This paper presents a stochastic model of capitalization which takes into account the financial risk in the actuarial processes.

We first introduce a stochastic differential equation which allows us to define the capitalization and actualization processes.

We use these concepts to present a new principle of premium calculation for the capitalization operations, based on the equality between backward reserve and conditional expectation of the forward reserve.

A generalization of the classical Thiele equation in life insurance is also given.

Numerical examples illustrate the model.

Chains of reinsurance were first modelled by Gerber, in a special case. It is shown that more general results can be obtained by applying Borch's theorem. The Pareto-optimal reinsurance indemnities are uniquely determined using the only assumption that the participating companies use exponential utility functions. A simple comparison then shows that Gerber's indemnities are not Pareto-optimal. Even if no assumption at all is introduced, the indemnities are shown to be closely linked to the risk aversions of the participants.

Estimation of pure premiums for alternative rate classes using regression methods requires the choice of a functional form for the statistical model. Common choices include linear and log-linear models. This paper considers maximum likelihood estimation and testing for functional form using the power transformation suggested by Box and Cox. The linear and log-linear models are special cases of this transformation. Application of the procedure is illustrated using auto insurance claims data from the state of Massachusetts and from the United Kingdom. The predictive accuracy of the method compares favorably to that for the linear and log-linear models for both data sets.

It has been argued in previous studies that the expected utility decision criterion provides useful insights for certain insurance problems, such as underwriting, reinsurance and portfolio optimization problems. In this study three new models are developed which extend and generalize previous results. The first model analyses modified stop-loss reinsurance. The second model analyses risk pooling where both inward and outward reinsurance occur. Expected utility calculations can be used to provide insight on the attractiveness of competing reinsurance and risk pooling options. The third model is for strategic planning, where risk/reward tradeoffs for all the insurer's business activities (underwriting, investment, reinsurance) can be considered in aggregate. The simpler models can often be solved analytically however the strategic planning model is relatively complex and uses Monte Carlo techniques to determine retained earnings distributions. The expected utility approach has been found to be powerful, flexible and comprehensive as a decision aiding mechanism. From a normative viewpoint, this approach accounts very well for all the important decision elements. Recent developments in decision support systems will allow these models to be made available to practitioners in user friendly forms.

Kornya-type higher order approximations are derived for the aggregate claims distribution and for stop loss premiums in the individual model with arbitrary positive claims. Absolute error bounds and error bounds based on concentration functions are given. In the Gerber portfolio containing 31 policies, second order approximations lead to an accuracy of 3 × 10−4, and third order approximations to 1.7 ×10−5.

Upper and lower bounds are derived for the stop-loss premium of compound distributions with fixed claim number distribution and known mean, variance and range for the claim severity distribution.

This paper deals with experience rating of claims processes of ARIMA structures. By experience rating we mean that future premiums should be only a function of past values of the claims process. The main emphasis is on demonstrating the usefulness of the control-theoretical approach in the search for optimal rating rules. Optimality is here defined to mean as smooth a flow of premiums as possible when the variation in the accumulated profit is restricted to a certain amount. First it is shown how the underlying model in its simplest form can be transformed into the state-space form. Then the Kalman filter technique is used to find the optimal rules. Also a time delay in information is taken into account. The optimal rules are illustrated by examples.

In the present paper the author investigates the problem of calculating the net premium for some versions of the largest claims reinsurance cover. A very handy recursive rating method is derived by applying some recursion formulas for the expectations of order statistics.

The distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous).

Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions.

It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.

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 consider a generalisation of a risk process under experience rating when the aggregation of claims up to time t is a Brownian motion (B.M.) with a drift. We prove that the distribution of ruin before time t is equivalent to the distribution of the first passage time of B.M. for parabolic boundary.

Using Wald identity for continuous time we give an explicit formula for this distribution. A connection is made with discounting risk model when the income process is a diffusion.

When the aggregation of claims is a mixture of B.M. and compound Poisson process, we give (using Gerber's result 1973) an upper bound for the distribution of finite time ruin probability.