The decision problem of acceptance or rejection of life insurance proposals is formulated as a two-person non cooperative game between the insurer and the set of the proposers. Using the minimax criterion or the Bayes criterion, it is shown how the value and the optimal strategies can be computed, and how an optimal set of medical informations can be selected and utilized.

We gıve a complete parametrıc solutıon of the followıng problem: Fınd a claım sıze dıstrıbutıon F on the fınıte ınterval [ο, ω], maxımizıng the stop-loss premıum correspondıng to a gıven excess e, under the constraınts that the fırst moment of F be at most equal to μ and the second at most equal to ν The method used ıs the dualıty technıque ın semı-contınuous lınear programmıng descrıbed in De Vylder (1978) Thıs technıque ıs summarızed, wıthout proofs, ın the fırst part of the paper.

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

This short note has as its starting point an interesting article by Taylor (1979) in which he considered the effects of inflation on a risk process. Taylor showed that if the premium density increased at the same rate as the cost of individual claims then, under certain conditions, ultimate ruin was certain. This raises a natural question, viz. “If the cost of individual claims is increasing how should the premiums be increased in order to keep the probability of ruin under control ?” It is this question that we shall be considering in this note.

In the next section we define the risk process that we shall be studying for the remainder of this note. Our process is essentially a compound Poisson process except that we allow the distribution function of an individual claim to depend on the time at which the claim occurs. We start the third section by deriving, with the help of a general result of Gerber (1973), a formula for the future premium density that will keep the probability of ruin for our process below a predetermined bound. We then derive a simple approximation to this formula that shows more clearly how we require the premium density to change in relation to the change in claims costs. Finally we show that if we consider the same process with annual premiums then the probability of ultimate ruin will be kept below a predetermined bound if the annual premium is calculated according to the principle of zero utility with an exponential utility function or, as a first approximation, according to the variance principle.

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.

Maximum likelihood estimation in case of a Poisson or Gamma distribution with loglinear parametrization for the mean is quite akin. The asymptotic variance-covariance matrix for the maximum likelihood estimator is derived as well as a linear estimator, which can serve as a starting value for the nonlinear search procedure.

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.

We consider a usual situation in risk theory for which the arrival process is a Poisson process and the claim process a positive (J — X) process inducing a semi-Markov process. The equivalent in queueing theory is the M/SM/1 model introduced for the first time by Neuts (1966).

For both models, we give an explicit expression of the probability of non-ruin on [o, t] starting with u as initial reserve and of the waiting time distribution of the last customer arrived before t. “Explicit expression” means in terms of the matrix of the aggregate claims distributions.

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.

(a) The notion of premium calculation principle has become fairly generally accepted in the risk theory literature. For completeness we repeat its definition:

A premium calculation principle is a functional assigning to a random variable X (or its distribution function Fx(x)) a real number P. In symbols

The interpretation is rather obvious. The random variable X stands for the possible claims of a risk whereas P is the premium charged for assuming this risk.

This is of course formalizing the way actuaries think about premiums. In actuarial terms, the premium is a property of the risk (and nothing else), e.g.

(b) Of course, in economics premiums are not only depending on the risk but also on market conditions. Let us assume for a moment that we can describe the risk by a random variable X (as under a)), describe the market conditions by a random variable Z.

Then we want to show how an economic premium principle

can be constructed. During the development of the paper we will also give a clear meaning to the random variable Z:

In the market we are considering agents i = 1, 2, …, n. They constitute buyers of insurance, insurance companies, reinsurance companies.

Each agent i is characterized by his

utility function ui(x) [as usual: ]

initial wealth wi.

In this section, the risk aspect is modelled by a finite (for simplicity) probability space with states s = 1, 2, …, S and probabilities πs of state s happening.

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.

It is commonly thought that the characteristic function (Fourier transform) of the Pareto distribution has no known functional form (e.g. Seal, 1978, pp. 14, 40, 57). This is quite untrue. Nevertheless the characteristic function of the Pareto density is conspicuously absent from standard reference works even when the Pareto distribution itself receives substantial comment (e.g. Haight, 1961; Johnson and Kotz, 1970, Ch. 19; Patel, Kapadia and Owen, 1976, § 1. 5).

The Pareto density may be written

with distribution function

mean = p/(v − 1) and variance = b2v(v − 1)2(v−2). These are infinite when v≤1 and v≤2, respectively. Its Laplace transform (s= c + iu)

where E is the generalized exponential integral (Pagurova, 1961) and can be written in terms of incomplete gamma or confluent hypergeometric functions (Slater, 1960, Sec. 5.6). When s = − it β(s) becomes the characteristic function (see Appendix I).

As Benktander (1970) tells us, the Pareto distribution has been particularly successful at representing the distribution of the larger claim amounts. In earlier years it was employed to represent the distribution of life insurance sums assured but more recently it has been used for the claim distributions of fire and automobile insurance. Table 1 provides the v-values we have been able to locate. Note that the variance of the distribution is infinite when v≤2 and if it were not for the anomalous v-values of Andersson (1971) we would have ventured the opinion that modern claim data encourage the assumption that v>2. In our numerical work we have used v = 2.7 and smaller values might change some of the computer rules we have proposed in Appendix II.

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).