Our discussions today will concern Subject A: Risk Theory, in particular the overall risk involved in operating an insurance concern. 7 papers have been handed in to the Colloquium on this subject.

To be quite candid, the referee would like to say that only one of these papers really treats the whole of the topic foreseen, and this is the paper by Mr. Colin M. Stewart (U.K.): The Assessment of Solvency. The other papers treat different special practical sides of the problem as well as general risk theoretical questions. One of the papers, that of Mr. Brichler, could have been attributed to Subject B as well as to Subject A, but it has been decided to take it up for treatment today.

Mr. Stewart delivers a lot of interesting points, which, I may say, take into account the solvency aspect as it looks to the supervisory authorities much more than has been done within the ASTIN discussions earlier. This, of course, is very related to the fact that regional as well as Common Market discussions have resulted in or will result in legislative rules for judging the solvency of insurance concerns, which rules will have great practical importance in the future. Mr. Stewart stresses the point that in case of non-life concerns it is the business on the books and its future contracted periods, short time business to be signed before judgment can be practically undertaken, claims reported, claims occurred but not yet reported, as compared with free reserves available that really decide the matter of sovency.

Effectively, a non-life insurance concern may be considered to be solvent if the supervisory authorities of the country or countries in which it operates allow it to continue operating. It is of no avail to claim that, by some other criterion, the concern may be considered to be solvent; it is by reference to the controls imposed by supervisory authorities that the concern must operate. For this reason, it is apposite to consider the principles of solvency assessment in the context of the financial statements and other documents generally available to supervisory authorities, as distinct from the theoretical mathematical concepts underlying insurance operation.

The primary purpose of supervision is to make sure that a concern does not enter into obligations which it will be unable to fulfil. It is thus essentially a dynamic standard, not a static one having regard only to the business already accepted, but the first requirement is nevertheless to test the concern's finances in relation to the business already on the books. If this test is passed, then at least it is known that the concern will not be depending upon profits from new business and renewals to help to meet the cost of its present liabilities.

This raises the question of what controls might be applied to new business. The two main types of control possible are on the extent of selection of risks permitted, and on the premiums to be charged, but while it may be feasible for such controls to be applied to the home portfolio of a particular class, for example motor vehicle insurance, it is difficult to envisage the operation of such controls internationally in, for example, the marine, aviation and transit class.

On sait que la loi de Poisson simple représente mal la distribution des sinistres d'un groupe observé d'automobilistes du fait que tout groupe que l'on peut étudier en pratique, même s'il est composé d'assurés présentant des caractéristiques communes (même zone de circulation, même type de véhicule, même utilisation de ce véhicule, …) est hétérogéne quant aux autres caractéristiques et surtout quant au comportement personnel des assurés, élément dont l'influence sur les résultats du risque est prépondérante.

M. Delaporte a obtenu une représentation intéressante du phénomène en supposant que les sinistres d'un véhicule se répartis-sent suivant une loi de Poisson de moyenne donnée, et que les moyennes de chaque véhicule du groupe étudié se distribuent selon une loi de Pearson type III d'equation:

où Γ(b) est la fonction eulérienne de 21ème espèce:

a, b, et S0 étant des paramètres dont la valeur est calculée en égalant les expressions des 3 premiers moments théoriques aux moments correspondants observés.

Cette formule conduit à des calculs assez longs.

M. Depoid a proposé une formule plus simple.

1. There is a general rule applicable to all insurance and reinsurance fields according to which the level of the so-called technical minimum premium should be fixed such that a certain stability criterion is satisfied for the portfolio under consideration. The two bestknown such criteria are

(i) the probability that there is a technical loss in any of the future years should be less than a given percentage

(ii) the probability that the company gets “ruined” i.e. initial reserves plus accumulated premiums minus accumulated claims becomes negative at any time of a given period in the future should be less than a tolerated percentage.

Confining ourselves to criterion (i) in the present paper we may then say that the problem of calculating technical minimum premiums is broadly spoken equivalent with the problem of estimating loss probabilities. Since an exact calculation of such probabilities is only possible for a few very simple and therefore mostly unrealistic risk models and since e.g. Esscher's method is not always very easy to apply in practice it might be worthwhile to describe in the following an alternative approach using results and techniques from Reliability Theory in order to establish bounds for unknown loss probabilities.

It would have been impossible for me to write this paper without having had the opportunity of numerous discussions with the Reliability experts R. Barlow and F. Proschan while I was at Stanford University. In particular I was told the elegant proof of theorem 3 given below by R. Barlow recently.

E. Sparre Andersen [1]) presented to the XVth International Congress of Actuaries, New York, 1957, a model of a collective risk process with a positive gross risk premium where the epochs of claims formed a renewal process. Let Ψ(u) (where u denotes the original risk reserve) denote the ruin probability in this model. Generalizing the classical result Sparre Andersen deduced the inequality

where R is a suitable positive number depending on the distribution function (continuous to the right), P(y), — ∞ < y < ∞, P(o) < 1, for the amounts of claims in case a claim occurs and also depending on the distribution function, K(t), t > o, K(o) = o, for the times between the epochs of successive claims, (The times between the epochs of successive claims, the inter-occurrence times, are assumed to be independent and identically distributed random variables. The time between the starting point and the epoch of the first claim is assumed to be independent of and to have the same distribution function as the inter-occurrence times. The amounts of claims are assumed to be independent of each other and of the epochs of claims and to be identically distributed.)

In this paper we are going to study some properties of a stochastic process, which has been proposed by Cramér (1968) as a model of the claims arising in an insurance company. This process has been studied by Cox in a different context. A few elementary results, concerning moments, are given by Cox and Lewis (1966). The present paper will be a survey of some results derived by the author (1970:1) and (1970:2). For detailed proofs we refer to these papers.

Let λ(t) be a real-valued stochastic process, such that P{λ(t) < o} = o. We further assume that Eλ(t) = 1 and that Eλ2(t) < ∞ for every fixed value of t. We denote the covariance

The process λ(t) will play the role of an intensity function. That means, that for every fixed realization of the process, the probability of

and that the number of events in disjoint intervals are independent.

We now define a point process N(t), where N(t) is the number of events which have occurred in (o, t]). With this definition we get

where

The integral is assumed to exist almost surely. This process will be called the N-process.

In many risk theoretical questions a central problem is the numerical evaluation of a convolution integral and much effort has been devoted over the years to mathematical and computational aspects. The paper presented to this colloquium by O. Thorin shows the subject to be topical but the present note stems from a recent paper by H. L. Seal, “Simulation of the ruin potential of non-life insurance companies”, published in the Transactions of the Society of Actuaries, Volume XXI page 563.

In this paper, amongst other topics, Seal has presented some simulations of ruin probabilities over a finite time interval. Some years ago (Journal Institute of Actuaries Students' Society, Volume 15, 1959). I pointed out that the form of ruin probability could be expressed as a successive product of values of a distribution function. To my knowledge no one has attempted to see if this product form was capable of development and the numerical values in Seal's paper prompted me to spend a little time on the problem. In the time I had available it has not been possible to do more than experiment, but the conclusions reached may be of interest to other workers in this field. They showed that calculation is feasible but laborious. However, the knowledge that it can be done may suggest methods of improving the techniques.

Seal's first simulation example is the calculation of ruin probabilities when the distribution of the interval of time between the claims is negative exponential and the individual claim distribution is also negative exponential. Instead of following Seal's method we can determine the “gain per interval” and find that if λ is the security loading the frequency function for the gain z is

i.e. a Laplace distribution.

In the wake of the technical development of our era we are increasingly faced with claims of extremely large amounts.

Whereas even in the past century large claims were mostly due to elemental or natural forces such as earthquakes, hurricanes and floods, there are today other causes, conditioned by human factors, that have considerably increased in importance. Another fact to be observed is that the risks that are passed on to the insurance market assume a more serious character year by year. Large building complexes, giant tankers, containers, atomic power stations, dams and jumbo jets pose for the insurers problems which are in no way easy to solve, since the risk covers required touch the limits of market capacity and their rating is subject to great uncertainties. At the same time, the readiness to underwrite such covers is a matter of utmost importance for the private insurance industry.

For the coverage of such large risks, various insurance forms have been developed in the past years within the range of non-proportional methods. Besides the excess of loss and stop loss covers in their usual form, the cumulative loss cover and the largest claim cover have attracted particular attention. The cumulative risk cover has, for example, been dealt with in [11]) and has also been frequently offered in practice. The coverage of the largest claim or, more generally, of the sum of the n largest claims does not appear to have gained a proper foothold in practice in spite of various valuable contributions in this field—considering particularly [2], [3], [5], [8] and [14]. The fact that it was during 1963/64 that various authors analysed these problems is no coincidence, for it was then that in two ASTIN Colloquia these topics were discussed.

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).

1. At an earlier ASTIN Colloquium participants were invited to present notes on problems which they considered as important but unsolved. There was little response to this invitation, presumably because a problem, once it is well formulated, is almost solved.

In this Note I do not present any new problems. In stead I try to outline a framework which may be useful for analysing different risk problems and seeing them in their proper perspective. In my view, a framework of this kind is urgently needed to place today's actuarial work on a sound foundation.

2. In general an insurance contract will define two stochastic processes. We lose little by assuming that the processes are discrete, and describing them in the following manner:

(i) The payment process: x0, x1 … xt …, where xt is the amount which the company pays to settle claims in period t, or at time t.

(ii) The premium process: p0, p1 … pt …, where pt is the premium which the company receives in period t, or at time t.

If the contract is concluded at time t = o, the Principle of Equivalence requires that

For the typical short-term contract with premium payable in advance (i) will reduce to

3. For a long-term insurance contract one usually requires that the inequality

shall hold for all τ. This means that the company must never be a net creditor of its customer.

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.

In order to make this report clear to those without experience in insurance matters, we first present some basic facts about the insurance business. In so doing we intentionally omit certain facts irrelevant to the present study. The most important omission of this kind is our assumption that the insurance business operates without administrative expenses and without sales costs. We also assume that the insurance business is run in such a way that no profit is made. It will be evident to readers already directly connected with insurance problems what further omissions we have made and why we have made them.

Insurance is the establishment of a contract between the insurance company and the insured person This contract is by tradition called the “insurance policy” and the insured person is called the “policyholder”. By agreeing to the insurance policy, the policyholder commits himself to paying certain premiums to the insurance company, and the insurance company commits itself to paying certain amounts to the policyholders. The conditions under which such payments are to be made are of many different kinds: the policy-holder dies, the policyholder becomes ill, a homeowner's house is burnt down, or the policyholder has a collision in his car. The circumstances under which a payment is to be made to the policy-holder are described in detail in the insurance policy. The amount to be paid is either fixed or variable: in life insurance the amount to be paid is always fixed and stated in the insurance policy, but in most other forms of insurance the intention is that the insurance compensate the policyholder for the losses he might incur as a consequence of the events covered by his policy.