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].

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.

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 original paper with this title was published in 1968, following presentation at a Casualty Actuarial Society meeting. This is another up-date of indexes in the economic model—first report of 1976 and a forecast for 1977. These LPI indexes measure economic factors affecting loss and loss adjustment settlement costs which are incurred after claims have been reported. The indexes measure direct claims costs and not the reinsured excess losses which would be more adversely affected by inflation.

This is a Claims Market Place concept of the loss settlement function of the insurance company. Claims Costs (our economic cost of production) result from purchasing of services under unusual circumstances in controversial, severe, hasty and often emergency situations. Claims settlements can take place in court rooms, lawyers' offices, repair garages and hospitals. The legalistic atmosphere is often one of friction and excessive demand rather than of a normal commercial esprit de corps. The procurement of these claims services requires dealing with high cost furnishers of services: doctors, clinics, hospitals, lawyers, repair garages or body shops, building trades, house furnishings.

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

Several “short cut” methods exist to approximate the total amount of claims ( = χ) of an insurance collective. The classical one is the normal approximation

where and σx are the mean value and standard deviation of x. Φ is the normal distribution function.

It is well-known that the normal approximation gives acceptable accuracy only when the volume of risk business is fairly large and the distribution of the amounts of the individual claims is not “too dangerous”, i.e. not too heterogeneous (cf. fig. 2).

One way to improve the normal approximation is the so called NP-method, which provides for the standardized variable a correction Δz

where

is the skewness of the distribution F(χ). Another variant (NP3) of the NP-method also makes use of the moment μ4, but, in the following, we limit our discussion mainly to the variant (2) (= NP2).

If Δz is small, a simpler formula

is available (cf. fig. 2).

Another approximation was introduced by Bohman and Esscher (1963). It is based on the incomplete gamma function

where

Experiments have been made with both formulae (2) and (4); they have been applied to various F functions, from which the exact (or at least controlled) values are otherwise known. It has been proved that the accuracy is satisfactory provided that the distribution F is not very “dangerous”.

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.

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.

Presented by the Central American Actuarial Association (Asociacion Acturial Centroamericana) to the 13th ASTIN Colloquium.

The Asociaciôn Actuarial Centroamericana (AAC) is a grouping of Actuaries from the Central American Republics of El Salvador, Guatemala, Honduras, Nicaragua, Costa Rica and Panama. The AAC is a contributing ASTIN member of the International Actuarial Association.

Earthquakes have been a constant scourge of mankind. Central America has not escaped this phenomenon, indeed the territory has been most affected by them precisely because of its condition as an isthmus that serves as a fragile union between the continental land masses of North and South America, and in consequence being subject to disturbances by the displacement of the continental plates. Our lands abound with beautiful volcanoes, which have also contributed to local seismic activity. Whatever their origin, the earthquakes that have struck our country have left their share of destruction of lives and property.

In Annex 1 a table is presented, showing a history of seismic activity and volcanic eruptions in the countries on the Caribbean Platform, in which we can appreciate in detail the catastrophes that occurred from the XVI Century until 1976.

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.

Earthquakes and Windstorm can range in size from small easily insurable instances up to full scale Natural Disasters. This note is particularly concerned with the problem of forecasting, rating and insuring earthquakes and windstorm at the natural disaster end of this scale.

Before we tackle the peculiar problems of natural disaster insurance it is worth having a look at why insurance works so well in practice.

Is it because actuaries and statisticians have precisely measured the statistical risks involved, and have developed a detailed and complex mathematical approach to insurance, a true “Technical Basis” for Insurance? Or is it really quite simple?

A quotation (the first of many) from R. Heller [1] is relevant.

“The great and fabled business empires with hardly an exception were built… on irresistable ideas of elemental simplicity.”

Insurance is one of these great empires, and some of the simple ideas which may explain why insurance works so well are suggested below.

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.

Japan is not only a well-known earthquake prone country, but also a land very liable to damage by windstorm, flood, tidal wave and highwater. Especially, it is visited every year by typhoons or tropical cyclones. These occur during the period from early summer to autumn with accompanying heavy rains, and severe damage is often suffered. Most of the serious disasters result from the following causes.

a) Flood from a rapid small river in mountainous areas or landslide occurring there.

b) Innundation of a large plain area caused by the collapse of embankments following heavy rain.

c) Highwater or tidal wave in a harbour or a bay, which is often caused by the visit of a typhoon particularly at full tide.

Generally speaking, damage by the wind is much less in severity than that by flood, tidal wave or highwater.

These natural calamities occur every year, but the extent of the damage differs greatly from year to year. Taking an example of the damage to buildings and contents during 10 years from 1965 to 1974, the largest annual damage estimated for the whole country was ¥ 161.875 million (approx. U.S. $ 578 million), the smallest ¥ 9.398 million (approx. U.S. $ 33,6 million) and the average ¥ 58.561 million (approx. U.S. $ 209 million). (These are the original figures not adjusted for inflation. Also, they do not include the damage to outdoor objects such as roads, bridges, railways, communication facilities, electricity facilities and farms). The greatest windstorm damage in recent years was brought about by the “Bay of Ise Typhoon” in September 1959, at which 153.893 buildings were lost or seriously damaged, 5.101 persons were killed or went missing and 2.430 ships were lost.