This paper provides an introduction to asset pricing theory and its applications in non-life insurance. The first part of the paper presents a basic review of asset pricing models, including discrete and continuous time capital asset pricing models (the CAPM and ICAPM), arbitrage pricing theory (APT), and option pricing theory (OPT). The second part discusses applications in non-life insurance. Among the insurance models reviewed are the insurance CAPM, discrete time discounted cash flow models, option pricing models, and more general continuous time models. The paper concludes that the integration of actuarial and financial theory can provide major advances in insurance pricing and financial management.

For a risk whose annual claim amounts are conditionally i.i.d. with respect to a risk parameter, it is known that the Bayes and credibility premiums are asymptotically optimal in terms of losses. In the present note it is shown that the Bayes and credibility premiums actually converge to the individual premium.

Minimax estimation procedures for the mean vector of a distribution on a compact set under squared error type loss functions are considered. In particular, a Dirichlet process prior is used to show that a linear function of is a minimax estimator in the class of all measurable estimators and all possible distributions. This effort extends some earlier work of Bühlmann to a more general setting.

Average absolute (instead of quadratic) deviation from median (instead of expectation) is better suited to determine the safety loading for insurance premiums than standard deviation: The corresponding premium functionals behave additive under the practically relevant risk sharing schemes between first insurer and reinsurer.

We consider exponential smoothing Yn = αXn + (1 − α)Yn−1, 0 < α < 1, in experience rating. Here the premium Yn is determined by the policy's own claims history (Xn). In order to uniformize the fluctuation of premiums, it is appropriate to use a bigger α for the big policies than for the small ones. When the size of the policy changes with time, a need arises to change α correspondingly. It has recently been shown that changing based on the size of the premiums Yn may lead to too low a tariff level. This result is presented here and illustrated by means of simulation. Further, some general results are given how the changing can be made without a decline in the tariff level. The results are applied to a tariff system in which the linking of the smoothing parameter to the size of the policy is particularly motivated.

In the present paper, different estimators of the Pareto parameter α will be proposed and compared to each others.

First traditional estimators of α as the maximum likelihood estimator and the moment estimator will be deduced and their statistical properties will be analyzed. It is shown that the maximum likelihood estimator is biased but it can easily be modified to an minimum-variance unbiased estimator of a. But still the coefficient of variance of this estimator is very large.

For similar portfolios containing same types of risks we will expect the estimated α-values to be at the same level. Therefore, credibility theory is used to obtain an alternative estimator of α which will be more stable and less sensitive to random fluctuations in the observed losses.

Finally, an estimator of the risk premium for an unlimited excess of loss cover will be proposed. It is shown that this estimator is a minimum-variance unbiased estimator of the risk premium. This estimator of the risk premium will be compared to the more traditional methods of calculating the risk premium.

The subject of predicting outstanding claims on a porfolio of general insurance policies is approached via the theory of hierarchical Bayesian linear models. This is particularly appropriate since the chain ladder technique can be expressed in the form of a linear model. The statistical methods which are applied allow the practitioner to use different modelling assumptions from those implied by a classical formulation, and to arrive at forecasts which have a greater degree of inherent stability. The results can also be used for other linear models. By using a statistical structure, a sound approach to the chain ladder technique can be derived. The Bayesian results allow the input of collateral information in a formal manner. Empirical Bayes results are derived which can be interpreted as credibility estimates. The statistical assumptions which are made in the modelling procedure are clearly set out and can be tested by the practitioner. The results based on the statistical theory form one part of the reserving procedure, and should be followed by expert interpretation and analysis. An illustration of the use of Bayesian and empirical Bayes estimation methods is given.