A l'encontre de la théorie classique de l'assurance collective, qui s'intéresse plus particulièrement aux problèmes asymptotiques, les praticiens de l'assurance „non life” et les autorités de contrôle souhaitent disposer d'une méthode effective de calcul permettant de chiffrer la valeur réelle des probabilités de ruine dans un avenir plus ou moins rapproché.

Telle fut la remarque de notre collègue Bichsel au colloque d'ASTIN en 1967 à Arnhem. Dans le bulletin des Actuaires Suisses nous avons publié deux notes sur „une théorie opérationnelle du risque”. A la base se trouve l'emploi des processus markoviens discrets. Ces études prolongeaient un premier essai, exposé à l'Université de Trieste en 1963; une conférence donnée à Tel Aviv en 1967 développait le même thème. Dans un autre domaine, celui de la Recherche Opérationnelle, les ingénieurs utilisent depuis longtemps les mêmes processus de Markov pour étudier, par la dynamique stochastique, l'évolution d'un système aléatoire.

Natural hazards—floods, hurricanes, tornadoes, earthquakes, windstorms and hailstorms—cause considerable property damage in various parts of the world. In the United States, average annual damage resulting from these hazards is increasing rapidly. A large percentage of the damages occur as a result of infrequent, but severe, geophysical events (individual storms or earthquakes). If aggregate damage resulting from the event is exceptionally large, the event is called a natural disaster. The number of natural disasters in the United States is increasing each year. Resultant property losses are increasing even more rapidly. Increased density of properties susceptible to damage, increased value of these properties, and increased cost of repair have raised the probability of natural disaster occurrence in recent years even though the magnitude and character of the natural hazards have not changed.

Insurance is one means of protection against the natural hazards for fixed property. In this report, one-to-four family dwelling structures represent fixed property. To provide protection, two components of risk must be evaluated: (1) risk per individual structure and (2) risk of a large number of simultaneous losses—catastrophe potential. The latter component has attained added importance recently with the increased number and magnitude of natural disasters.

Information available for risk evaluation is (1) past damage experience; (2) data on the damage susceptibility of structures to be insured and the cost of repair; and (3) knowledge of physical characteristics of the natural hazards from the natural sciences. Damage experience results from the interaction of a natural hazard (frequency, location and severity of a geophysical event—storm, earthquake or flood) with characteristics and geographical distribution of exposed properties. Occasionally, interaction between the geophysical event and the distribution of properties leads to the creation of catastrophic losses resulting in a natural disaster. Evaluation of risk must be based either upon a retrospective or prospective measure of this damage experience.

Past damage experience, a retrospective measure of loss potential, is a poor measure of future risk because of (1) non-stationarity of property characteristics; (2) bias introduced by chance interactions of hazard and property array; and (3) by random occurrence (or non-occurrence) of a severe geophysical event during the short sampling period of years that is usually available for study. A pure extrapolation into the future of past loss experience, including the chance combinations of events that led to past natural disasters, does not provide a great amount of insight into the character of future risk. What is needed is not actual damage that occurred as a result of past geophysical events, but damage resulting to the present distribution of properties from a recurrence of these past events. For example, to estimate future earthquake risk in California, emphasis should not be on what the 1906 San Francisco earthquake cost, but what it would cost if a comparable earthquake occurred today and affected the present type, distribution, and value of properties.

A supplementary approach to the use of loss experience (called Natural Hazard Simulation) is presented which provides a prospective measure of risk. A mathematical approximation of the natural hazard mechanism is constructed which artificially produces geophysical events that mathematically interact with a given geographical array of properties. Natural Hazard Simulation utilizes and ties together available pertinent information. Use of an electronic computer permits calculation of a large number of, say, 25-year sequences of synthetic loss experience which can be used to estimate the two measures of natural hazard risk. “Natural disasters” occur at irregular intervals in the simulation analysis when a severe geophysical event occurs near a center of population.

Using this approach, effects of a recurrence of past geophysical events or simulated future events upon present or hypothetical future distributions and types of properties can be estimated. It produces, in effect, a weighted measure of the many possible interactions between natural hazard and property array which, because of his short life span, man cannot afford to wait for nature to produce. Characteristics of an insurance operation needed to cover the hazard (rating, underwriting, claim settlement, loss reserving, and reinsurance) also are simulated.

Examples of the application of Natural Hazard Simulation to flood, earthquake, hurricane wind and tides, winter windstorm, and thunderstormspawned hazards (such as tornadoes, wind and hail hazard) are presented. Purpose of the application to flood hazard was to (1) estimate magnitude of the hazard to more than fifty-million dwelling structures in the United States for which very little damage experience was available and (2) determine characteristics of a joint Insurance Industry/Federal Government flood insurance program need to cover the hazard. Characteristics of a joint program were needed to establish relationships and financial arrangements between the Federal Government and the Insurance Industry. This work was done as consultants to the U.S. Department of Housing and Urban Development during development of a National Flood Insurance Program which is now operational. In this plan, the Federal Government assumes a portion of risk by acting as a reinsurer against excessive losses on industry's share of the Program.

An application of Natural Hazard Simulation to the earthquake hazard on the West Coast of the United States has been made. Examples of mathematically produced earthshock patterns are given. Correspondence between calculated and observed patterns is good. Measures of both components of risk are discussed for the present array of 625,000 dwellings in the San Francisco Metropolitan area when a recurrence of all earthquakes in the historical past (170 years) is used as a measure of earthquake hazard. A similar type of analysis also has been made for the Los Angeles Metropolitan area.

An application of the approach to the hurricane wind hazard is illustrated using computer printouts of the geographical pattern of highest wind expected during a hurricane's passage as obtained from the computerized mathematical model. Calculated patterns of wind speed severity provide realistic approximations of observed patterns. An example of the interaction between natural hazard and property array in producing a “natural disaster” is illustrated by calculating “loss experience” to dwelling properties in Louisiana from an intense hurricane whose path is successively changed relative to centers of population. The effect of changing the intensity of a hurricane upon resulting damage when the path is held constant is also shown. Both measures of hurricane wind risk—expected loss per exposure and catastrophe potential—are being estimated by developing “loss experience” to the present array of dwelling properties in the Gulf and Atlantic States based upon two measures of the magnitude of the hurricane wind hazard; namely, (1) a recurrence of hurricanes of various intensities and paths which have been recorded in the historical past and (2) a number of series of 25-year sequences of “synthetic loss experience” based upon computer simulation techniques.

An application of Natural Hazard Simulation to winter windstorm and thunderstorm-spawned tornadoes, wind, and hail in the Middle Western United States has been carried out for the first measure of risk. The mathematical model for obtaining a measure of catastrophe potential for these hazards is currently being developed. Future applications will include development of an integrated procedure for simulating “loss experience” from all of the natural hazards to a given array of structures in various geographical areas.

Natural hazard simulation offers a supplementary approach to the sole use of past loss experience for (1) estimating the two components of natural hazard risk and (2) developing characteristics of an insurance program needed to cover the natural hazards at a time when average annual property damages caused by natural hazards are increasing rapidly because of the increased number and magnitude of natural disasters.

For a number of reasons it is important for an insurance company to estimate the claims costs of a year within the different branches of non-life insurance as soon as possible after the end of the year. The claims cost of a year is hereby defined as the total cost, before taking reinsurance into account, of all claims generated by events that have occurred during the year. When the estimation has to be done, part of these claims will be reported and closed, others will be reported and still open, and the remaining ones will be incurred but not yet reported. The total cost of the claims is defined as the sum of all payments that have been made or will be made on account of the claims. Thus, in this definition no regard is paid to interest, i.e. no discount factors are applied to payments to be made in the future.

Instead of considering a year, we could consider an arbitrary period of twelve consecutive months. The estimation problem is the same, and estimates of the claims costs of consecutive twelve months periods will allow a closer following up of trends and yield predictions for the present year.

For estimates to be available quickly, it is necessary that the estimation procedure be founded on data that are available immediately at the end of the year or the latest twelve months period. This means a.o. that for the bulk of the open claims, individual estimates of reserves by claims adjusters are out of the question. In other words, the estimation procedure has to be basically of a statistical character. In addition, for continuous estimates to be produced it has to be well adapted to electronic data processing. Indata to the procedure have to be stored in the memories of the computer.

The rapid economic growth in the last decade and the fierce competition have forced industry to raise its output, to develop new manufacturing methods and, where possible, to lower the fixed costs per unit of output. Consequently bigger factories and warehouses have been and are being built. Furthermore increasing labour costs have speeded up rationalisation and the introduction of efficient machinery.

Often, in the course of this development, too little attention has been paid to safety. This is reflected in the increasing number of Fire and Consequential Loss claims which have become so costly that the premium income has proved inadequate. Insurers have therefore adjusted their tariffs and increased their rates.

Of course, this is no solution to the problem. Insurers have to insist on adequate fire prevention and fire protection measures. Progress in the right direction can certainly be expedited by realistic tariffs which take account of all the positive features (sprinklers, inspection reports, etc.) and the negative features of a given risk (in commerce or industry).

Rating experts say that the tariffs of industrialized countries do not always point to the actual risk and claims fluctuations; one very important factor here, the size of the risk or of the building (i.e. its insured value, volume, surface area), brings me to the main theme of my today's talk.

In this paper the authors remind of the known formulas for the double Laplace-Stieltjes transforms of the ruin probabilities ψ(u, t), where u is the initial risk reserve and t stands for the operational time, in the case of independent interoccurence times and claim amounts such that the interoccurrence times are identically distributed Κ(t), t ≥ o, Κ(o) = o, and the claim amounts are identically distributed P(y), — ∞ < y < ∞. For some cases, where I — P(y) and I — Κ(t) are exponential polynomials, numerical inversions of the said Laplace-Stieltjes transforms are made for a selection of u- and t-values in combination with safety loadings of various sizes and signs. Moreover, some values are given when I — P(y) or I — Κ(t) are of Pareto type and comparisons are made with the results when the Pareto distributions are approximated by suitable exponential polynomials.

Lorsque le secrétaire du Comité d'organisation du IXe colloque „ASTIN” — Randers 1970 — le Dr. Paul Johansen, a pris l'initiative d'inviter, dès septembre 1969, les participants roumains aux travaux du VIIIe colloque „ASTIN” de Sopot, de présenter à Randers „un rapport sur le système national d'assurances”, la manière même dont il en a suggéré le thème prouve qu'il avait nettement conscience qu'une simple présentation des assurances pour les personnes, les biens et la responsabilité civile d'un pays serait insuffisamment éloquente si toutes ces assurances n'étaient englobées dans un cadre plus vaste, autrement dit si elles n'étaient pas rendues vivantes dans l'ensemble général des formes que l'idée d'assurance adopte au sein de la vie de la société respective. On sait que, déjà au XIXe siècle, du tronc plus ancien des assurances individuelles, se sont détachées les assurances sociales qui, en se développant vertigineusement, ont obtenu de longue date un statut et un profil propres, tout en acquérant entre temps la prépondérance en matière d'assurances pour les personnes et en conditionnant de manière décisive ce secteur des assurances.

Ce n'est qu'en présentant les assurances dans ce vaste cadre que l'on peut avoir une image adéquate de la manière dont les assurances sont utilisées afin de protéger les habitants d'un pays.

Par ailleurs, la présentation du champ tout entier d'action de l'idée d'assurance est également exigée par un intérêt scientifique. Car ce n'est que dans une semblable vaste perspective que l'on peut élucider de manière satisfaisante des processus déjà consommés, des processus en cours, ainsi que la tendance au renouvellement qui se manifeste au sein de l'évolution des assurances.

Etant donné, d'ailleurs, les conditions existant en Roumanie, même si nous voulions procéder à une présentation exclusive des assurances individuelles, cette tentative rencontrerait de graves difficultés et ne pourrait aboutir sans de substantielles amputations.

This paper contains little which can be considered as new. It gives a survey of results which have been presented over the last 10-15 years. At one time these results seemed very promising, but in retrospect it is doubtful if they have fulfilled the expectations they raised. In this situation it may be useful to retrace one's steps and see if problems can be reformulated or if new approaches can be found.

Mathematical models have been used in insurance for a long time. One of the first was the Gompertz mortality law; a more recent model, which has been intensively studied is the Compound Poisson Distribution in Lundberg's risk theory.

When a model is introduced, one usually proceeds by stages. The first step is to see if the model appears acceptable on a priori reasons. If it does, the second step is to examine the implications of the model, to see if any of these are in obvious contradiction with observations. If the result of this examination is satisfactory, the third step is usually a statistical analysis to find out how well the model approximates the situation in real life, which one wants to analyse. If the model passes this second examination, the next and final step may be to estimate the parameters of the model, and use it in practice, i.e. to make decisions in the real world.

The advantage of working with a model is that it gives an overall purpose to the collection and analysis of data. A good model should tell us which data we need, and why.

Models for the risk business of an insurance company are often constructed by weighting pure Poisson models. In this paper it is verified that it is possible to calculate the probability of ruin in such weighted models by weighting ruin probabilities of pure Poisson models.

Stimulated by the paper of W. S. Jewell) on multidimensional credibility I should like to show that at level 2 assumptions (Jewell's terminology) one can obtain explicit formulae for forecasting total losses in the future based upon total losses and number of claims observed in the past.

Information relating to the expected number of losses is of importance in automobile insurance systems. The distribution of risks by number of losses per year may be based on the following model

with λ representing the average number of losses per year. This distribution is the Poisson distribution. Tests of this model versus actual observations often indicate significant deviation. This discrepency can result from the constancy of λ which makes the model appropriate for an individual but would require an isohazardous population when applied to a group of individuals. In reality, however, λ will vary from individual to individual. A model accounting for this spread in λ is given in

where z(λ) is a distribution describing the spread of λ. The results of model (2) certainly will depend on the form of z(λ). It has been hypothesized that z(λ) can be represented by (3)

which is a Pearson Type III [1, 5, 7]. With this assumption model (2) becomes the negative binominal distribution with a mean of and a variance of

If the observed mean is and the observed variance it is possible to determine a and b by solving the above equations for mean and variance. Thus and

The results indicate an improved fit to actual observations [1, 5, 6].

Ohlin [7] showed optimality properties of Stop Loss reinsurance when the ceding insurer uses a continuous loss function to evaluate his risks. We generalize this property to the range of the distribution; we then show that Stop Loss reinsurance is no longer the best form when the company uses a percentile parameter. Finally we prove an optimality theorem concerning chance games which allows us to determine the retention as a function of the size of the portfolio.

L'assurance est un jeu. Le rôle principal de l'actuaire est d'aider l'assureur à „trouver son adversaire”. En assurance-vie, le risque est défini par la table de mortalité et l'assuré est trouvé quant il déclare son âge et son état de santé. En assurance non life, la question n'est pas aussi simple. Mais la pratique commerciale a introduit la notion de classe du tarif. Les risques qui appartiennent à une classe de tarif n'appartiennent pas à une classe homogène, mais les conséquences d'un sinistre sont comparables au sens du calcul des probalités. La valeur moyenne d'un sinistre résultant de la classe de tarif est statistiquement stable. Ce qui fondamentalement a garanti le fonctionnement de l'assurance non life est l'application de la loi des grands nombres. La régularité statistique sera toujours l'élément de classement et de décision. Nous considérons donc la classe de tarif comme étant l'élément primaire du risk-business, même si la valeur moyenne d'un sinistre varie dans le temps.

Etablir un modèle est toujours une approximation plus au moins bien réussie de la réalité. Le problème de la représentation d'un risque couvert par un contrat d'assurance ou un ensemble de contrats (réassurance) est un problème théorique. Autre chose est le problème de l'efficacité c'est-à-dire la possibilité d'effectuer des calculs pratiques. Le modèle gaussien de la loi des erreurs de mesure n'est pas théoriquement exact, mais il permet les calculs de compensation. Le modèle de Dodson en assurance-vie est une première approximation de la réalité mais il permet à l'actuaire d'effectuer ses tarifications.

Consider the set of functions

Obviously, π1(x) represents the net premium of the excess cover over the priority x, and the variance thereof.

If a distribution function F(x) = 1 − π0(x) is given, the set (1) can be generated by means of the recursion formulae

Let us study the special class of d.fs. F(x) which satisfy

where

If these conditions are met, the integrals (1) have the solution:

Aj(y) and Bj(y), respectively, are polynomials of rank jk and jk − 1. Their coefficients are determined by the equations:

The system (5) is obtained by differentiation of (4) with respect to y, and observing (2).

We will in this paper consider the risk process from the point of view of random walk in one dimension. The particle starts out at the origin. Each claim is equivalent to a step in the random walk. The length of the step is equal to the amount of the claim minus the amount of the premium which has been obtained since the preceding claim. If the difference is positive the particle advances to the right and if the difference is negative to the left. At distance U to the right from the origin there is a barrier. The problem is to find the distribution function of X, the time it takes the particle to cross the barrier for the first time.

In most practical applications of risk theory U is large in comparison to the individual steps of the particle. We will in this paper assume that U is large in comparison to the individual steps and draw certain conclusions about the risk processes from this assumption.

The individual steps of the particle have a certain distribution. The corresponding characteristic function is ϕ. For reasons which will be seen later we will consider ϕ to be a function of it = θ instead of t. This means that

The mean value and the standard deviation of each step is equal to m and σ respectively. We now write

We now define two random variables X and Y.

X = time to cross the barrier for the first time

Y = X σ2/U2.

A premium calculation principle is a general rule that assigns a premium P to any given risk S. Intuitively, P is what the insurance carrier charges (apart from an expense allowance) for taking over the risk S (see [3], p. 85-87). Mathematically, S is a random variable, and P depends on S through its distribution function. The value of P may be finite or infinite; in the latter case we speak of an uninsurable risk.

A premium calculation principle is called additive, if the premium assigned to the sum of two independent risks is the sum of the premiums that are assigned to the two risks individually. For example, the variance principle, P = E[S] + β Var [S] (β > o), is additive, because the variance of the sum of independent random variables equals the sum of the variances. Additivity is a very desirable property, from a theoretical as well as from a practical point of view (as pointed out by Borch [2], p. 429).

The variance principle is not entirely satisfactory for various reasons. For one thing it does not take account of the skewness of S (a risk whose distribution is skewed to the right seems to be more dangerous than one with a symmetrical distribution). Furthermore, it produces in some cases a premium P that exceeds S with probability one (example: β = .3, S = o or 10, each with probability 1/2).

In Bohman the following model is considered. Our notation follows Bohman.

Let Z1, Z2, … be a sequence of independent random variables with distribution function F and put

Put

and define X by

X = inf{n; Sn > U, Sk ≤ U for k = 1, …, n − 1}.

Bohman shows that if U → ∞ in such a way that U/σ → ∞ and then

where G(α, x) is the distribution function for the time when a Wiener process X(t) with EX(t) = αt and Var X(t) = t first crosses the level 1.

Let N be an integer, which in a certain sense corresponds to “time”, and consider P(X ≤ N). This is thus the probability of ruin before the N:th claim.