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.

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.

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.

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.

(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.

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.