The approximation of the individual risk model by a compound Poisson model plays an important role in computational risk theory. It is thus desirable to have sharp lower and upper bounds for the error resulting from this approximation if the aggregate claims distribution, related probabilities or stop-loss premiums are calculated.

The aim of this paper is to unify the ideas and to extend to a more general setting the work done in this connection by Bühlmann et al. (1977), Gerber (1984) and others. The quality of the presented bounds is discussed and a comparison with the results of Hipp (1985) and Hipp & Michel (1990) is made.

The theory of linear filtering of stochastic processes provides continuous time analogues of finite-dimensional linear Bayes estimators known to actuaries as credibility methods. In the present paper a selfcontained theory is built for processes of bounded variation, which are of particular relevance to insurance. Two methods for constructing the optimal estimator and its mean squared error are deviced. Explicit solutions are obtained in a continuous time variation of Hachemeister's regression model and in a homogeneous doubly stochastic generalized Poisson process. The traditional discrete time set-up is compared to the one with continuous time, and some merits of the latter are pointed out.

In the present paper we study the question of how to allocate the reinsurance premium between the sub-portfolios when an excess of loss treaty is to be shared between several sub-portfolios. Several allocation schemes based on the expected value principle and the standard deviation principle are suggested. The calculations are relatively simple with unlimited free reinstatements. However, with limited and/or paid reinstatements the situation becomes rather tricky, and we therefore suggest a simulation scheme.

In this paper we present algorithms to calculate the probability and severity of ruin in both finite and infinite time for a discrete time risk model. We show how the algorithms can be applied to give approximate values for the same quantities in the classical continuous time risk model.

The estimation of outstanding claims is one of the important aspects in the management of the insurance business. Various methods have been widely dealt with in the actuarial literature. Exploration of the inaccuracies involved is traditionally based on a post-facto comparison of the estimates against the actual outcomes of the settled claims. However, until recent years it has not been usual to consider the inaccuracies inherent in claims reserving in the context of more comprehensive (risk theoretical) models, the purpose of which is to analyse the insurer as a whole. Important parts of the technique which will be outlined in this paper can be incorporated into over-all risk theory models to introduce the uncertainty involved with technical reserves as one of the components in solvency and other analyses (Pentikäinen et al. (1989)).

The idea in this paper is to describe a procedure by which one can explore how various reserving methods react to fictitious variations, fluctuations, trends, etc. which might influence the claims process, and, what is most important, how they reflect on the variables indicating the financial position of the insurer. For this purpose, a claims process is first postulated and claims are simulated and ordered to correspond to an actual handling of the observed claims of a fictitious insurer. Next, the simulation program will ‘mime’ an actuary who is calculating the claims reserve on the basis of these ‘observed’ claims data. Finally, the simulation is further continued thus generating the settlement of the reserved claims. The difference between reserved amounts and settled amounts gives the reserving (run-off) error in this particular simulated case. By repeating the simulation numerous times (Monte Carlo method) the distribution of the error can be estimated as well as its effect on the total outcome of the insurer.

Under certain conditions, a Bonus-Malus system can be interpreted as a Markov chain whose n-step transition probabilities converge to a limit probability distribution. In this paper, the rate of the convergence is studied by means of the eigenvalues of the transition probability matrix of the Markov chain.

This paper describes a technique to find the maximal stop-loss premiums in a given retention for a compound Poisson risk with known parameter, and known mean and variance of the claims. Restricting to an arithmetic and finite support of the claims, one gets an optimization problem of a non-linear function with a computable gradient, under linear constraints.

Numeraical results are given contrasting the method with the method of a previous paper, where only diatomic distributions were considered.

The sensitivity of the ruin probability depending on the claim size distribution has been the topic of several discussion papers in recent ASTIN Bulletins. This discussion was initiated by a question raised by Schmitter at the ASTIN Colloquium 1990 and attempts to make further contributions to this problem. We find the necessary and sufficient conditions for fitting three given moments by diatomic and diexponential distributions. We consider three examples drawn from fire (large spread), individual life (medium spread) and group life (small spread) insurance data, fit them with diatomics and diexponentials whenever the necessary and sufficient conditions are met, and compute the ruin probabilities using well known formulas for discrete and for combination of exponentials claim amounts. We then compare our approximations with the exact values that appeared in the literature. Finally we propose using diatomic and diexponential claim distributions as tools to study the Schmitter problem.

Stimulated by a recent contribution by G. Venter in this journal the adequate-ness of (re-)insurance premium calculation based on the hypothesis of arbitrage free (re-)insurance markets is questioned. It is argued that—in contrast to the theory of financial markets—it is not reasonable to demand that insurance markets are arbitrage free. In addition the adjusted distribution principles put forward by Venter are claimed to be invalid.

Albrecht's interesting discussion raises two major points: is an arbitrage free insurance market a reasonable assumption, and are adjusted distribution principles appropriate?

I had assumed an arbitrage free insurance market without really discussing reasons in support of it. Certainly there have been arbitrage profits made through reinsurance, possibly, for example, by selling excess on excess at rates based on regular excess policies, and perhaps in the London Market Excess arena. But even if arbitrage occurs frequently, I feel that an arbitrage free market is still a reasonable assumption for theoretical developments, if we take it to mean that systematic ongoing arbitrage is not possible. Although some arbitrage situations have probably lasted a few years, all I am aware of eventually have disappeared as they became better known. Either entrepreneurs try to outbid each other to take advantage of them, or the sellers lose their willingness to play. A list of insurers that are regularly making arbitrage profits would be an interesting refutation of this assumption.

Albrecht gives some examples of the well known and quite true proposition that larger portfolios are more stable. Large insurers can maintain the same security level at a lower price. Our risk theory training leads actuaries to believe that the smaller needed security premium for large insurers will induce them to charge lower prices. This is not necessarily true in the market, however. Larger insurers may in fact charge the market price and make more profits, for example. Also, it would be fairly surprising if small insurers actually do get away with charging more for the same risks. If they do, they should buy prorata reinsurance heavily from the large insurers at the large insurers' risk price and pocket the difference. There are of course transaction costs to buying prorata reinsurance; those may in fact be the limit of what the smaller insurers can demand as the supposed bonus price they get from the market for being small.