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.

Astin is formally twenty years old, or should I say young, and the occasion of the 13th Colloquium being held in the country where it was constituted seems a very suitable opportunity for making a record of the ten years prior to its formal constitution. This is a personal account and sets out facts and impressions which are now becoming coloured by the vagaries of memory; others will have their own impressions of some of the events and I hope will be stimulated to correct any biasses or to supplement the historical record.

Inevitably this record will not be complete and I apologise in advance for any significant omissions. Over the years I have accumulated a large mass of correspondence and had hoped to codify this in anticipation of this article but unfortunately a series of complications arising from my move to a part of rural England as a further stage in my retirement programme have delayed sorting through the mass of paper.

My interest in non-life insurance was first stimulated when I was involved in the early 1930's in an analysis of the results of Fire and Auto Property business written in the US. This brought me face to face with the differences between the UK and US approach to profit measurement, in particular the real meaning of unearned premiums. The conventional UK method of basing the provision for unexpired risks on 40% of the premiums written during the year led to significant distortion in the emergence of profits (or losses) with an increasing volume of business, a large proportion of which was for periods of three years. Equally the US method of basing a statutory result without any allowance for the equity in the unearned premiums led to further distortions of the true profit, quite apart from the practice, then permitted, of substantial portfolio reinsurance transactions at the close of accounting periods.

There is mounting pressure on (re)insurers to quantify the impacts of climate change, notably on the frequency and severity of claims due to weather events such as flooding. This is however a very challenging task for (re)insurers as it requires modeling at the scale of a portfolio and at a high enough spatial resolution to incorporate local climate change effects. In this paper, we introduce a data science approach to climate change risk assessment of pluvial flooding for insurance portfolios over Canada and the United States (US). The underlying flood occurrence model quantifies the financial impacts of short-term (12–48 h) precipitation dynamics over the present (2010–2030) and future climate (2040–2060) by leveraging statistical/machine learning and regional climate models. The flood occurrence model is designed for applications that do not require street-level precision as is often the case for scenario and trend analyses. It is applied at the full scale of Canada and the US over 10–25 km grids. Our analyses show that climate change and urbanization will typically increase losses over Canada and the US, while impacts are strongly heterogeneous from one state or province to another, or even within a territory. Portfolio applications highlight the importance for a (re)insurer to differentiate between future changes in hazard and exposure, as the latter may magnify or attenuate the impacts of climate change on losses.

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.

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)

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.

In this paper, the establishment of a new insurance organization, the International Insurance Fund (IIF), is proposed. This organization should undertake to deal with extra-hazard risks which are beyond the capacity of both the commercial insurance industry and national governments. It is proposed that the IIF be designed along the basic lines of the International Monetary Fund (IMF) and its sister organizations.

The hypothetical scope of the IIF operations during recent years is illustrated. Certain practical aspects of its actual operations are discussed; some of them call for further examination.

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 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.

Il a été créé en France, en 1948, dans le cadre de l'Association Générale des Sociétés d'Assurances contre les Accidents, un pool dénommé „Section de Réassurance Mutuelle” ayant pour objet la réassurance en mutualité, en excédent de sinistres, des risques de Responsabilité Civile Automobile, de Responsabilité Civile Générale et d'Accidents du Travail.

Le seuil au dela duquel les sinistres ont été répartis en mutualité a été:

Etaient réassurés par la Section tous les sinistres survenus en France, Afrique du Nord, Union Française et Belgique, mais la part relative aux risques assurés hors de la France métropolitaine était faible.

Nous n'avons tenu compte dans la présente étude que des sinistres de Responsabilité Civile Automobile supérieurs à:

L'adhésion à ce pool était libre. A titre d'indication le tableau ci-après donne par année les encaissements globaux en France et Algérie de toutes les Sociétés d'Assurances pratiquant l'Assurance Automobile. (Rapport annuel de la Direction des Assurances) et le montant des primes Auto et R.C. diverses ayant servi de base au calcul de la cotisation des Sociétés adhérentes à la Section de Réassurance Mutuelle.

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.

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.

Quadratic programming means maximizing or minimizing a quadratic function of one or more variables subject to linear restrictions i.e. linear equations and/or inequalities.

Among the numerous insurance problems which can be formulated as quadratic programs we shall only discuss four, namely the Credibility, Retention, IBNR and the Cost Distribution problems.

Generally, there is no explicite solution to quadratic optimization problems, only statements about the existence of a solution can be made or some algorithm may be recommended in order to get exact or approximate numerical solutions. Restricting ourselves to typical problems of the above mentioned type, however, enables us to give an explicit solution in terms of general formulae for quite a number of cases, such as the onedimensional credibility problem, the retention problem and—under relatively week assumptions— for the IBNR-problem.

The results given here are by no means new. The only goal of this paper is to describe a few fundamental insurance problems from a common mathematical standpoint, namely that of quadratic programming and at the same time, to draw attention to a few special aspects and open questions in this field.

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.

Commentaires sur les constatations expérimentales d'une Compagnie Française en matière d'Assurance Responsabilité Civile Automobile pour la période de 1949 à 1959 et méthode préconisée pour la détermination du prix de revient moyen d'une catégorie de risques bien définie.

De nombreux travaux actuariels ou statistiques ont été effectués dans tous les pays pour rechercher les lois pouvant régir les primes d'assurance Automobile en fonction de certaines variables.

Il ne semble pas toutefois que ces travaux théoriques soient facilement exploitables dans la pratique, étant donné les difficultés de saisir d'une manière suffisamment approchée les valeurs réelles des variables.

Notamment, en matière d'assurance Responsabilité Civile Automobile, l'expérience française montre que dans les éléments pouvant servir au calcul des primes, seul parait suffisamment connu à ce jour, le nombre des véhicules exposés au risque.

Dans une Conférence faite à l'Institut des Actuaires Français M. Constant a exposé ses recherches d'une loi de distribution permettant de déterminer les variations du coût moyen des sinistres en fonction des variations d'indices économiques, mais il ne semble pas qu'une utilisation pratique de cette théorie ait pu être obtenue dans le cadre des Entreprises d'Assurances Françaises.

C'est pourquoi il m'a paru bon d'exposer ci-après les résultats d'une méthode expérimentale utilisée dans une importante Compagnie d'Assurance Française pour apprécier si des corrélations peuvent apparaître entre la variation du coût moyen des sinistres et la variation du ou de plusieurs indices économiques. Cette méthode par ailleurs, donne également un moyen pour déterminer un nombre théorique de sinistres qui soit à l'abri des modifications importantes que les déclarations de sinistres — seul élément connu dans les Compagnies — peuvent subir selon certaines conjonctures.