This paper considers a particular credibility model for the claim numbers N1, N2, …, Nn, … of a single risk within a collective in successive periods 1, 2, …, n, … In the terminology of Jewell (1975) the model is an evolutionary credibility model, which means that the underlying risk parameter Λ is allowed to vary in successive periods (the structure function is allowed to be time dependent). Evolutionary credibility models for claim amounts have been studied by Bühlmann (1969, pp. 164–165), Gerber and Jones (1975), Jewell (1975, 1976), Taylor (1975), Sundt (1979, 1981, 1983) and Kremer (1982). Again in Jewell's terminology the considered model is on the other hand stationary, in the sense that the conditional distribution of Ni given the underlying risk parameter does not vary with i.

The computation of the credibility estimate of Nn+1 involves the considerable labor of inverting an n × n covariance matrix (n is the number of observations). The above mentioned papers have therefore typically looked for model structures for which this inversion is unnecessary and instead a recursive formula for the credibility forecast can be obtained. Typically nth order stationary a priori sequences (e.g., ARMA (p, q)-processes) lead to an nth order recursive scheme. In this paper we impose the restriction that the conditional distribution of Ni is Poisson (which by the way leads to a model identical to the so called “doubly stochastic Poisson sequences” considered in the theory of stochastic point processes). What we gain is a recursive formula for the coefficients of the credibility estimate (not for the estimate itself!) in case of an arbitrary weakly stationary a priori sequence. In addition to this central result the estimation of the structural parameters is considered in this case and some more special models are analyzed. Among them are EARMA-processes (which are positive-valued stationary sequences possessing exponentially distributed marginals and the same autocorrelation structure as ARMA-processes) as a priori sequence and models which can be considered as (discrete) generalizations of the Pólya process.

Recursive credibility estimation is discussed from the viewpoint of linear filtering theory. A conjunction of geometric interpretation and the innovation approach leads to general algorithms not developed before. Moreover, covariance characterizations considered by other researchers drop our elegantly as a result of geometric considerations. Examples are presented of Kalman type filters valid for non-Gaussian measurements.

The paper deals with the renewal equation governing the infinite-time ruin probability. It is emphasized as intended to be no more than a pleasant ramble through a few scattered results. An interesting connection between ruin probability and a recursion formula for computation of the aggregate claims distribution is noted and discussed. The relation between danger of the claim size distribution and ruin probability is reexamined in the light of some recent results on stochastic dominance. Finally, suggestions are made as to the way in which the formula for ruin probability leads easily to conclusions about the effect on that probability of the long-tailedness of the claim size distribution. Stable distributions, in particular, are examined.

The unbayesed credibility procedure proposed by Gerber is revisited. Its performance is discussed, connections are drawn to earlier literature, and some possible ideas of generalizations are investigated (and found fruitless).

This paper is intended to show how premiums are related to the stability criterion imposed on a portfolio of risks and to the dividend requirements for the capital invested into the insurance operation. The point is that premium calculation should be seen as a consequence of the strategic concepts adopted by the insurance carrier.

We improve on some results of Sundt (1982) on the asymptotic behaviour of compound negative binomial distributions.

Credibility theory refers to the use of linear least-squares theory to approximate the Bayesian forecast of the mean of a future observation; families are known where the credibility formula is exact Bayesian. Second-moment forecasts are also of interest, for example, in assessing the precision of the mean estimate. For some of these same families, the second-moment forecast is exact in linear and quadratic functions of the sample mean. On the other hand, for the normal distribution with normal-gamma prior on the mean and variance, the exact forecast of the variance is a linear function of the sample variance and the squared deviation of the sample mean from the prior mean. Bühlmann has given a credibility approximation to the variance in terms of the sample mean and sample variance.

In this paper, we present a unified approach to estimating both first and second moments of future observations using linear functions of the sample mean and two sample second moments; the resulting least-squares analysis requires the solution of a 3 × 3 linear system, using 11 prior moments from the collective and giving joint predictions of all moments of interest. Previously developed special cases follow immediately. For many analytic models of interest, 3-dimensional joint prediction is significantly better than independent forecasts using the “natural” statistics for each moment when the number of samples is small. However, the expected squared-errors of the forecasts become comparable as the sample size increases.

This paper considers reinsurance retention limits in cases where the cedent has a choice between a pure quota-share treaty, a pure excess of loss treaty or a combination of the two. Our primary aim is to find the combination of retention limits which minimizes the skewness coefficient of the insurer's retained risk subject to constraints on the variance and the expected value of his retained risk. The results are given without specifying precisely how the excess of loss reinsurance premium is calculated. It is also shown that, depending to some extent on the constraint on the variance, the solution to the problem is a pure excess of loss treaty if the excess of loss premium is calculated using the expected value or standard deviation principle but that this need not be true if the variance principle is used.

We consider particular semi-Markov risk models M/SM and /SM for which the interarrival distributions are exponential with parameters depending of the risk type. We obtain theoretical expressions for the ruin probabilities on an infinite horizon.

In the special case of exponential distributions for the claim amounts, ruin probabilities are in both models solutions of linear differential systems. These systems are explicitly solved when there are only two risk types.

A simple recursion for the n-fold convolution of an arithmetic distribution with itself is developed and its relation to Panjer's algorithm for compound distributions is shown.

The reinsurer has a monopoly in the following sense: He will select a random variable P that determines the reinsurance premiums. The first insurer can purchase a payment of R (a random variable) for a premium of π = E[PR]. For known P, the first insurer chooses R to maximize his expected utility. Knowing this, i.e., the demand for reinsurance as a function of P, the reinsurer chooses P to maximize his utility. The resulting pair (P, R) is called the Bowley solution. Assuming exponential, quadratic and/or linear utility functions, some explicit results are obtained.

An outline is given of a proposed system for solvency control in non-life insurance that has recently been discussed within a Working Party appointed by the Norwegian supervisory authorities. According to this system the factual technical reserves must at any time be sufficient to meet, with high probability, all future liabilities stipulated by insurance contracts that have either expired or are currently in force. The system is applied to a provisional, simple model that has been fitted to claims data assembled from Norwegian non-life companies. The numerical examples illustrate, inter alia, how the required reserve depends on the volume of the business, the portfolio mix, and the reinsurance cover.

The run off-pattern of long-term reinsurance treaties is described by means and standard deviations of logarithmic increments of premiums and loss ratios in a normal distribution. From this description forecasts of ultimate claims and current IBNR-reserves are derived, with associated distributions and confidence limits. Aggregation from individual treaties to portfolio level is proposed by normal approximation. Security loading of IBNR-reserves is proposed by a contingency reserve at portfolio level.