Nous avons pensé que les assurances contre l'incendie pouvaient aussi s'étayer du calcul. (T. J. Barrois, 1835)

To a mathematician 25 is a very uninteresting number: Just the square of a small prime. For a historian it means an epoch. If 25 years bring us back to one of the cataclysms of recent history, then it is worth looking back at them. (S. Vajda, 1964)

Various methods for developing recursive formulae for compound distributions have been reported recently by Panjer (1980, including discussion), Panjer (1981), Sundt and Jewell (1981) and Gerber (1982) for a class of claim frequency distributions and arbitrary claim amount distributions. The recursions are particularly useful for computational purposes since the number of calculations required to obtain the distribution function of total claims and related values such as net stop-loss premiums may be greatly reduced when compared with the usual method based on convolutions.

In this paper a broader class of claims frequency distributions is considered and methods for developing recursions for the corresponding compound distributions are examined. The methods make use of the Laplace transform of the density of the compound distribution.

Consider the class of claim frequency distributions which has the property that successive probabilities may be written as the ratio of two polynomials. For convenience we write the polynomials in terms of descending factorial powers. For obvious reasons, only distributions on the non-negative integers are considered.

My assistants at ETH have a wall calendar—not with the usual pictures of Swiss mountains, hills and lakes, but with “quotations for intelligent people”. Recently, the quotation for the week read as follows: “Even the future is no longer what it used to be in the past”.

Observe that also in this supposedly intelligent approach it seems impossible to speak about the future without referring to the past. I shall not deviate from this rule. Of course, my task is greatly simplified by the fact that Paul Johansen has just entertained you in a charming way about the past 25 years of ASTIN and the earlier endeavors leading to the foundation of ASTIN.

In the year 1693, Edmond Halley constructed the first mortality table based on mortality data from Breslau which he had obtained through the intervention of Leibniz. This can be regarded as the starting point of actuarial science. In my opinion it can however not be considered as the starting point of the actuarial profession. Why? Yes, Halley's table was used for some eighty years because subsequent information coincided with his estimate of mortality. Yes, De Moivre, in his classic textbook of 1725, performed ingenious calculations of annuities, based on the same table. Yes, Süssmilch published the first basic and substantial work of demography in 1741 but—here comes the big but—no government (and nobody else sold annuity insurance at that time) made use of the available scientific method to calculate annuities. Perhaps the first statistical results to be taken seriously were the Northampton tables of 1780, devised by Richard Price. Incidentally, this date coincides reasonably well with the first valuation by William Morgan in 1786. Hence, I think that either of these dates may be taken, at the earliest, as the start of the actuarial profession, a profession being by definition a dedicated group of people accepted by society for the performance of a particular skill. Let me make my point explicit: We have historical evidence of the existence of actuarial science about 90 years prior to the emergence of the actuarial profession.

We shall define a mutual insurance firm as a firm whose stockholders are the bearers of the insurance contracts issued by the firm. The firm's insurance is then viewed as a collective process of say N persons seeking to protect themselves against claims that may occur to any one of them. For example, large employers protecting their employees by pooling risks and deducting for protection given amounts from salaries may be a case in point. In this latter case, the employer may match withdrawals from employees salaries and provide in the process a fringe benefit and increase employees loyalty to the firm. Alternatively, agricultural collectives have in some cases established mutual insurance firms whose purposes are to protect them, at a cost, from the uncertainty implicit in their production processes and the fluctuations of agricultural markets. Since these firms do not work for profit, contingent payments, or fund reimbursement in case of excess cash holdings are typical control policies which help cover extraordinary claims and at the same time are assumed the best investment policies. To further protect themselves against extraordinary claims, mutual insurers can turn to reinsurance firms, “selling” for example the excess claims of, say, a given amount R (e.g., see Tapiero and Zuckerman (1982)). The purpose of this paper is to consider such a mutual insurance firm facing a jump stochastic claims process, as is often assumed in the insurance literature (e.g., Feller (1971), Borch (1974)). For example, Poisson and Compound Poisson processes are typical jump processes treated in this paper, although other processes could be considered as well (see Srinivasan (1973) and Srinivasan and Mehata (1976), Bensoussan and Tapiero (1982)). First we define the mutual insurer problem as a jump process stochastic control problem which we solve analytically under the assumption of a gamma density claim sizes distribution. A solution is obtained by applying arguments from stochastic dynamic programming and by solving the resultant functional equation by application of Laplace transforms (e.g., Colombo and Lavoine (1972), Miller (1956), Tapiero, Chapter V). Subsequently, the effects of reinsurance are introduced and preliminary results obtained. As in Tapiero and Zuckerman (1982), we assume that the mutual insurance firm is a direct underwriter and that the reinsurer is a leader in a Stackleberg game (Stackleberg (1952), Simman and Cruz (1973), Luce and Raiffa (1957)).

I take great pleasure in addressing this audience. As you might know I'm a mathematician with a deep interest in insurance mathematics. As such, it is my sincere opinion that the gap between practising actuaries and theoretical researchers can be made substantially smaller. If my contribution can help in bridging the gap, I will feel fully compensated for the effort it took to prepare this lecture and the results contained therein.

The simple fact that we meet on the occasion of the sixteenth ASTIN Colloquium gives me a challenging opportunity to help in creating a platform on which both theoreticians and practitioners can meet.

The subject of my lecture stems from a long interest in large claims: What are they? Are they really dangerous? Is there a way to get them under control? Can one recognize them in practical situations?

I like to express in simple mathematical terms some results that might help in acquiring better insight on the impact of large claims in insurance mathematics. Perhaps, no result will be of immediate applicability as reality is too complicated to be described by the simplicity of the results to follow. Nevertheless the latter can be considered as building blocks of a real world in which one has to tackle large claims in theory as well as in practice.

In case of a stop-loss treaty the reinsurer takes over that part of the risk that exceeds a given amount y1. We will deduce bounds on a modified stop-loss treaty where the liability of the reinsurer is limited to y2–y1 in case the claim amount exceeds y2. Upper and lower bounds of this modified stop-loss premium are obtained as a simple application of results obtained earlier by the first author.

Some comments are given on a recent paper by de Wit and Kastelijn (1980) and alternative methods for analysing loss ratios are proposed in connection with the determination of the necessary solvency margins of non-life insurance companies. The methods are illustrated by a numerical example.

The paper gives some asymptotic results for the compound distribution of aggregate claims when the claim number distribution is negative binomial. The case when the claim numbers are geometrically distributed, is treated separately.

We give some actual possibilities for computing numerical values in the classical risk models both in transient and asymptotical cases by introducing the concept of normed model. Some recent approximations are tested on numerical examples.

We also emphasize the interest of these methods to compute waiting time distributions (transient and stationary cases) in queueing theory.

In the present paper we deal with the problem of calculating a premium for the largest claims and ECOMOR reinsurance treaties. Ammeter derived already in 1964 formulas for calculating the premiums of the largest claims and ECOMOR reinsurance treaties (compare also Seal (1969), Thépaut (1950)), which we will restate in the following Section 2. Lately Benktander (1978) has established an interesting connection between the premiums of the largest claims and excess of loss reinsurance treaties. He proved that the net risk premium of the largest claims treaty covering the p largest claims is bounded by the risk premium of an excess of loss treaty plus p times its priority, which has to be determined such that the mean number of excess claims equals p. Furthermore Benktander showed in examples that the upper bound is quite good in case of the Poisson-Pareto risk process. Nevertheless he did not give a formal proof for the quality of the bound in the Poisson-Pareto case nor for other risk processes.

In the following note we take up this last point and prove that for general risk process Benktander's upper bound is equivalent to the premium of the largest claims reinsurance cover when the size of the collective approaches infinity. Consequently, for large portfolios the risk premium of the largest claims cover may be replaced by the upper bound, e.i., calculated from the premium of the corresponding excess of loss treaty. Moreover we state a similar result for the ECOMOR treaty.

Two premium calculation principles by negotiation. Using, as main tools,

the classical risk exchange model by Borch and

the bargaining models of Nash and Kalai-Smorodinsky,

we define two new premium calculation principles, whose main goal is to take explicitly into account the attitude towards risk of the policy-holders. Those principles are neither additive nor iterative, but they nevertheless possess several important properties: the premium is translation-invariant, it does not depend neither on the reserves nor on the portfolio of the company; it takes into account all the moments of the claim distribution; it is independent of the policy-holder's wealth but increases with his risk aversion.

Coalition against an insurance company. While computing the core of this risk exchange, we show that it can be of the policy-holder's interest to coalize in order to obtain premium cuts.

Le modèle d'échange de risques de Borch entre plusieurs compagnies d'assurances soucieuses d'améliorer leur situation en formant un pool de réassurance a fait l'objet de très nombreuses publications. Ce n'est que depuis quelques années cependant que l'on semble s'être aperçu que le même modèle pouvait être utilisé pour décrire toute économie d'échange, en particulier le contrat d'assurance simple entre un assuré et sa compagnie. Si l'on suppose que les préférences de l'assuré peuvent être décrites par une fonction d'utilité exponentielle, et que l'assureur est indifférent au risque en première approximation, les contrats Pareto-optimaux consistent en une couverture complète du risque, moyennant le paiement d'une prime que le critère de Pareto-optimalité ne permet pas de déterminer.

One of the central problems in risk theory is the calculation of the distribution function F of aggregate claims of a portfolio. Whereas formerly mainly approximation methods could be used, nowadays the increased speed of the computers allows application of iterative methods of numerical mathematics (see Bertram (1981), Küpper (1971) and Strauss (1976)). Nevertheless some of the classical approximation methods are still of some interest, especially a method developed by Esscher (1932).

The idea of this so called Esscher-approximation (see Esscher (1932), Grandell and Widaeus (1969) and Gerber (1980)) is rather simple:

In order to calculate 1 –F(x) for large x one transforms F into a distribution function such that the mean value of is equal to x and applies the Edgeworth expansion to the density of The reason for applying the transformation is the fact that the Edgeworth expansion produces good results for x near the mean value, but poor results in the tail (compare also Daniels (1954)).

Let u(x) be a utility function, i.e., a function with u′(x)>0, u″(x)<0 for all x. If S is a risk to be insured (a random variable), the premium P = P(x) is obtained as the solution of the equation

which is the condition that the premium is fair in terms of utility. It is clear that an affine transformation of u generates the same principle of premium calculation. To avoid this ambiguity, one can standardize the utility function in the sense that

for an arbitrarily chosen point y. Alternatively, one can consider the risk aversion

which is the same for all affine transformations of a utility function.

Given the risk aversion r(x), the standardized utility function can be retrieved from the formula

It is easily verified that this expression satisfies (2) and (3).

The following lemma states that the greater the risk aversion the greater the premium, a result that does not surprise.

A great attention has been devoted, in the actuarial literature, to premium calculation principles and it has been often emphasized that these principles should not only be defined in strictly actuarial terms, but should also take into account the market conditions (Bühlmann (1980), de Jong (1981)).

In this paper we propose a decision model to define the pricing policy of an insurance company that operates in a market which is stratified in k risk classes .

It is assumed that any class constitutes a homogeneous collective containing independent risks Sj(t) of compound Poisson type, with the same intensity λj. The number nj of risks of that are held in the insurance portfolio depends on the premium charged to the class by means of a demand function which captures the concept of risk aversion and represents the fraction of individuals of , that insure themselves at the annual premium xj.

With these assumptions, the return Y on the portfolio is a function of the vector x = (x1, x2, …, xk) of the prices charged to the single classes (and of the time) and x is therefore the decision policy instrument adopted by the company for the selection of the portfolio, whose optimal composition is evaluated according to a risk-return type performance criterion.

As a measure of risk we adopt the ultimate ruin probability q(w) that, in the assumptions of our model, can be related to a safety index τ, by means of Lundberg-de Finetti inequality. Even though it has been widely debated in the actuarial field, the use of q(w) offers undeniable operational advantages. In our case the safety index τ can be expressed as a function of x and therefore, in the phase of selecting an efficient portfolio, it becomes the function to be maximized, for a given level M of the expected return.