Parameter estimation in case of loglinear modelled claim cost distribution characteristics is mathematically tractable, especially with the Inverse Gaussian and Lognormal distribution.

When setting or modifying insurance rates on the basis of observed data, adjustments should be made for inflation between the average claim date of the observation period and the average claim date covered by policies to which the new rates will apply. In automobile insurance ratemaking in the United States, the adjustment is made as follows:

where Lα = trended losses and loss adjustment expenses incurred,

L0 = observed losses and loss adjustment expenses incurred during the experience period,

= the predicted rate of inflation per quarter, and

k = the number of quarters for which losses are being trended.

The inflation coefficient, β, is an exponential growth factor that is estimated using the following equation:

where = average paid claim costs for the year ending in quarter t,

t = time, in quarters, t = 1, 2, …, 12, and

εt = a random disturbance term.

The estimation period for the equation is the twelve quarters immediately preceding the first forecast period, and the estimation is conducted utilizing ordinary least squares. Annual averages of claim costs are used as the dependent variable to smooth random and seasonal fluctuations.

It is demonstrated that the problems of balancing a reinsurance network and finding the maximum flow in a graph are identical. Gale's theorem is applied first in order to prove a conjecture of Sousselier concerning simple first order networks, next to extend those results to any network. The balanced reinsurance scheme can effectively be constructed by means of Ford and Fulkerson's algorithm, as is shown by an example. Finally, we demonstrate that graph theory provides tools able to tackle more general problems than the one considered by Sousselier: capacity constraints and reinsurance costs are introduced in the model.

There are many reasons why an insurer may choose to reinsure a part of his portfolio (see, for example, Carter (1979, p. 5 ff.)) and many ways in which he can assess the effectiveness of the reinsurance arrangements he makes. In this paper we assume the insurer wishes to reinsure a part of his portfolio in order to reduce its “riskiness”. We take as given a portfolio consisting of n independent risks together with the total premium charged to insure these risks and we investigate the effect on the degree of risk associated with the portfolio (see §3 for a definition) of varying the excess of loss or proportional reinsurance limits for each risk.

We are given an insurance portfolio consisting of n independent risks. A risk may consist of a single policy or a group of policies: the essential points being that a reinsurance limit, either excess of loss or proportional, is the same for all claims arising from a particular risk, although reinsurance limits may vary from one risk to another. We assume the claims arising from each risk have a compound Poisson distribution. To be more precise, we assume the number of claims arising from the i-th risk is a Poisson process with mean Pi claims each year and the size of each claim has distribution function Fi. As usual, the size of a claim is independent of the time at which it occurs and of all other claims. We also assume that Fi(O) = O for each i, so that we consider only positive claims amounts. We take as given the total annual premium, P, charged by the insurer in respect of these risks. We make no assumption about the way in which P is calculated but we do assume that

Much attention has been paid in the last ten years to the necessary solvency margin especially as influenced by risk theory considerations. In these calculations, by which the solvency margin is determined in such a way that the probability of ruin remains under a specified norm, such factors as type of insurance, size of portfolio, reinsurance etc. play an essential role.

In these calculations, based on pure risk theory, there is thus no attention paid to risks in the investment field, in the area of costs, etc. Yet one must not forget that the solvency margin must also act as a buffer against these uncertainties, in other words the solvency margin must be large enough to cover all risks to which the concern is liable, within stipulated limits of certainty.

One drawback of a pure risk theoretical approach, as interpreted above, is that the bases of the risk theory are often difficult to translate into practical terms, and furthermore will differ from company to company. The latter is also the case with the size of portfolio, reinsurance, etc. However, if this could all be translated into practical terms in the right way, the results would upset relations between competitors, so that it seems natural to make the required solvency margin the same for all companies.

Seen in this light Campagne's approach is of great importance—already in 1948 he realized that a scientific basis for the size of the solvency margin was necessary. He confined himself to the life insurance industry, and based his calculations on the idea that the total results of the companies working in a country should be such that, ceteris paribus, the chance of bankruptcy remains below a stipulated norm.

Some of the results obtained in an earlier paper entitled “An Invariance property of the Swiss premium calculation principle” by F. De Vylder and M. Goovaerts (1979) are generalized. For that purpose the notion of additivity and iterativity is extended. Some rather general characterization theorems for some premium calculation principles are obtained.

In a recent paper Seal (1980) calculated numerically survival probabilities based on Pareto claim distributions.

The Pareto density may be written as

Generalizing, the Pareto distribution may be regarded as a special case of the so-called beta-prime distribution (Keeping, 1962, p. 83) with density function

where B(p, q) = is the beta function.

In his paper Seal (1980, Appendix 1) arrived at a contradiction concerning this beta-prime distribution. He found on one side that all derivatives of the characteristic function exist at the origin and on the other side that only the moments of order n < q exist. In this note we will show that this contradiction is due to the use of an incorrect expression for the characteristic function of the beta-prime distribution, which was taken over from Johnson and Kotz (1970, Ch. 26) and Oberhettinger (1973, Table A).