The study and analysis of the various factors influencing insurance risks constitutes an intricate and usually quite extensive problem. We have to consider on the one hand the nature and heterogeneity of the elements we have been able to measure, and on the other the problem of deciding—without knowing exactly what results to expect—on the types of analysis to carry out and the form in which to present the results.

These difficulties, essentially stemming from the fact that we cannot easily define “a priori” a measure of influence, can be overcome only by using highly sophisticated mathematical models. The researcher must define his objectives clearly if he is to avoid spending too much of his time in exploring such models.

Either for these reasons or for lack of our experience in this field we were led to the study of three models, presenting entirely different characteristics though based on the analysis and behaviour of mean value fluctuations, measured by their variances or by the least-squares method.

Our first model, described in II. 1, associates the notion of influence with the notion of variance. It analyses in detail the alteration of the mean values variance, when what we refer to as a “margination” is executed in the parameter space, taking each of the parameters in turn. We start off by having n distinct parameters, reducing them by one with each step.

1. In an article on such processes in ASTIN Bull. (II-3, pp. 450–451) the author has derived a relation between cPp i.n.s. (compound Poisson processes in the narrow sense) and composed Poisson processes, which is quite general. In this note this relation shall be simplified under the rather weak condition that Po(t), Pn(t) being the distribution of the number of changes occurring in the interval (o, t), for every fixed value of t, tends to a positive limit less than unity when t tends to infinity.

2. By the theory expounded by Thyrion (Bull, de l'Assoc. Act. Beiges, 1959 and 1963) the characteristic function of the number of events in the interval (o, t) can for each fixed value of t in this case be written

where η is a real variable and i the imaginary unity.

It is easily seen that (1) is equal to the following expression

which has the form of a characteristic function of the number of changes in a composed Poisson process.

Some reasons are given for paying special attention to the gross cost of catastrophe claims in planning and control. A method is then described of defining catastrophe claims and estimating their expected cost. The various steps in applying the method to real data and its performance for planning and control are discussed and illustrated in conjunction with an investigation carried out on a company portfolio.

Les 5 contributions suivantes ont été présentées sur le thème no 3:

— par R. E. Beard:

2 contributions intitulées “Calculation of Reserves for Non-Life Insurance” (Le calcul des réserves pour les branches non-vie)

— par Karl Borch:)

“Control of a Portfolio of Insurance Contracts” (Le contrôle d'un portefeuille d'assurances)

—par B. H. Jongh:)

“The Insurer's Ruin” (La ruine de l'assureur)

— par T. Pentikäinen, Helsinki:

“On the Solvency of Insurance Companies” (De la solvabilité des compagnies d'assurance)

— par Erkki Pesonen, Helsinki

“Magnitude Control of Technical Reserves in Finland” (Le contrôle du montant des réserves techniques en Finlande)

Les questions traitées par ces auteurs peuvent être groupées comme suit:

1. Donnée la situation d'une compagnie d'assurance, quelles sont les affaires qu'elle devrait accepter?

2. Donnée la situation d'une compagnie d'assurance, dans quelle mesure doit-elle réassurer son portefeuille?

3. Le calcul des réserves techniques, c'est-á-dire de la réserve pour risques en cours et pour les sinistres non régiés.

4. La détermination des réserves de sécurité nécessaires en plus des réserves techniques proprement dites.

Nielsen (1999) showed the surprising fact that a nonparametric one-dimensional hazard as a function of time can be estimated -consistently if a high quality marker is observed. In this paper we show that the hazard relevant for predicting remaining duration time, given the current status of a high quality marker, can be estimated -consistently if a Markov type property holds for the high quality marker.

The survival energy model (SEM) is a recently introduced novel approach to mortality prediction, which offers a cohort-wise distribution function of the time of death as the first hitting time of a “survival energy” diffusion process to zero. In this study, we propose a novel SEM that can serve as a suitable candidate in the family of prediction models. We also proposed a method to improve the prediction in an earlier work. We further examine the practical advantages of SEM over existing mortality models.

In this paper the dynamic portfolio selection problem is studied for the first time in a dual utility theory framework. The Wang transform is used as distortion function and well diversified optimal portfolios result both with and without short sales allowed.

A few years ago Scurfield (JSS 18) indicated a method used inone U.K. non-Life Office for estimating claims liability for the Motor class of business. Since that paper was written a considerable amount of development has taken place and it is now used in the office as an effective continuous method, using computer techniques, for estimating claims liability. It has also been used for projecting expected cash flow for claims arising in the past.

Whilst it is accepted that the method has limitations, it has been found that an automatic method is required especially in the production of financial models in the non-Life field, and its limitations are outweighed by the fact that there are, in our opinion, no easier methods of producing a satisfactory working model.

In the following paper it is the intention to set out briefly the details of the method, then concentrating on the known limitations and the methods so far devised to combat these limitations. In the appendices we have reproduced some computer printout and figures which will provide practical information on the method.

The principle behind the method is that the “Run-off” of claims payments for any “year of claim”, or similar cohort of claims, follows a particular pattern which experience has shown to be reasonably stable. We can thus study the pattern of claims payments at each stage of “run-off”, say each month and thus ascertain the average and range of values that the historic pattern shows (Fig. 1).

Fig. 1.

Period of delay since beginning of claims Period (Years)

A method is introduced for testing the distribution of yield curves that are produced by asset scenario generators. The method is based on historical relationships in the conditional distributions of yield spreads given the short-term rate. As an illustration, this method is used to test a few selected models. To provide background, stochastic modeling for interest rates and fitting methods are briefly discussed.

De nombreux modèles, en actuariat, se basent sur l'existence d'une fonction d'utilité additive. Le but de cet article est de montrer que cette hypothèse enlève au problème traité son caractère dynamique et que la suppression de l'additivité conduit à une solution plus réaliste.

Trois traits caractérisent, me semble-t-il, le développement récent de la théorie du risque et influencent done l'exposé dont notre Président m'a chargé, avec une amabilité tenement pressante que je n'ai pu m'y dérober.

Le temps n'est plus où l'on pouvait déplorer un manque d'intérêt des actuaires pour la théorie du risque: tel est le trait le plus immédiatement évident. ASTIN y a certainement contribué. Le cercle restreint d'adeptes — dirais-je d'initiés — a éclaté. Les idées traversent les frontières et même l'océan. Ce faisant, elles s'enrichissent au passage, se généralisent, ou bien encore rencontrent une idée-soeur qui leur ressemble parfois comme une jumelle mais porte un autre nom. On voit les idées “reςues” se heurter à des conceptions rajeunissantes. On voit aussi les nécessités pratiques infléchir la théorie. Bref il se produit un véritable bouillonnement de recherches et d'applications, des plus heureux pour la théorie du risque, mais des plus pénibles pour le conférencier qui voudrait en cerner convenablement l'évolution. Pour ce faire, l'ampleur et la diversité des publications exigeraient un professionnel en la matière, disposant du temps voulu pour s'informer complètement. Ce n'est pas mon cas et cela explique — et excusera, je l'espère — le cadre délibérément restreint de mon exposé. Car je n'ai pas l'intention de suivre la théorie dans ses multiples démarches réeentes et dans tous ses perfectionnements les plus subtils. Je vous propose plutôt de jeter quelques regards sur le développement récent de la théorie du risque et ainsi de tenter de dégager et d'illustrer quelques idées essentielles sur cette évolution et d'essayer d'entrevoir son déroulement futur. Telles sont les limites de mon objectif.

In this paper, we present a method for generating a copula by composing two arbitrary n-dimensional copulas via a vector of bivariate functions, where the resulting copula is named as the multivariate composite copula. A necessary and sufficient condition on the vector guaranteeing the composite function to be a copula is given, and a general approach to construct the vector satisfying this necessary and sufficient condition via bivariate copulas is provided. The multivariate composite copula proposes a new framework for the construction of flexible multivariate copula from existing ones, and it also includes some known classes of copulas. It is shown that the multivariate composite copula has a clear probability structure, and it satisfies the characteristic of uniform convergence as well as the reproduction property for its component copulas. Some properties of multivariate composite copulas are discussed. Finally, numerical illustrations and an empirical example on financial data are provided to show the advantages of the multivariate composite copula, especially in capturing the tail dependence.

An insurance company aims to increase the volume of its business.

The volume is measured by the premium income during one year.

The company makes use of N tariff groups.

pn = premium charged by the company per unit of insurance in tariff group n.

qn = premium per unit of insurance in tariff group n corresponding to the statistical experience in the said group.

Ln = The size of the market in tariff group n. If all insurances in the group pay the same premium and if the company is able to cover the whole of its potential market then Ln is equal to the number of policies. If the premium per policy is equal to pn times the sum insured and if the company is able to cover the whole of its potential market then Ln is equal to the total sum insured.

Fn(p) = the proportion of Ln that actually buys insurance in tariff group n if the company charges the premium p per unit of insurance.

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.