A method is presented for projecting the development of reported losses, or of paid losses, or the number of reported cases, for each of the first 12 months of an accident year. The method is based on the relationship between development patterns after the first 12 months of an accident year and development patterns during the first 12 months. A way of deriving the development pattern of an individual accident quarter or accident month is also described.

It is pointed out that the method can be useful for underwriting decisions for which the most recent experience is relevant, or for estimating loss reserves at the end of a quarter or when earned premium differs greatly by month or quarter.

Consider a risk process which is characterised by three stochastic variables

(1) the number of accidents, N,

(2) the number of claims per accident, C, and

(3) the amount of a claim, X.

Let Y be a random variable denoting the total loss in a given period.

Suppose that

and

If Pr represents the probability that exactly r claims occur in the period, then Kupper [4] has shown on certain simplifying assumptions that

where , the probability of exactly r claims in n accidents, is given by

and for γ < n

Further

“War is the father of all things”. This sentence of Heraklit (between 540 and 535 BC) has a lot of truth in general. I am quoting it in connection with World War II (1939-1945) and inventions originating in this period.

The best known examples of inventions (or first practical use of such invention) are

• Nuclear Fission leading to the construction of the atomic bomb and nuclear reactors,

• The Programmable Electronic Computer. Best known is the machine of John von Neumann as Los Alamos: Mathematical Analyser Numerical Integrator and Computer (MANIAC),

• Radar to guide the airplanes particularly at night and in bad weather conditions,

• Penicillin to fight bacteriological diseases.

Some of these inventions relied on original discoveries already made before the war, but during the war they were for the first time used on a large scale.

In dem vorliegenden Artikel werden verschiedene Berechnungsmethoden diskutiert. Mit Hilfe mathematischer Tests werden die den Näherungsverfahren zugrundeliegenden Annahmen überprüft. Ausserdem ergibt ein Vergleich der Ergebnisse nach den verschiedenen Methoden einen Überblick bezüglich der Grenzen, in denen die Schwankungsrückstellung liegen sollte.

Die exakte Berechnung der benötigten Reserve ist nach den Methoden der kollektiven und auch der individuellen Risikotheorie nur möglich über die Verteilungsfunktion des Gesamtschadens. Wegen der auftretenden Faltungspotenzen der Verteilungsfunktion der Schadenshöhe sind diese Formeln sehr kompliziert. Man ist daher gezwungen bei praktischen Berechnungen von Näherungsformeln auszugehen.

Die einzelnen Verfahren unterscheiden sich wesentlich in den über die Schadensverteilung gemachten Annahmen. Es wird entweder die Verteilungsfunktion unmittelbar aus den Beobachtungsergebnissen hergeleitet oder von bekannten Verteilungsgesetzen, wie z.B. dem Poisson-Gesetz, ausgegangen.

According to the editorial rules of treating discussion situations in the ASTIN Bulletin the paper by J. Lemaire & H. Zi being somewhat a discussion on Holtan's paper was sent to the author of the original paper, who was given the opportunity to make an additional comment. The editors then received the following note by Jon Holtan:

In this note I want to give some general comments on the papers by Lemaire & Zi (1994) and Holtan (1994).

Interpret henceforth a bonus-malus (BM) principle as consisting of two basic components:

(a) The BM design.

(b) The BM tariff parameters.

Traditional actuarial literature has basically been preoccupied with component (b). Or more precisely, the tariff parameters of an initial accepted BM design have usually been mathematically optimalized within different criteria of succes like e.g. high efficiency and financial balance. In my opinion, however, this strategy seems to be too narrow if the aim is to construct a BM principle which is totally optimalized in favour of both the insurer and the insured. In our strive for maximizing BM advantages and minimizing BM disadvantages, actuarial BM research should instead simultaneously focus on both components (a) and (b). The construction of the High-Deductible System (HDS) in Holtan (1994) is an example of this strategy. However, as pointed out in Lemaire & Zi (1994) (see Section 1 and 4) and Holtan (1994) (see Section 3, 5 and 6), a HDS compared with existing BM systems both eliminates and generates important disadvantages which are linked to component (a). Based on some mathematical model assumptions, Lemaire & Zi moreover concludes (see Section 3 and 5) that this two-sided conclusion is in principle also valid within some mathematical criteria of success linked to component (b). These complex, and perhaps confusing, conclusions make it difficult for us to decide whether to prefer the existing BM systems or the HDS. However, the solution to this problem of decision seems to be naturally dependent on some strategic questions like: What kind of BM advantages and what kind of BM disadvantages will be the most important to focus on in the future automobile insurance market?

Besides its well known applications, the collective risk theory has recently also been applied to problems connected with the so called Experience Rating. This term is used to define a method of premium calculation in insurance business which is based partially or totally on the individual experience of the particular risk involved. It is obvious that Experience Rating is essentially applicable to collective insurances which contain no saving element. In practical applications various possibilities may be considered.

The collective theory of risk provides an efficient calculus for the analysis of the various forms of Experience Rating and in paper [3] a particular form of Experience Rating for collective insurances is examined. It is there assumed that the collective risk premium is based on experience derived from non-individual observations. If any cost loading is disregarded, the net premium is given by the relation P′ = (I + λ) P, where λ is a security factor and P the part of the premium covering the expected claims cost.

A premium refund

is to be deducted from the basic net premium P′. In this formula α′ and β are suitable numerical values and S means the due sum to be paid out for claims. Hence the net cost to the group considered depends on the actual claims S and therefore takes into account the individual claims experience. It may be shown that for β = I the form of Experience Rating considered is equivalent to a stop loss cover. The general case with β ≠ I represents a combination of ordinary insurance cover and stop loss cover.

Ce sujet n'a pas été traité, peut-être parce qu'il paraît trop simple; pourtant il mérite examen vu son utilité pratique certaine.

Il est impossible de négliger le jeu de la réserve de Risques en Cours dans l'estimation du taux des sinistres c'est-à-dire dans l'ap-préciation de la tarification et de l'équilibre d'une branche d'assurances.

D'autre part, comme le souligne justement la note introductrice aux thèmes proposés établie par le Comité Astin, le problème est moins immédiat qu'il ne paraît:

On considère généralement la réserve de Risques en Cours (l'expression anglaise risques non expirés est peut-être meilleure) comme un simple pourcentage des primes totales: c'est la fraction de primes non acquise à l'exercice mais ceci suppose que le risque est uniformément réparti dans le temps et que les primes correspondent exactement aux risques couverts.

En fait deux conceptions sont possibles:

— méthode rétrospective: on considère le déroulement passé des opérations: schématiquement on prend la différence entre les primes perҁues et les sinistres survenus; ce qui hypothèque l'avenir.

— méthode prospective: on fixe la somme nécessaire pour faire face aux sinistres à venir dans l'intervalle de temps encore couvert par les polices en cours, ceci compte tenu du taux actuel de sinistres et même de son évolution probable dans le futur immédiat.

La méthode prospective est-elle plus correcte? Il peut le sembler. Mais c'est seulement viser à l'équilibre probable du bilan futur, au détriment du bilan actuel. Pourquoi une insuffisance de tarification, par exemple, doit-elle peser sur le seul compte de l'exercice clos?

In this paper, we question the traditional independence assumption between mortality risk and financial risk and model the correlation between these two risks, estimating its impact on the price of different life insurance products. The interest rate and the mortality intensity are modelled as two correlated Hull and White models in an affine set-up. We introduce two building blocks, namely the zero-coupon survival bond and the mortality density, calculate them in closed form and perform an investigation about their dependence on the correlation between mortality and financial risk, both with theoretical results and numerical analysis. We study the impact of correlation also for more structured insurance products, such as pure endowment, annuity, term insurance, whole life insurance and mixed endowment. We show that in some cases, the inclusion of correlation can lead to a severe underestimation or overestimation of the best estimate. Finally, we illustrate that the results obtained using a traditional affine diffusive set-up can be generalised to affine jump diffusion by computing the price of the zero-coupon survival bond in the presence of jumps.

In the discussion in the ASTIN Colloquium 1962, which followed my lecture on the numerical evaluation of the distribution functions defining some compound Poisson processes, a remark was made, which drew my attention to the paper quoted here below under reference number (1), On composed Poisson distributions I.

As I have the impression that this remark may induce some confusion of the terms “compound” and “composed”, the more as the same French word (composé) is used for the two terms, a comparison between the two kinds of processes shall be made.

1. The most general propositions of (1) are a theorem which concerns a general homogeneous Markoff process (1.c. § 2) and a theorem for the family {P(k,p)} of distribution functions of positive integervalued variables with mean p, where p runs over all non-negative numbers, and where the convolution of P(k,p1) and P(k,p2) is equal to P(k,p1 + p2) (l.c. § 3). By these propositions the characteristic functions corresponding in the 1st case to the distribution functions defining the homogeneous Markoff process, and in the 2nd case to the distribution functions belonging to the family {P(k,p)} can all be written in the same form, namely

where u is an entirely imaginary variable, p is the parameter of the process, respectively of P(k,p) and c1, c2 … non-negative constants such that converges. If, in the 1st case, converges, and as, in the 2nd case, this series converges, the distributions defined by these characteristic functions are called composed Poisson distributions, which define homogeneous composed Poisson processes.

As we grapple with the problems of conducting non-life insurance in a world of high inflation, we should perhaps pause for a moment to look for historical analogies for our present predicament. Let us, then, look back at a distressing period many years ago in the history of the Kingdom of Carmania.

Inflation was by no means new to the citizens of Carmania. Year after year, all prices and incomes had been rising at a steady rate of 5% per annum. The economy functioned quite well, albeit in a rather uninteresting way, and the people were accustomed to the idea that the purchasing power of the Carmanian dollar would fall by roughly 5% every year. There were, however, some economists who argued that inflation was both undesirable and unnecessary, and prevailed upon the King to adopt some measures which, they assured him, would quickly reduce the rate of inflation to zero. The measures were adopted, and the following year the rate of inflation rose to 10%.

At this point a rival group of economists explained that this unfortunate development was simply what should have been expected, and they set out an alternative policy which would undoubtedly reverse the trend and bring inflation quickly under control. Convinced by the weight of their arguments, the King introduced what became known as Phase 2 of the counter-inflation policy. By the next year, the rate of inflation had risen to 20%.

This paper describes a rather simple application of credibility which was used to develop experience rating plans for Blue Cross and Blue Shield organizations in the United States. These are non-profit corporations which provide prepaid coverage for the cost of hospitalization and medical-surgical care. The benefits are in many instances full service benefits so that there is no additional cost to the subscribing member.

While coverage is being granted to individual applicants, the bulk of the unterwriting is composed of groups, that is employees of a given commercial entity. Since the plans are voluntary and, in many instances, the employer either acts only as collecting agent for his employees or else pays only a fraction of the premium, certain minimum participation percentages have been established to prevent anti-selection.

Because of the competion of Insurance Companies which have concentrated their efforts on larger and more profitable groups (especially those where the employer pays all of the premiums) it was early recognized that a uniform or “community” rate will result in a gradual loss of groups with good experience thereby requiring substantial increases in the average rates. The original program of experience rating which the author has developed for the Massachusetts Blue Cross in 1949 has been since adopted “mutatis mutandis” by a number of other Blue Cross and Blue Shield organizations.

By intuition, the subdivision of an insurance portfolio into a number of classes is said to be good if it reflects the heterogeneity of the portfolio in an efficient way. To illustrate this rather vague statement we take the following very simple example:

The portfolio consists of 20 independent risks, 10 of them producing an expected loss ratio of say 30% each (type A risks) and 80% each (type B risks) respectively.

This “natural” subdivision is certainly better than, for instance, no subdivision at all;

or, the finest possible subdivision with 20 classes consisting of only 1 risk each (because there is no point in differentiating between risks of the same type);

or, 5 classes each containing two A- and two B-risks (here the number of classes is unnecessarily high and the heterogeneity has been completely wiped out; statistics based on this subdivision would even make us believe that the portfolio is totally homogeneous).

As a matter of fact, the above “natural” subdivision is—of course!—the best of all subdivisions, it is the optimal subdivision in this case.

In practice, however, as we all know, it is not easy to find the optimal subdivision. For one thing, the inherent structure or “natural” subdivision is not known a priori and secondly, for many different reasons, we can only choose from a limited number of subdivisions and not from all theoretically possible solutions. Note that even with only 20 risks there are 58.1012 possibilities of subdividing the portfolio. Thus, in practice, there is only a relatively small number of admissible subdivisions and the optimal one may not be among them, but we still need some sort of statistical criterion to choose the best one from these admissible subdivisions.

1.1. — Le titre de cette note indique que le problème étudié est très général et par le fait même applicable à de nombreux domaines; physique, biologie, recherche opérationnelle, assurances, téléphonie, etc …

C'est évidemment parce que certains problèmes ont la même structure que leur considération sous forme abstraite, est plus importante. C'est la raison pour laquelle nous traitons le problème de structure, qu'il suffit de particulariser dans chaque domaine d'application.

a) Nous supposerons une période de référence de durée constante.

b) Un évènement E, généralement quelconque, mais invariable, peut se produire pendant cette période. Le nombre de fois que cet évènement se réalise est une variable aléatoire I, qui sera définie par la suite

qi désignant la probabilité pour que E se réalise i fois pendant la période de référence. Nous conviendrons de dire que I mesure l' “intensite” du phénomène E.

c) Lorsque l'évènement E se réalise, on effectue une expérience relativement à. une autre variable aléatoire quantitative N (positive ou non). Cette variable stochastique est completement définie par sa fonction de répartition.

F(n) = prob{N<n}

généralement quelconque; nous conviendrons de dire que n est le “niveau” atteint par la variable N au cours de l'expérience.

1.3. — De nombreux problèmes de la vie courante peuvent être synthétisés sous cette forme et pour ne mentionner que quelques dans le domaine de la vie des organismes d'assurances, citons sous forme de tableau

Cette simple énumération indique la grande variété de problèmes de même structure que l'on rencontre journellement; il est par suite utile de les traiter en synthèse.

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.

Fair valuation is becoming a major concern for actuaries, especially in the perspective of IAS norms. One of the key aspects in this context is the simultaneous analysis of assets and liabilities in any sound actuarial valuation. The aim of this paper is to illustrate these concepts, by comparing three common ways of giving bonus in life insurance with profit: reversionary, cash or terminal. For each participation scheme, we compute the fair value of the contract taking into account liability parameters (guaranteed interest rate and participation level) as well as asset parameters (market conditions and investment strategy). We find some equilibrium conditions between all those coefficients and compare, from an analytical and numerical point of view, the systems of bonus. Developments are made first in the classical binomial model and then extended in a Black and Scholes economy.