in a practical manner.

The paper analyses the questions: Should – or should not – an individual buy insurance? And if so, what insurance coverage should he or she prefer? Unlike classical studies of optimal insurance coverage, this paper analyses these questions from a bonus-malus point of view, that is, for insurance contracts with individual bonus-malus (experience rating or no-claim) adjustments. The paper outlines a set of new statements for bonus-malus contracts and compares them with corresponding classical statements for standard insurance contracts. The theoretical framework is an expected utility model, and both optimal coverage for a fixed premium function and Pareto optimal coverage are analyzed. The paper is an extension of another paper by the author, see Holtan (2001), where the necessary insight to – and concepts of – bonus-malus contracts are outlined.

The normal inverted gamma mixture or generalized Student t and the symmetric double Weibull, as well as their logarithmic counterparts, are proposed for modeling some loss distributions in non-life insurance and daily index return distributions in financial markets. For three specific data sets, the overall goodness-offit from these models, as measured simultaneously by the negative log-likelihood, chi-square and minimum distance statistics, is found to be superior to that of various “good” competitive models including the log-normal, the Burr, and the symmetric α-stable distribution. Furthermore, the study justifies on a statistical basis different important models of financial returns like the model of Black-Scholes (1973), the log-Laplace model of Hürlimann (1995), the normal mixture by Praetz (1972), the symmetric α-stable model by Mandelbrot (1963) and Fama (1965), and the recent double Weibull as limiting geometric-multiplication stable scheme in Mittnik and Rachev (1993). As an application, the prediction of one-year index returns from daily index returns is discussed.

In the last few years we have witnessed growing interest in Dynamic Financial Analysis (DFA) in the nonlife insurance industry. DFA combines many economic and mathematical concepts and methods. It is almost impossible to identify and describe a unique DFA methodology. There are some DFA software products for nonlife companies available in the market, each of them relying on its own approach to DFA. Our goal is to give an introduction into this field by presenting a model framework comprising those components many DFA models have in common. By explicit reference to mathematical language we introduce an up-and-running model that can easily be implemented and adjusted to individual needs. An application of this model is presented as well.