We give an extension of the Economic Premium Principle treated in Astin Bulletin, Volume 11 where only exponential utility functions were admitted. The case of arbitrary risk averse utility functions leads to similar quantitative results. The role of risk aversion in the treatment is essential. It also permits an easy proof for the existence of equilibrium. KEYWORDSMathematical economics, equilibrium theory, premium principles. THE PROBLEMIn BUHLMANN (1980) it was argued that in many real situations premiums are not only depending on the risk to be covered but also on the surrounding market conditions. The standard actuarial techniques are not geared to produce such a dependency and one has to construct a model for the whole market, if one wants to study the interrelationships between market conditions and premiums.Such models exist in mathematical economics. For the purpose of this paper we borrow the model of mathematical economics for a pure exchange economy and we use the usual Walrasian equilibrium concept.The more practically oriented reader might consider the model as an idealization of e.g., a reinsurance market where premiums of the contracts are determined by the market. Of course, the Walrasian model is not the only way to describe a reinsurance market. In oligopolistic situations one would rather have to rely on the theoretical framework provided by game theory. On the other hand the model used in this paper extends far beyond reinsurance.The more theoretically minded reader will note that the model of an exchange economy used in the following has infinitely many commodities. The classical result of existence of equilibrium [see e.g., DEBREU (1959DEBREU ( , 1974] therefore does not hold. The existence proof given here is the theoretically most important aspect of the present paper. THE MODEL FOR THE MARKETWe have agents i, i = 1, 2 , . . . , n (typically reinsurers, insurers, buyers of direct insurance etc.).

Let me begin with some practical examples of experience rating.a) Swiss Automobile Tariff 1963— Within each tariff-position there are 22 grades: — The new owner of a car starts at grade 9— The basic premium is determined on the basis of objective characteristics of the risk and essentially depends on the horse-power classification of the car— The 22 grades are experience-rated as follows: For each accident one rises three grades and for each accident-free year one falls one grade. A driver who has I accident in every 4 years hence remains within four adjacent grades.b) Sliding Scale Premiums in ReinsuranceExcess of Loss Contracts often stipulate that:The rate of premium to be applied to the subject premium volume is determined at the end of the cover period as follows: subject to a minimum of 0,04and a maximum of 0,08c) Participation in Mortality Profit in Group Life InsuranceA group life insurance covers the members of the group on a one year term basis. It is often agreed that at the end of the year mortality profits are given back to the group according to the formularefund = x% gross premiums — y% claims (where x < y)All these examples fall under the heading “Experience Rating”. What do they have in common?Definition: A system by which the premium of the individual risk depends upon the claims experience of this same individual risk.

(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 symbolsThe 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 principlecan 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 hisutility 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.

Classical statistics deals with the following standard problem of estimation:Given: random variables X1, X2 … Xn independent, identically distributed, andobservations x1, X2 … xn,Estimate: parameter (or function thereof) of the distribution function common to all Xi.It is not surprising that the “classical actuary” has mostly been involved in solving the actuarial equivalent of this problem in insurance, namelyGiven: risks R1R2 … Rn no contagion, homogeneous group,Find: the proper (common) rate for all risks in the given class.There have, of course, always been actuaries who have questioned the assumptions of independence (no contagion) and/or identical distribution (homogeneity). As long as ratemaking is considered equivalent to the determination of the mean, there seem to be no additional difficulties if the hypothesis of independence is dropped. But is there a way to drop the condition of homogeneity (identical distribution)?