bluuebunny/crossref_metadata_2025 huggingface dataset:
L'écrêtement des sinistres “automobile”
Authors: Moreau Par Lionel
Published: July 2016

French insurance companies usually classify their agents according to their results by branch and to this effect they calculate their respective claim ratios.

As regards motor insurance many agents have argued against this practice as they believed it was unfair to be adversely classified when they had the misfortune to incur a large claim.

In this article, the merits of various statistical systems for attenuating the effects of large claims on the portfolio are considered in order to come to the most equitable solution possible, i.e., to ascertain the best method and system for attenuating the effects of large claims. The conclusion is that the excess of loss technique is best suited to this effect.

To arrive at this conclusion the author of this article relates in detail the various stages of the study, the various systems envisaged and the tests which have allowed to select the most appropriate system. The main results obtained are also given together with their respective drawbacks.

At the end of the article the author mentions the still fairly limited bibliography which deals with this question.

Dans le but d'améliorer les résultats, les compagnies d'assurance ont l'habitude en France de publier un classement de leurs Agents Généraux basé sur le ratio S/P (sinistres/primes) et de tenir compte de ce classement pour l'attribution de certains avantages commerciaux.

Or, en assurance automobile, un malaise profond s'était emparé d'un certain nombre d'entre eux, et ils avaient mis l'accent sur un point qu'ils estimaient injuste:

“Nous ne pouvons pas être pénalisés” disaient-ils, “si un très gros sinistre, atteignant 500 ou 1000 fois la prime annuelle, frappe un de nos clients, car il s'agit alors d'un phénomène aléatoire dont nous ne sommes pas responsables”.

Telle est l'origine de la politique de l'écrêtement.
Sur la determination d'un contrat optimal de reassurance
Authors: Lemaire Jean
Published: July 2016

Ohlin [7] showed optimality properties of Stop Loss reinsurance when the ceding insurer uses a continuous loss function to evaluate his risks. We generalize this property to the range of the distribution; we then show that Stop Loss reinsurance is no longer the best form when the company uses a percentile parameter. Finally we prove an optimality theorem concerning chance games which allows us to determine the retention as a function of the size of the portfolio.
Balancing International Insurance Portfolios and Exchange Risks
Authors: Agmon Tamir , Kahane Yehuda
Published: July 2016

This paper examines the balancing of the investment and liability portfolios of a (re)insurance firm operating in the international market. The model captures two effects which are ignored by traditional analysts.

(a) According to the Interest Parity Theorem, the expected changes in the exchange rates should already be reflected in the expected rate of return on foreign investments. Therefore, an insurer operating in a perfect market should be indifferent to the currency denomination of its financial assets.

(b) A second effect which has often been ignored is related to the additional risk resulting from the unexpected fluctuations of the exchange rates.

The multi-index model suggested in this paper is capable of capturing these effects. The model can be used to examine and analyze alternative policies of the firm operating in international markets. For example, the model can be used to examine whether an insurer should take positions in certain currencies, or rather take a “full-hedge” policy.
Some Personal Reminiscences of the First Twenty Plus Ten Years
Authors: Beard R. E.
Published: July 2016

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.
Towards an “International Insurance Fund”: A Tool for Mitigation of Extra-Hazard Risks
Authors: Shenhav Yehuda
Published: July 2016

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.
Problems in the Economic Theory of Insurance
Authors: Borch Karl
Published: July 2016

More than ten years ago I wrote a paper with the title “The Economic Theory of Insurance” [6]. I was not particularly happy about this paper, and I do not think it contributed much to the development of a sastifactory theory. The paper did however make me—and I hope some readers—acutely aware of the difficulties and problems which must be overcome before a proper theory can be constructed. These problems are still unsolved, so I have on the present occassion chosen a more modest title for a paper on substantially the same subject.

Insurance is an economic activity of some importance, and there is an obvious need for a theory to explain and analyse the activity in the insurance sector of the economy. During the last decade many economists seem to have felt the need, and to have taken it as a challenge. The results have been a fair amount of research, and a number of publications, which I shall not try to review here. It may however be useful to refer to three very recent survey articles by Farny [10], Ferry [11] and Rosa [14], which give extensive bibliographies. The three articles seem to indicate that the economics of insurance is becoming a fashionable subject of research.
Pareto-Optimal Risk Exchanges and Related Decision Problems
Authors: Gerber Hans U.
Published: July 2016

In various branches of applied mathematics the problem arises of making decisions to reconcile conflicting criteria. One example is the classical statistical problem, where a type 1 error cannot be arbitrarily reduced without increasing the probability for a type 2 error. Another example, quite familiar to actuaries, is graduation, where a compromise between smoothness and fit has to be reached. This motivates the concept of Pareto-optimal decisions, which is discussed in section 2. There is a simple method, maximizing a weighted average of the scores, to obtain certain Pareto-optimal decisions. In section 3 a condition is given, which is satisfied in most applications, that guarantees that all the Pareto-optimal decisions can be found by this method. This is applied in section 4, where the problem of risk exchange between n insurance companies is considered. The original model of Borch is generalized: it is assumed that some of the companies are not willing to contribute more than a certain fixed amount towards the aggregate loss of the other companies. The theorem in section 4 gives a characterization of all the Pareto-optimal risk exchanges. Because of the restrictions, these risk exchanges do not just depend on the combined surplus (which would amount to pooling) in general, and can be found by an algorithm. One benefit of this generalization of Borch's Theorem is that two seemingly unrelated results (optimality of a stop loss contract, and optimality of certain dividend formulas in group insurance) follow from it as special cases.
Testing Goodness-of-Fit of an Estimated Run-Off Triangle
Authors: Taylor G. C.
Published: July 2016

By the term actual run-off triangle we shall mean the two-way tabulation—according to year of origin and year of payment—of claims paid to date, which has the following form:

where Cij is the amount paid during development year j in respect of claims whose year of origin is i.

The information relating to the area below and/or to the right of this triangle is unknown since it represents the future development of various cohorts of claims.

Now in seeking to use this triangle as a basis for projection of claims in future development years for each of the years of origin 0, 1, 2, etc., we must recognise that the entries Cij in the above triangle, being random variables, contain random deviations from their expected values uij. It is the corresponding triangle of these expected values in which we are interested, and which shall be called the expected run-off triangle.
Parameter Estimation in Credibility Theory
Authors: De Vylder Fl.
Published: July 2016

The problem of distribution-free parameter estimation in recent credibility theory is discussed in the papers [1], [3] and [4] of the bibliography. Here, we consider a multiclass model with regression assumption. In that case, already treated by Ch. Hachemeister, [3], this author obtains an unsymmetrical matrix as an estimator of a covariance matrix. Of course, for practical use, this matrix is symmetrized in the obvious way. We show that this procedure can be avoided and that a lot of symmetrical unbiased estimators can be obtained at once.

By particularisations to the 1-rank model, we find the estimators given by Bühlmann and Straub, [1], [4].

In the multirank case, a generalization of the minimumvariance principle (minimization of the trace of the covariance matrix) leads to an optimal estimator of the mean regression vector. A noteworthy conclusion of our discussion is that there is no difference at all between the various credibility formulae (the inhomogenous formula, the homogeneous formula, the meanfree formula) if the mean regression vector is estimated optimally.

Finally we show that it must not be hoped to find, in a large set of unbiased estimators of the covariance matrix, one estimator furnishing, always, a semidefinite positive estimate.
A Non Symmetrical Value for Games without Transferable Utilities; Application to Reinsurance
Authors: Lemaire Jean
Published: July 2016

We define axiomatically a concept of value for games without transferable utilities, without introducing the usual symmetry axiom. The model—a generalization of a previous paper [6] extending Nash's bargaining problem—attempts to take into account the affinities between the players, defined by an a priori set of “distances”. The general solution of all three- and four-person games is described, and various examples are discussed, like the classical “Me and my Aunt” and a reinsurance model.

Nous définissons de manière axiomatique un concept de valeur pour les jeux à utilités non-transférables, sans introduire l'axiome classique de symétrie. Le modèle — une généralisation d'un concept de valeur [6] étendant à plusieurs joueurs le problème de marchandage de Nash — tient compte des affinités entre les joueurs, données sous forme d'une matrice de “distances” a priori. Nous donnons la solution générale de tous les jeux à trois et quatre joueurs, et discutons plusieurs exemples classiques, dont le célèbre “Ma tante et moi” et le modèle de réassurance de Borch.
On the Loglinear Poisson and Gamma Model
Authors: ter Berg Peter
Published: July 2016

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.
An Economic Premium Principle
Authors: Hans Bühlmann
Published: July 2016

(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.
Survival Probabilities Based on Pareto Claim Distributions: Comment
Authors: Goovaerts Marc J. , de Pril Nelson
Published: July 2016

In a recent paper Seal (1980) calculated numerically survival probabilities based on Pareto claim distributions.

The Pareto density may be written as

Generalizing, the Pareto distribution may be regarded as a special case of the so-called beta-prime distribution (Keeping, 1962, p. 83) with density function

where B(p, q) = is the beta function.

In his paper Seal (1980, Appendix 1) arrived at a contradiction concerning this beta-prime distribution. He found on one side that all derivatives of the characteristic function exist at the origin and on the other side that only the moments of order n < q exist. In this note we will show that this contradiction is due to the use of an incorrect expression for the characteristic function of the beta-prime distribution, which was taken over from Johnson and Kotz (1970, Ch. 26) and Oberhettinger (1973, Table A).
The Optimal Control of a Jump Mutual Insurance Process
Authors: Tapiero Charles S.
Published: July 2016

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 sayNpersons 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 amountR(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)).
Speed of Finalization of Claims and Claims Runoff Analysis
Authors: Taylor G. C.
Published: July 2016

To some extent, this paper provides theoretical back-up for Taylor's (1980b) see-saw method of analysis of outstanding claims.

Section 3 deals with the payments per claim finalized (PPCF) method. This assumes PPCF to be a function of development year only.

Section 4 carries out a theoretical investigation of the validity of this assumption. It is found that the assumption is not justified in general; that PPCF can (and usually will) be sensitive to changes in speed of finalization of claims; and that this sensitivity will depend on the convexity properties of PPCF as a function of development year. The analysis is made on the basis of the so-called hypothesis of invariant order.

Section 5 develops the see-saw method from this theoretical basis, although Section 7 is at pains to point to situations in which that method remains applicable despite violation of the hypothesis of invariant order.

Section 6 provides a numerical example.
The Esscher Premium Principle: A Criticism. Comment.
Authors: Gerber Hans U.
Published: July 2016

Zehnwirth (1981) contains some flaws. If

is the Esscher premium for a risk X, the loading is H(X) — E(X) and not h as Zehnwirth states. The first and third formulas on page 78 are wrong, since o(h) is a quantity such that

A correct statement would have been that

or simply that H(X) is a continuous function of the parameter h. However, this continuity is not uniform in all risks, which is illustrated by (3). No matter how small h is, there is always an X such that the difference between H(X) and E(X) is substantial. In view of this what is the meaning of a statement like “… the Esscher premium is a small perturbation of the linearized credibility premium”?
The Swedish Systems of Bonus
Authors: Philipson Carl
Published: July 2016

According to the provisions of the Swedish tables of rates the premium payable on a certain policy both for Third Party Liability and for Car Damage Insurance (without clause for particular self retention) is equal to the initial premium reduced with regard to points earned as related to years with no indemnities. One such point is earned for each year during which no indemnity has been paid on the policy; upon an indemnity one or two points are lost. The detailed rules for the reduction have been altered from time to time, the existing rules have been valid from the year 1952, these rules will here be called system II, while system I denotes the rules valid during the years immediately previous to 1952. The systems are described in the following table.

The probabilities for the occurrence of a claim and for the occurrence of a lapse for a policy can be considered functions of two variables, the points obtained in the bonus scale, b, and the “duration” i.e. the time elapsed since the policyholder signed his first policy, τ. For simplicity two kinds of models have been constructed, one assuming the probabilities to depend only on τ, the other assuming the probabilities to depend only on b.
Remarks on Statistical Distribution of Intensity of Chance Damages
Authors: Banasinski Antoni , Wanatowski Antoni
Published: July 2016

Actuaries of Polish National Insurance are in the happy position that they have at their disposal a wealth of complete statistics which permits study of the level of chance damage (the loss-rate) and its elements such as frequency of occurrence, spread, intensity, etc. This welcomed situation stems from the fact that in Poland there is general insurance in certain sectors of the national economy. For instance, in the case of cooperative and individual (private) farms the entire property (buildings, products, livestock, real estate and machines, household furniture, etc.) is insured according to uniform principles against all hazards—fire, winds, floods, hail, death of animals and others.

This, among other things, makes possible the statistical determination of the intensity of chance damages (i.e. the extent of damage to a given object during one accident) in various elements of farm property.

Studies of this type have been of great practical importance in the activities of Polish National Insurance, first and foremost in the reconstruction of the insurance rates in connection with a change in the system to the “first risk” system (employed almost exclusively at present in farm insurance). Moreover, it is interesting that these studies have also served as auxiliary material in the elaboration of the optimum programme for the utilization of the funds allocated for preventing chance damages.

Studies on the distribution of the intensity of chance damages on insured objects have led to interesting results which have confirmed once again that this distribution is dependent on the structure of the objects insured and on the risks (hazards) covered by insurance protection; any hypotheses concerning these distributions must be verified in each case by trial statistical studies.
A propos de la distribution des cas de maladie entre les assurés et par rapport à la durée
Authors: Coppini Mario Alberto
Published: July 2016

Il est un fait bien connu que la technique des assurances, qu'elles soient privées ou sociales, qui versent une indemnité en cas de maladie, se base premièrement sur la connaissance des fréquences et des durées moyennes de maladie, ou bien des coefficients de morbidité. Pour résoudre des problèmes particuliers, des instruments plus raffinés sont toutefois nécessaires, savoir:

a) la distribution des assurés d'après le nombre de cas dont ils ont été frappés au cours d'une année;

b) la distribution des cas d'après la durée en jours;

c) la distribution des assurés d'après les jours de maladie dont ils ont été frappés au cours d'une annèe.

Nous avons eu la possibilité d'elaborer les fiches de 44.829 travailleurs salariés (36.134 hommes et 8.695 femmes âgés de 15 à 65 ans) résidant à Rome et assurés obligatoirement auprès de l'Institut National d'assurance-maladie (Istituto Nazionale per l'assicurazione contro le malattie — INAM), et de relever les cas de maladie qui les ont frappés au cours de l'année 1960, en même temps que les durées correspondantes.).

Puisqu'il s'agit, comme on le verra, de données très détaillées, il nous a paru utile d'étudier les caractéristiques principales des trois distributions en question et les relations existant entre elles: l'exposé de certains résultats obtenus dans ce sens constitue l' objet de la présente note.
A Numerical Method of Finding a Suitable Bonus Scale
Authors: Pesonen Erkki
Published: July 2016

The bonus system used by Finnish insurers is as follows: Policies are divided into 4 bonus classes, C1, C2, C3, C4, each of them being initially placed in class C1. After one claimless year, a policy placed in class Ch is transferred to class Ch+1 (h ≤ 3) from the beginning of the following year. After a claim year, each policy is entered back to class C1. (Whether the number of claims is one or more does not play any role in this transition system.) From the initial premium, i.e. the premium in class C1, a reduction of pi, per cent is granted in class Ci. In numbers these percentages are as follows:

In present investigations there is a trend to replace this scale by another in which mere contingency would not cause too great a change in the premium and in which the „test period” would be longer.

Let us suppose that we want to find a suitable percentage scale P = (P1, P2, …, Pm) for a bonus system (Cl, C2, …, Cm) with a given transition mechanism according to which for each bonus class Ch two bonus classes Ci and Cj are given to which the policy is transferred after one year from Ch according as there have been claims or not.
“Estimating incurred claims”
Authors: Benedikt V. , Feay Herbert L.
Published: July 2016

Mr. Benedikt uses “chain relatives” based on the incurred claim totals included in Part 5 of Schedule “P” of the annual statement required for fire and casualty companies in the United States. Each total is for the losses as developed to end of calendar year (j) for claims incurred because of accidents in calendar year (i). Each total is the sum of the actual payments made before the end of year (j) plus the reserve for estimated payments to be made after the end of year (j) for claims incurred in year (i). The “chain relatives” are ratios. The “chain relative” ai,j is the ratio of developed losses to end of (j + 1) to the developed losses at the end of year (j).

Each total of Part 5 of Schedule “P” equals the sum of the total payments to date plus the total reserves for future payments for the corresponding classification of claims. Separate totals for these amounts are given in Part 1 of Schedule “P”. The totals of Part 5 are not secured directly from Part 1 because Part 1 gives totals by policy year of issue only and Part 5 separates the totals by policy of issue by calendar year in which claims are incurred. The two parts are prepared from the same basic claim information and agree in total.

The accumulated total paid losses for most casualty lines increase with passage of time. This accumulated total for paid losses can be reduced only if there are recoveries for losses previously paid, such as can occur for auto collision. For auto collision, the insurance company for this insurance can pay the insured for the damage to his car and then later recover from the insurance company that provided the liability insurance for another car involved in the same accident. Such substantial recoveries normally do not occur for auto liability insurance for bodily injury and property damage.
Une note sur des systémes de tarification basés sur des modéles du type Poisson composé
Authors: Lundberg Ove
Published: July 2016

Hans Bühlmann traite dans son ouvrage “Optimale Prämien-stufensysteme” (1964) principalement le problème de trouver l'estimation optimale de la prime de la n-ième période à partir des observations de sinistres au cours des périodes précédentes.

Max Gürtler a traité dans le “Bonus ou Malus” du Astin Bulletin vol. III part I deux systèmes différents pour la détermination de la prime individuelle pour l'assurance automobile, c'est-à-dire

a) système “Absence de sinistre”

b) système “Nombre de sinistre”.

A l'aide de simples modèles (examples page 56) pour le groupement de portefeuilles de risques bons et lourds Gürtler étudie l'effet des deux systèmes et en tire des conclusions générales sur l'applicabilité des systèmes. Gürtler souligne d'abord que le système b) utilise considérablement mieux l'information, parce que tous les sinistres y sont considéréd et non pas, comme dans le système a), seulement le premier sinistre étant registré pour chaque assurance.

Marcel Derron traite dans “A theoretical study of the no-claim bonus problem” du même cahier de la publication la quote-part du ER (Error ratio) qu'il définit comme une quote-part égale au montant absolut de la différence entre la prime payée et la vraie prime divisée par la prime payée. Il calcule la quote-part du ER pour le modële de Gürtler et pour des modèles alternatifs et cherche le système de bonification qui rend la quote-part du ER aussi petite que possible pour le modèle choisi.
A Note On Optimal Reinsurance
Authors: Benktander Gunnar
Published: July 2016

In the past the subject of optimal reinsurance has been dealt with by various authors such as Borch (1), Kahn (2) and Verbeek (3). Borch tries to look at the problem from the point of view of a company which acts as both Insurer and Reinsurer whereas the viewpoint of Kahn and Verbeek is that of a Ceding Company.

Let us study the simplest possible market situation of one Insurance Company C and one Reinsurer R. C writes a portfolio P and seeks reinsurance protection at R.

In the present note I will start with the viewpoint of optimizing simultaneously the situation of C and R.

For both C and R the variance is used as a measure of risk. The variance of P is V. C makes an effort to reduce V to a level Vc by way of a reinsurance treaty T. The corresponding variance of R is VR.

Out of the set of possible reinsurance treaty arrangements {T} we assume that the parties are first of all interested in such arrangements for which

In other words they will look for solutions which will lead to a substantial reduction of the variance.

It is possible to think of “nonsense” arrangements where no substantial reduction is achieved and in the extreme case even of arrangements for which

In a further paper we might deal with such a case.
Note sur la gestion automatique de la réserve pour sinistres en assurance automobile
Authors: Petit Jean
Published: July 2016

„… La tendance la plus significative qui se dégage … consiste à doter les systèmes automatiques des facultés d'adaptation qui sont le propre de l'homme. „On constitue ainsi un ensemble doué de mémoire dont la structure se modifie progressivement au fur et à mesure qu'il prend conscience des résultats de ses expériences successives”.

P. Naslin(Automatisme, Nov. 1959)

Cette note est présentée par un actuaire qui n'a eu que depuis peu à s'intéresser aux branches d'assurances autres que la Vie et seulement du point de vue de la gestion administrative, à l'occasion de la mise en œuvre d'un grand ensemble électronique de traitement de l'information. Il ne faut done pas s'attendre à trouver ici une théorie sur la détermination des montants à réserver pour les sinistres à régler, encore moins sur la construction des tarifs d'assurance automobile.

Néanmoins, la gestion administrative n'ayant pas pour seul but de recueillir et de classer des données comptables, nous serons bien naturellement conduits à prévoir comment ces données peuvent être liées à des recherches sur ces deux questions fondamentales.
La théorie des plus grandes valeurs et son application aux problèmes de l'assurance
Authors: Finetti M. Bruno De
Published: July 2016

L'intérêt de ce sujet en soi-même, au point de vue des mathématiques, du calcul des probabilités et de la statistique — les différents aspects et questions qui peuvent le relier à des applications dans le domaine des assurances — le fait que des discussions à cet égard ont été commencées l'année dernière à Juan-les-Pins avec l'importante contribution de M. Franckx et favorisées ensuite par les notes introductives de M. Beard si riches en indications et suggestions — toutes ces choses auraient dû conduire, sinon à épuiser le sujet dans le présent colloque — à achever une base de discussion suffisamment complète et variée.

Sans doute, cela aurait été un miracle, et c'est loin d'être étonnant si cela ne s'est pas passé. En effet, c'est bien naturel que les communications à un congrès portent aux sujets en programme des contributions fragmentaires concernant des aspects particuliers choisis par chaque auteur d'après ses intérêts et son point de vue. Et ce fait en soi est fécond en ce qui concerne la pluralité des intérêts et des points de vue: seulement il serait souhaitable d'éviter le caractère partiel et fragmentaire du résultat d'ensemble. Je crois que la voie suivie par ASTIN pourrait amener à ce résultat heureux, et qu'il suffirait à ce but de la perfectionner quelque peu davantage: par exemple, en cherchant à favoriser par des contacts préalables, chaque année pour l'année suivante, la préparation (toujours libre et indépendante) d'un ensemble minimum de contributions suffisant pour le recouvrement d'au moins tous les traits essentiels du sujet. Toute suggestion de ce genre, il est bien entendu plus aisée de l'avancer que de la réaliser; mais, si elle n'est pas totalement utopique (et s'il n'y a pas de raison de craindre qu'elle décourage la présentation d'autres contributions), je pense que ASTIN, qui se trouve dans la phase de formation d'une tradition et qui n'a besoin que de perfectionner des procédés déjà spontanément adoptés, est dans les conditions les meilleures pour y réussir.
On the Use of Extreme Values to Estimate the Premium for an Excess of Loss Reinsurance
Authors: Jung Jan
Published: July 2016

I. Suppose that the claims experienced by a portfolio could be represented as independent random variables with a distribution function F(x). The net premium per claim for an excess loss cover above an amount of L is then

If we have no information about F(x) except a number M of independent claims, we might compute the observed “staircase” distribution function SM (x) which is for every x an unbiased estimate of F(x), and could thus compute an unbiased estimate for P(L) with the variance [7])

2. In real life we have some qualitative knowledge of F(x) and very limited information about the claims. In his introduction to this subject Beard treats the case where the only information about F(x) consists of the largest claim xi and the number of claims ni (i = 1, 2, …, N) observed during N periods (Reference No 2). It is known from the theory of extreme values [3] that for large ni the distribution of xi depends mainly on the parameters uni and αni( defined by

Beard further assumes that F(x) belongs to what is called by Gumbel “the exponential type” of distribution functions, which have an unlimited tail and finite moments. This class is strictly defined by Gnedenko's necessary and sufficient condition [4, p. 68]:
Methode de sondage pour l'etude de l'influence de nouveaux criteres sur le tarif “automobile”
Authors: Montreynaud B. Dubois De
Published: July 2016

Pour construire un tarif “Automobile” il est nécessaire d'étudier la charge des sinistres causés par les véhicules appartenant à des groupes homogènes divers: il est commode de procéder à un échantillonnage.

On peut construire un échantillon où figurent des proportions constantes ou connues de chacun des groupes constituant la population à étudier.

En pratique l'échantillonnage, doit, pour avoir une efficacité admissible, être particulièrement important dans les groupes à faible fréquence pour donner un nombre significatif de sinistres.

Or, comme pour déterminer la méthode de sélection des échantillons on ne connait pas le groupe des véhicules sélectionnés, on serait contraint de prendre un échantillon très important.

En France, le Groupement Technique Accidents fait des études sur la totalité des véhicules assurés par les Compagnies adhérentes à la statistique commune et qui représentent un tiers du Pare National.

L'inconvénient de la méthode actuelle est l'impossibilité d'étudier rapidement un nouveau critère — car il faut attendre les mouvements sur le portefeuille à l'occasion desquels on enregistre le critère à étudier; les mouvements, en outre, frappent probablement moins les véliicules à faible fréquence. Le délai nécessaire pour l'étude d'un nouveau critère se compte en années.

Il peut être souhaitable pour suivre l'évolution économique ou sociale d'étudier rapidement dans un délai chiffré en mois, l'influence de certains critères nouveaux, pour éviter sous la pression d'événements extérieurs d'être amené à. prendre des décisions dites “politiques” — cet euphémisme justifiant, dans le doute, des réductions de tarif explicables uniquement par le fait qu'un concurrent dangereux est peut-être sérieux — à ce moment il est inévitable de procéder à un sondage.
Exemplification of Ruin Probabilities
Authors: Wikstad Nils
Published: July 2016

The following numerical values of ruin probabilities, Ψ(u, T) for finite times T, have been calculated by the method proposed in “Analytical steps towards a numerical calculation of the ruin probability for a finite period when the risk process is of the Poisson type or of the more general type studied by Sparre Andersen”, presented to this colloquium by Olof Thorin. The notations used in the sequel follow those of Thorin.

Two distributions of the individual claims are considered, viz.,

The latter distribution is a rather crude attempt to interprete the extremely skew distribution (Swedish non-industry fire insurance 1948-1951) considered by Cramér in his treatise “Collective Risk Theory”, Jubilee volume of Försäkringsaktiebolaget Skandia (1955) pp. 43-45.

Likewise two distributions of the interoccurence times are considered, viz.,

The d.f.B was considered by Sparre Andersen (TICA 1957 vol. II pp. 225-227).

Note that the first moment equals one in all the d.f. mentioned.

Though the analytical machinery also seems to work for o ≤ c ≤ 1 the Ψ values are indicated only for some values of c > 1. As known from Thorin's paper c stands for 1 + λ, where λ is the premium

loading, which means that for c = o Ψ(u, T) corresponds to the tail of the d.f. for the total amount of claims during the period (o, T).
Risque et Rentabilite
Authors: Croset Gérard
Published: July 2016

1. M. Karl Borch a présenté, dans plusieurs de ses communications à l'ASTIN), un modèle de comportement de l'assureur, qu'il qualifie lui-même de simplifié, mais qui met néanmoins en évidence une intéressante propriété d'optimalité. L'assureur maximiserait la valeur actuelle des dividendes futurs en laissant les bénéfices éventuels s'ajouter aux fonds propres jusqu'au niveau Z; les montants qui porteraient alors les richesses libres au dessus du niveau Z seraient intégralement distribués. L'espérance mathématique optimale V(S, Z) de la valeur actuelle des dividendes futurs dépend, dans ces conditions, du capital S initial. D'autre part, il reste entendu que si S > Z, alors (S — Z) est immédiatement „remboursé”, et, si S = o, la société cesse, en principe, ses opérations, toutefois, M. Karl Borch a envisagé la possibilité d'un renflouement).

2. L'utilisation pratique de ce modèle pose deux problémes particulièrement importants:

— la société a-t-elle le droit de fonctionner avec le capital S?

— les capitalistes accepteraient-ils d'investir S?

2.1. Le premier de ces problèmes a été envisagé par nous-même dans une étude précédente). Il apparaît que la contrainte imposée par l'autorité de tutelle peut-être sommairement résumée par l'obligation d'un capital minimum T. La valeur actuelle optimale V(S, T, ZT) des dividendes futurs est alors diminuée par rapport à la valeur précédemment considérée (correspondant à T = o), soit V(S, o, Z0), tandis que le niveau ZT de la barrière reflexive est placé plus haut que Z0.

2.2. Il reste à considérer le second problème, c'est-à-dire, finalement, celui de l'opportunité du placement des capitaux dans une société d'assurances dommages. Cela nécessite que soit défini un critère de choix des placements.
Applications to a theory of bonus systems
Authors: Vepsäläinen S.
Published: July 2016

This paper deals with bonus systems used in Denmark, Finland, Norway, Sweden, Switzerland and West Germany. These systems are studied by methods given by Mr. Loimaranta). The bonus rules of Denmark and Sweden have been modified because they contradict to one of the assumptions of the theory.

It is assumed that the number of claims in a year follows the Poisson distribution with a mean λ. Further it is assumed that the value of λ is independent of time.

In each case bonus rules are given in form of transformations Tk defined in ref. 1., i.e. Tk(i) = j when a policy moves from class i to class j after k claims. The class where a new policy starts from is called the initial class. Bonus scales, the vectors B, are normed so that the premium of the initial class is 100.

For each bonus system the efficiency η of the system and the discrimination power d of the bonus rules as a function of the mean claim frequency λ have been calculated. The graphs of these functions are presented in the figures 2 to 4 respectively. In the following pages, different bonus rules are described in detail and some simple analysis based on the curves on pp. 211-214 has been made. The calculations were performed quite recently and any deeper analysis of the results has not been possible because of lack of time.

In order to be able to apply the theory of Markov chains to a bonus system we must require among others that the transition to a certain class depends only on the number of claims occurred during last period ignoring possible former claims. The Danish system has originally four bonus classes labeled from 0 to 3.
Some statistical aspects of catastrophic risks
Authors: Anantapadmanabhan C. S.
Published: July 2016

1. Risks of a catastrophic nature, such as war or earthquake, can differ, in a statistical sense, from other ordinarily insured perils only in respect of the distribution of the probability function. For example, although the probability of a loss occurring may be small, its magnitude once it occurs may be very large. One reason for the latter feature is that the sum insured may itself be heavy. But still more important is the fact that there would be strong positive correlation between similar contiguously situated risks. This is the main reason for the loss taking on a catastrophic character.

2. The fact that the probability of a loss is ordinarily small, leads to the result that the expected loss and hence the pure premium for the risk is small. Equally, and for the same reason, the “loss strain at risk” is large. If the latter could be considerably reduced—which can only be done by charging impracticably high premiums—the risk becomes more easily insurable.

3. If there are n risks of this nature, 1, 2, 3 … n and νi(x) is the conditional frequency function of the ith risk we may write

While we may imagine this as a frequency curve, the complementary probability Ϸi of a loss not occurring can be represented as a vertical line at the origin—the different methods of representation of Ϸ and q are actually convenient. With this representation, the expansion of the binomial product

shows that the convolution of the n curves can be expressed as the sum of n + I functions

where

(ii) the term A1(x) comprises n separate curves of the form νi(x) each multiplied by its complementary probability, being of the form

(iii) the term A2(x) comprises terms each consisting of the convolution of the curves taken two at a time and multiplied by the complementary probability, being of the form

and so on.
Insurance as a cybernetic institution of self-regulation of the national economy
Authors: Banasinski Antoni
Published: July 2016

The present referendum is only indirectly connected with the themes of the fifth Colloquium of ASTIN. It is more or less a memorandum about the actual work of Polish actuaries in the sector of the Insurance Theory and trying out a new conception and solution of social and economic problems of Property and Capital Insurances.

The principles and hypotheses of the cybernetic insurance theory posed in this referendum are a result of social situations in a planned economy. However, it is assumed that the cybernetic analysis of the economic role of insurance, and the methods of calculation resulting from this analysis, can also be applied to other social and economic situations.

In the judgment of the author, the problem under consideration would be of interest to the members of ASTIN and could possibly be included in the syllabus of one of the subsequent colloquia of ASTIN.
The Calculation Of A Fluctuation Loading For An Excess Of Loss Cover
Authors: Benktander Gunnar
Published: July 2016

A diminishing number of Reinsurers still seem to work with percentage loadings which are applied with more or less sophistication to the pure risk premium rate.

A more sensible approach is to study the distribution of the total claims amount of the layer reinsured (standard deviation, variance, skewness).

In practice this distribution is not known and we have to work with estimates E* and σ*2 of the first moment which is the pure risk premium (E) and of the second central moment (σ2). Because of the positive skewness of the distribution, E* will fall below E in more than half of the cases and sometimes considerably below.

One security loading principle could be to add a proportion k of σ* to E* in such a way that

If ε is put equal to 0.25 this reflects our wish to ensure that our security loaded rates are too low only in 25% of the cases. A σ- loading here emerges as a result of the statistical uncertainty.

If we analyse the result fluctuations we are led to apply a variance loading (1), which can be interpreted as a price for capacity (2). (See also discussion contributions by B. Ajne, G. Benktander, G. Berger, H. Bühlmann; Transactions of the Congress, 5, Oslo 1972, p. 169 ff.)

An underwriter who is confronted with two portfolios having the same first and second moments should, if conditions are the same, prefer the one with the lower third moment. A special loading for skewness is thus indicated.
The Surplus Process As A Fair Game—Utilitywise
Authors: Gerber Hans U.
Published: July 2016

The concept of utility is twofold. One may think of utility:

1) as a tool to describe a “fair game”

2) as a quantity that ought to be maximized.

The first line of thought was initiated by Daniel Bernoulli in connection with the St. Petersburg Paradox. In recent decades, actuaries, economists, operations researchers and statisticians (this order is alphabetical) have been concerned mostly with optimization problems, which belong to the second category. Most of the actuarial models can be found in a paper by Borch [4] as well as in the texts by Beard, Pesonen and Pentikainen [3], Bühlmann [6], Seal [15], and Wolff [17].

We shall adopt the first variant and stipulate the existence of a utility function such that the surplus process of an insurance company is a fair game in terms of utility. This condition is naturally satisfied under the following procedure: a) a utility function is selected, possibly resulting from a compromise between an insurance company and supervising authorities, b) whenever the company makes a decision that affects the surplus, it should not affect the expected utility of the surplus.

Mathematically, this simply means that the utility of the surplus is a martingale. Therefore martingale theory (that was initiated by Doob) is the natural framework in which we shall study the model. We shall utilize one of the most powerful tools provided by this theory, the Martingale Convergence Theorem.