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.

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.

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.

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.