The question of large claims in insurance is, evidently, a very important one, chiefly if we consider it in relation with reinsurance. To a statistician it seems that it can be approached, essentially, in two different ways.

The first one can be the study of overpassing of a large bound, considered to be a critical one. If N(t) is the Poisson process of events (claims) of intensity v, each claim having amounts Yi, independent and identically distributed with distribution function F(x), the compound Poisson process

where a denotes the critical level, can describe the behaviour of some problems connected with the overpassing of the critical level. For instance, if h(Y, a) = H(Y − a), where H(x) denotes the Heavside jump function (H(x) = o if x < o, H(x) = 1 if x ≥ o), M(t) is then the number of claims overpassing a; if h(Y, a) = Y H(Y − a), M(t) denotes the total amount of claims exceeding the critical level; if h(Y, a) = (Y − a) H(Y − a), M(t) denotes the total claims reinsured for some reinsurance policy, etc.

Taking the year as unit of time, the random variables M(1), M(2) − M(1), … are evidently independent and identically distributed; its distribution function is easy to obtain through the computation of the characteristic function of M(1). For details see Parzen (1964) and the papers on The ASTIN Bulletin on compound processes; for the use of distribution functions F(x), it seems that the ones developed recently by Pickands III (1975) can be useful, as they are, in some way, pre-asymptotic forms associated with tails, leading easily to the asymptotic distributions of extremes.

Sous la responsabilité de l'Association Générale des Sociétés d'Assurances contre les Accidents, un sondage au 1/50e a été effectué en 1971 dans les portefeuilles d'un grand nombre de Compagnies ou Mutuelles pratiquant en France l'assurance de responsabilité civile automobile.

Trois objectifs étaient visés:

Etudier l'influence de critères de tarification tels que l'âge du véhicule, l'âge du souscripteur ou l'ancienneté du permis, zone, groupe et usage.

Déterminer, pour chaque classe du tarif, les primes pures. Tirer les enseignements des résultats de ces travaux, en particulier, étudier les possibilités d'amélioration du tarif à la fois sur le plan technique (choix des critères et méthodes de calcul des primes) et sur le plan politique (réalisation effective du tarif).

Les informations apportées par le sondage se présentaient comme suit:

l'uuité statislique était constituée par l'ensemble souscripteur — véhicule — durée d'observation. Au cours de cette durée, les caractéristiques du souscripteur, du véhicule et de l'environnement, étaient inchangées. A chaque unité statistique étaient rattachés les renseignements concernant la zone, le groupe de tarification du véhicule, les clauses d'usage et de garantie souscrites, l'état matrimonial, le sexe et l'âge du souscripteur, l'année de première mise en circulation du véhicule, l'année d'obtention du permis de conduire. On disposait, d'autre part, du nombre de sinistres de chaque sorte (matériels ou mixtes) et le coût au titre de la responsabilité civile de ceux-là.

In reference [1] Dr. G. C. Taylor has described a useful advance in the techniques available for verification of outstanding claims estimates when the data provided is the cohort development of numbers and amounts of claims. In this note it is assumed that the numbers relate to settled claims and that the amounts relate to claim payments, so there is an implicit assumption that the pattern of partial payments is constant. If the amounts of settled claims were to be used, there would be a one/one relationship between the numbers and amounts, but the effect of the exogeneous factor would be blurred because the settlements in a year other than the first include partial payments made some time previously, and, by hypothesis, based on different factors. If information relating to partial payments is available the data can be examined for any major fluctuation in the pattern and allowance made accordingly.

In paragraph (2) of reference [1] a brief description is given of a standard routine calculation in which the average distribution function of claim payments in time is estimated from the triangle of payments by a chain ladder technique. This distribution function is then used to estimate the expected development of the incomplete cohorts, the implicit assumption being made that the function was stable in time. With a constant rate of inflation the results obtained by this technique were found to be satisfactory but with a rapid increase in the rate of inflation the distribution function changed so that projection led to underestimates of the future claims payments. Various methods of adjusting the projections to allow for the change in the rate of inflation have been investigated, but they all involve an important element of subjective judgment and so far no generally suitable basis for “automatic” verification by this particular technique has been discovered. See however reference [2].

Stop Loss reinsurance has attracted the interest of ASTIN members for years. May I recall the paper of Borch [1] in which he demonstrates some optimality qualities of the stop loss reinsurance from the ceding company's point of view, the contribution of Kahn [2] and the paper of Pesonen [3]. I also mention the paper of Esscher [4] and Verbeek's contribution [5]. Going back to the pre-ASTIN days we find a paper of Dubois [6].

The rating problems have been dealt with by several authors. Let me recall the rating formula worked out by a group of Dutch Actuaries some 20 years ago. This was based on the assumption that the mean and the standard deviation were known. Based on Chebycheff's inequality an approximation formula was worked out which, of course, was heavily on the safe side.

Even younger members of ASTIN are probably familiar with the studies made in the early sixties by a group of Swedish Actuaries, the results of which were presented by Bohman at the Actuarial Congress in London in 1964. Partly based on this, Bühlmann worked out some tables which he used for rating purposes.

My present contribution to the subject may not justify the above reviews, particularly as I will deal with a very special retention situation which a practical underwriter will rightly not accept, namely a stop-loss point as low as equal to the mean value of the distribution.

My excuse for this is that the formula deduced is very handy and that it is of value to the underwriter to know the stop loss risk rate also at this low level.

The qualifications of the semivariance as a useful risk measure are examined and compared to those of the variance. Although on first sight the semivariance may seem more appropriate from the insured's point of view the analysis of this paper leads to a preference for the variance as a risk measure.

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

An inflation index is essential when constructing claim payment models from past payment data, and when projecting these results to give estimates of the provisions for outstanding claims and of necessary premiums.

This paper examines the choice of inflation indices for compulsory third party insurance in two Australian states. Two different indices, one based on average weekly earnings per employed male unit and the other based on consumer prices, were tested. The index based on average weekly earnings was considered to be superior in that past claim payment data, together with this index, gave reasonably stable claim payment models.

Some experiments were made for an actual office to illustrate the effects of different inflation rate assumptions.

In this paper any given risk S (a random variable) is assumed to have a (finite or infinite) mean. We enforce this by imposing E[S−] < ∞.

Let then v(t) be a twice differentiate function with

and let z be a constant with o ≤ z ≤ 1.

We define the premium P as follows

or equivalently

Notation: v−(∞) = ∞.

The definitions (1) and (equivalently) (2) are meaningful because of the

Lemma: a) E[v(S − zQ)] exists for all Q∈(− ∞, + ∞).

b) The set{Q∣ − ∞ < Q < + ∞, E[v(S−zQ)]>v((1−z)Q)} is not empty.

Proof: a)

b) Because of a) E[v(S−zQ)] is always finite or equal to + ∞ If v(− ∞) = − ∞ then E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q. The left hand side of the inequality is a nonincreasing continuous function in P (strictly decreasing if z > 0), while the right hand side is a nondecreasing continuous function in Q (strictly increasing if z > 1).

If v(− ∞) = c finite then E[v(S − zQ)] > c

(otherwise S would need to be equal to − ∞ with probability 1) and again E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q.

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.

En matière d'assurance de la Responsabilité Civile Automobile, chacun connait la difficulté de fixer un tarif qui doit prévoir l'évolution à court terme, aussi bien de la fréquence, que du coût des sinistres mais, à notre avis, un problème bien plus important et d'une actualité brûlante réside dans l'étude de la répercussion sur la liquidation des sinistres, des facteurs monétaires et économiques nationaux et peut-être dans la remise en cause du système en vigueur dans notre pays.

Le compte d'exploitation est établi en tenant compte des provisions pour sinistres qui sont de l'ordre de grandeur des primes.

Or, l'incertitude de l'évolution de la valeur de la monnaie courante dans laquelle sont établis les comptes entraîne une variation possible sur les provisions techniques, d'un ordre de grandeur supérieur au résultat de l'exploitation.

Notre propos est de montrer l'ampleur des répercussions d'une période de stagflation sur l'évolution des provisions pour sinistres de Responsabilité Civile Automobile et de donner une règle empirique permettant une meilleure évaluation.

Nous ne parlerons pas des sinistres matériels qui sont réglés rapidement et pour lesquels l'indemnité est fixée au jour du sinistre; l'inflation n'a donc pratiquement pas d'influence sur le montant du règlement. La part des sinistres matériels dans les provisions ne dépasse du reste pas 15%.

Nous nous bornerons à l'étude des sinistres corporels. Les indemnités pour ces sinistres sont fixées au jour du règlement; elles correspondent à des salaires et subissent donc à la fois l'augmentation des prix et la progression du pouvoir d'achat. Les frais de gestion et de justice qui accompagnent le règlement des indemnités sont également homogènes à des salaires.

There are two phases of difficulties in estimating a claims distribution. If we are going to estimate the claims distribution as accurately as possible, we should gather considerably long terms statistics. While economical and social environment will change. As a result the statistics gathered should be amended by a kind of trend value. One of difficulties here is the estimation of that trend value. Another difficulty is the estimation of claims distribution as being the stochastic distribution. In the case of considering claim size, the estimation becomes more difficult.

The intention of this paper is to propose an actual way of estimating stochastic claims distribution considering various kind of claim size by the use of a computer. Regarding the problem of amending claims distribution by a trend Value I will discuss at another time.

Let Sn be the set of motor claims Si, i = 1, 2, …, N occurred during a given year n, namely Sn represents the set of claims relevant to the generation (or cohort) n.

If Tn is a subset (even empty) of claims resulting without payement (that is the set of zero-claims), the set Pn = Sn − Tn shall denote the set of claims that should be settled.

For every si ∈ Pn, we can define the r.v. Xi which represents the period of time required for its settlement (namely the lag of settlement).

It is not sensible to deem that the r.v. Xi are equally distributed: as a matter of fact we know that the larger is the claim, the longer the lag of payement.

However, we can assume that in a subset U of Pn, the r.v. Xi(U) have the same distribution function, which will be denoted by FU(x) or in short F(x).

As F(x) represents the probability that a claim sj ∈ U is settled within a period o − x, the function 1 − F(x) = l(x) denotes the probability that the claim results unsettled after a lag x, that is the survival function of the claim.

In this study we intend to find an analytical expression of the function 1(x) on the basis of particular assumptions about the behaviour of the adjuster with regard to the settlement of claims.

The assumptions will be tested by fitting the function to some observed data.

One of the basic problems in life is: Given information (from the past), make decisions (that will affect the future). One of the classical actuarial examples is the adaptive ratemaking (or credibility) procedures; here the premium of a given risk is sequentially adjusted, taking into account the claims experience available when the decisions are made.

In some cases, the rates are fixed and the premiums cannot be adjusted. Then the actuary faces the question: Should a given risk be underwritten in the first place, and if yes, what is the criterion (in terms of claims performance) for cancellation of the policy at a later time?

Recently, Cozzolino and Freifelder [6] developed a model in an attempt to answer these questions. They assumed a discrete time, finite horizon, Poisson model. While the results lend themselves to straightforward numerical evaluation, their analytical form is not too attractive. Here we shall present a continuous time, infinite horizon, diffusion model. At the expense of being somewhat less realistic, this model is very appealing from an analytical point of view.

Mathematically, the cancellation of policies amounts to an optimal stopping problem, see [8], [4], or chapter 13 in [7], and (more generally) should be viewed within the framework of discounted dynamic programming [1], [2].

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%.

A theorem of Borch characterizing Pareto-optimal treaties in a reinsurance market is extended to the case of non-differentiable utilities. Sufficient conditions for the existence of a solution to the equations are established. The problem is then shown to be identical to the determination of the value of a cooperative non-transferable m-person game. We show how to compute the Shapley value of this game, then we introduce a new value concept. An example illustrates both methods.

In motorcar insurance is widely used a merit rating system characterized by the fact that only the number of claims occurred (and not their amount) modifies the premium. This system induces the insured drivers to support themselves the cost of the cheap claims. We analyze this “hunger for bonus” and solve this decision problem by means of an algorithm related to dynamic programming. The method is then applied to the Belgian bonus system.

One of the most important properties of a distribution function is that it fits the data well enough for the decision-makers' or analysts' purposes. The statisticians' problem is to select a specific form for the distribution function and to determine its parameters from the available data. Various methods (graphical method, method of moments, maximum likelihood method) are available for that purpose.

In many real world situations a single distribution function, however, may not be appropriate over the entire range of the available data. This suggests that the underlying process changes over the range of the respective variable. This fact should be considered in curve fitting. A typical example of such a situation is given in Figure 1 representing third party liability losses for trucks.

It is interesting to speculate about the different raisons d'être (Seal [5]) for the observed discontinuity. It may be the result of out-of-court or in-court settlements or could stem from differences between bodily injury and property damages.

By experience rating the main problem is to estimate the credibilities. We have for the credibility αk the famous formula)

but it is often troublesome to find suitable estimates for the variances and . In the present paper a general method to estimate them from the actual statistics is given.

A disadvantage of the method is that good estimates require relatively extensive statistical material. If one of the variances is known, the method can be easily modified to give the other variance from statistics of moderate size.

The method is based on the Maximum Likelihood principle and leads to a system of non-linear equations. The equations can be solved by an iterative process, easily programmable for computers.

The mathematical model underlying the experience rating problem differs in our case lightly from the usual one.

This note is an attempt to put the problems referring to the reserves for outstanding claims into a simple understandable form in order to facilitate the discussion of the difficult questions. In that purpose I have taken up some of Harald Bohman's ideas of the subject). I find it convenient to start with the simplest case where the liability consists of index-regulated payments at fixed epochs. My presentation is restricted to reserves of incurred and reported claims.

It seems that there are people who are prepared to accept what the numerical analyst would regard as a shockingly poor approximation to F (x, t), the distribution function of aggregate claims in the interval of time (o, t), provided it can be quickly produced on a desk or pocket computer with the use of standard statistical tables. The so-called NP (Normal Power) approximation has acquired an undeserved reputation for accuracy among the various possibilities and we propose to show why it should be abandoned in favour of a simple gamma function approximation.

Discounting encomiums on the NP method such as Bühlmann's (1974): “Everybody known to me who has worked with it has been surprised by its unexpectedly good accuracy”, we believe there are only three sources of original published material on the approximation, namely Kauppi et al (1969), Pesonen (1969) and Berger (1972). Only the last two authors calculated values of F(x, t) by the NP method and compared them with “true” four or five decimal values obtained by inverting the characteristic function of F(x, t) on an electronic computer.