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?

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 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.

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.

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%.

We propose a capital allocation method for insurance companies. The amount of capital is directly related to the default risk. The expected value of default can be distributed among the liabilities based on the rule of asset payoff at the time of default. We derive a capital allocation scheme from this allocation of the expected default. Assets, liabilities, and other risky items on the balance sheet are treated in a uniform framework. The insurer’s capital is allocated among all these risk contributors. The allocated capitals are given in closed-form formulas, which have straightforward interpretations and are easy to compute. Connections with other allocation methods are also discussed.

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.

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.

The paper discusses the problem of estimating the mean of a long-tailed claim size distribution when the investigator's knowledge of the distribution is only vague.

One method of dealing with this problem, the method developed by Johnson and Hey, is examined and found to produce strongly biased estimators.

The situation in which a sufficient statistic (but nothing else) for the claim size distribution is known is examined, and an approximately unbiased estimator developed. This estimator is substantially more efficient than the arithmetic mean in some cases. It appears to be quite successful when the sufficient statistic is real-valued. It is of limited use when the sufficient statistic is vector-valued.

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.

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.

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.

Solvency regulations require financial institutions to hold initial capital so that ruin is a rare event. An important practical problem is to estimate the regulatory capital so the ruin probability is at the regulatory level, typically with less than 0.1% over a finite-time horizon. Estimating probabilities of rare events is challenging, since naive estimations via direct simulations of the surplus process is not feasible. In this paper, we present a stratified sampling algorithm for estimating finite-time ruin probabilities. We further introduce a sequence of measure changes to remove the pathwise discontinuities of the estimator, and compute unbiased first and second-order derivative estimates of the finite-time ruin probabilities with respect to both distributional and structural parameters. We then estimate the regulatory capital and its sensitivities. These estimates provide information to insurance companies for meeting prudential regulations as well as designing risk management strategies. Numerical examples are presented for the classical model, the Sparre Andersen model with interest and the periodic risk model with interest to demonstrate the speed and efficacy of our methodology.