bluuebunny/crossref_metadata_2025_split huggingface dataset:
Random processes and their application to demography and insurance
Authors: Gnedenko B. W. 
Published: July 2016

Mr. Chairman, Honourable Colleagues!Let me thank the organisers of this symposium for the honour they have done me by inviting me to present my report before this eminent audience. I would also like to express my thanks to all who are present here for the confidence shown to me as rapporteur.The genesis and development of the theory of random processes is closely related to the problems of demography and insurance. Numerous problems stimulating the development of the theory of probability and the theory of random processes—the latter constituting one of the essential parts of the modern theory of probability—are connected with these two fields of research and activity.The theory of random processes, being stimulated—similarly to other fields of mathematics—by concrete practical tasks, has created the general ideas and methods of research, achieving many valuable results. With the passage of time, it appeared that the theory of random processes can be usefully applied, by its conceptions, results and methods, to a number of scientific fields and practical activity. Its conceptions have found wide and important application to the earlier crystallized fields of human knowledge, as well as to the newly appearing ones.These new fields of mathematics serve us as an example which once more corroborate the thesis, well known to us, according to which the course of the development of human knowledge starts from empirical observations, then passes up to abstract thinking and finally to practice.
The ruin probability in a special case
Authors: Bohman H. 
Published: July 2016

It is fantastic how the computer has changed our attitude to numerical problems. In the old days when our numerical tools were paper, pencil, desk calculator and logarithm tables we had to stay away from formulas and methods which led to too lengthy calculations. A consequence is that we have a tendency to think of numerical analysis in terms of the classical tools. If we go back to the results of earlier writers it seems, however, very likely that many results and formulas developed by them which had earlier a theoretical interest only could nowadays be applied successfully in numerical analysis.As an example I take the ruin probability ψ(x). The Laplace transform of ψ(x) is given by the following expressionwhere c > 1. In fact (c — 1) is equal to the “security loading”. The function p(y) is equal to the Laplace transform of the claim distribution. We assume that the mean claim amount is equal to one, i.e. p′(0) = — 1.In his book from 1955 [1] Cramer points out that this formula will be more easy to handle if the claim distribution is an exponential polynomial. In this case we havewhereCramér's results are given on pages 81-83 in his book. We reproduce them here with a slight change of notations only.
Hedging by Sequential Regression: an Introduction to the Mathematics of Option Trading
Authors: Föllmer H. , Schweizer M. 
Published: July 2016

It is widely acknowledge that there has been a major breakthrough in the mathematical theory of option trading. This breakthrough, which is usually summarized by the Black–Scholes formula, has generated a lot of excitement and a certain mystique. On the mathematical side, it involves advanced probabilistic techniques from martingale theory and stochastic calculus which are accessible only to a small group of experts with a high degree of mathematical sophistication; hence the mystique. In its practical implications it offers exciting prospects. Its promise is that, by a suitable choice of a trading strategy, the risk involved in handling an option can be eliminated completely.Since October 1987, the mood has become more sober. But there are also mathematical reasons which suggest that expectations should be lowered. This will be the main point of the present expository account. We argue that, typically, the risk involved in handling an option has an irreducible intrinsic part. This intrinsic risk may be much smaller than the a priori risk, but in general one should not expect it to vanish completely. In this more sober perspective, the mathematical technique behind the Black–Scholes formula does not lose any of its importance. In fact, it should be seen as a sequential regression scheme whose purpose is to reduce the a priori risk to its intrinsic core.We begin with a short introduction to the Black–Scholes formula in terms of currency options. Then we develop a general regression scheme in discrete time, first in an elementary two-period model, and then in a multiperiod model which involves martingale considerations and sets the stage for extensions to continuous time. Our method is based on the interpretation and extension of the Black–Scholes formula in terms of martingale theory. This was initiated by Kreps and Harrison; see, e.g. the excellent survey of Harrison and Pliska (1981,1983). The idea of embedding the Black-Scholes approach into a sequential regression scheme goes back to joint work of the first author with D. Sondermann. In continuous time and under martingale assumptions, this was worked out in Schweizer (1984) and Föllmer and Sondermann (1986). Schweizer (1988) deals with these problems in a general semimartingale model.
The Random Walk of a Simple Risk Business
Authors: Seal H. L. 
Published: July 2016

We suppose that a risk business issues a single type of contract under which, in return for a unit “premium”, it will pay a “sum insured”m(mbeing an integer) on the occurrence of a contingency of probabilityq˂ ½. The expected gain on each contract is i—mqand is assumed to be positive. This type of enterprise is conveniently denominated asimple risk business. Two “real life” situations it simulates are those of a group life policy for a uniform amount covering a number of young lives (e.g., university students), and a roulette casino where the stakes are uniform and the bets are limited to the single numbers o to 36.The risk business is supposed to commence its operations with a “risk reserve” ofKunits. Each premium is added to this reserve as it is received and all claims by contract holders are paid there-from. We say that the business is “ruined” as soon as the risk reserve becomeszeroornegative(though it could be argued that it would be unethical to accept a premium once the risk reserve is less thanm— 1). On the other hand, if the reserve reaches an amountMunits no further premiums are paid into it until a claim occurs to reduce it belowM(de Finetti, 1957). The risk business intends to continue its operations for a long, but finite, period unless ruined in the meantime.We consider two probabilities: (i) υx, the chance of eventual ruin given that the risk reserve is nowx, and (ii) υx, n, the probability that ruin occurs as a result of thenth contract (simultaneous contracts being ranked in a prearranged, e.g., alphabetical, order). Clearlyand we will write I — νx=μx
Experience Rating in Subsets of Risks
Authors: Bichsel Fritz 
Published: July 2016

We consider a set of risks B, which is divided into subsets Bi according to some property as profession, region etc. The subsets Bi are thought to be too small to permit a calculation of a premium from their own experience. On the other hand we assume that there are differences between the Bi. This paper tries to give a solution to the problem how the experience of the single Bi should be combined with that of the whole B.The problem is treated only with respect to frequencies of claims. The amounts of the claims are not taken into consideration. I suppose that an extension of the theory to amounts is possible and highly desirable for practical applications. Lack of time did not permit me however to proceed to such an extension in this paper.I think that the methods developed in the sequel might be applied to the following practical problems:— rate making— experience rating in the restrictive sense of the word, i.e. adjusting periodically the premium or granting premium refunds according to the experience of every single Bi— judging the experience of a. Bi.I do not give a list of references which would have to be very long. The basic idea was, as far as I know, first developed by Lundberg.
On the solvency of insurance companies
Authors: Pentikäinen T. 
Published: July 2016

This report is a contribution to the discussion on the solvency problem, which has been taking place at ASTIN-meetings. In his report in Edinburgh 1964 Beard referred to many aspects which are closely connected with the problem. Such aspects are1. the evaluation of liabilities;2. the evaluation of assets;3. the level of the premiums of long term policies and4. reinsurance.If all of these are not in order, there is no sense in speaking about solvency. E.g. a solvency margin defined as the difference between assets and the expected value of liabilities would not be a reliable measure of the financial state of an insurance company, if either of these—or maybe both—are not evaluated in a reliable way. The fixing of solvency margins is not an isolated problem, on the contrary it is only part of the security measures which must all be managed at the same time. The ultimate purpose of the security system prescribed by legislation must be to safeguard policyholders and claimants against losses.
Credible Means are exact Bayesian for Exponential Families
Authors: Jewell William S. 
Published: July 2016

AbstractThe credibility formula used in casualty insurance experience rating is known to be exact for certain prior-likelihood distributions, and is the minimum least-squares unbiased estimator for all others. We show that credibility is, in fact, exact for all simple exponential families where the mean is the sufficient statistic, and is also exact in an extended sense for all regular distributions with their natural conjugate priors where there is a fixed-dimensional sufficient statistic.
The Mincing Machine Revisited
Authors: Bühlmann Hans 
Published: July 2016

For the 4th ASTIN colloquium in Trieste (1963) Robert E. Beard [1] had initiated the topic of extreme value theory and its application to actuarial problems. The impetus came from Gumbel's book [2], published in 1958, and the idea of applying extreme value theory ran about as follows:Problem: Given: The size of the largest claim observed = XNThe number of observed claims = N.Find: The excess of loss premium for a portfolio of independent risks identical with those under observation.The idea was to use Gnedenko's limit distribution for the wide class of distribution functions with an unlimited tail and finite moments, the “exponential type” in Gumbel's terminology, and to calculate the excess of loss premium on the basis of this limit distribution.The discussion in Trieste was very lively and culminated in Jan Jung's citation of Jan Jung [3]: “There is a natural law which states that you can never get more out of a mincing machine, than what you have put into it. That is, if the reinsurance people want actuarially sound premiums, they must get a decent information about the claim distribution”. In more mathematical language: The brilliant idea of Bobbie Beard is unfortunately leading to a non-robust procedure. A deviation in the true (but unknown) underlying distribution may lead to completely different results.In any case the mincing machine argument seemed so powerful that we have had little reconsideration of the above problem at subsequent ASTIN colloquia. It may therefore be worthwhile to have another look at the problem now, i.e. more than ten years after the Trieste colloquium.
Caisse nationale suisse d'assurance en cas d'accidents: résultats de la Statistique des Accidents de la Huitième Période Quinquennale 1953–1957
Authors: Koch Hans 
Published: July 2016

La Caisse nationale suisse d'assurance en cas d'accidents à Lucerne, chargée du service de l'assurance obligatoire en cas d'accidents en Suisse, a publié il y a quelque temps les „Résultats de la statistique des accidents de la huitième période quinquennale 1953-1957”. Cette publication, qui est la huitième, paraît tous les cinq ans en français et en allemand). Ces différents rapports donnent un aperçu intéressant de la situation de l'assurance-accidents obligatoire en Suisse. Ils constituent dans leur ensemble une documentation précieuse touchant l'un des aspects du développement de la sécurité sociale en Suisse depuis 1918. C'est en effet à cette date que l'assurance obligatoire en cas d'accidents est entrée en vigueur et que la Caisse nationale suisse d'assurance en cas d'accidents, établissement de droit public autonome, a commencé son activité.Le dernier rapport sera aussi consulté avec fruit par quiconque s'intéresse aux problèmes de l'assurance sociale. Cette publication d'environ 180 pages, illustrée par de nombreux tableaux, graphiques et illustrations, fournit une quantité d'indications très utiles présentées de façon claire. Les premiers chapitres sont surtout consacrés aux statistiques tirées du règlement des sinistres, tandis que les derniers chapitres traitent de quelques problèmes d'assurance spéciaux qui sont particulièrement actuels.
Bemerkungen zur Berechnung der Schwankungsrückstellung in der Brandversicherung
Authors: Hofmeister Emmi 
Published: July 2016

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.
Application of Game Theory to Some Problems in Automobile Insurance
Authors: Borch Karl 
Published: July 2016

In this paper we shall study the problem of determining “correct” premium rates for sub-groups of an insurance collective. This problem obviously occurs in all branches of insurance. However, it seems at present to be a really burning issue in automobile insurance. We shall show that the problem can be formulated as a conflict between groups which can gain by co-operating, although their interests are opposed. When formulated in this way, the problem evidently can be analysed and solved by the help ot the “Game Theory” of Von Neumann and Morgenstern (5).We shall first illustrate the problem by a simple example. We consider a group of n1 = 100 persons, each of whom may suffer a loss of 1, with probability p1 = 0.1. We assume that these persons consider forming an insurance company to cover themselves against this risk. We further assume that for some reason, government regulations or prejudices of managers, an insurance company must be organized so that the probability of ruin is less than 0.001If such a company is formed, expected claim payment will beand the standard deviation of the claim payments will beIf the government inspection (or the company's actuary) agrees that the ruin probability can be calculated with sufficient approximation by assuming that the claim payments have a normal distribution, the company must have funds amounting to
Note Concerning the Distribution Function of the Total Loss Excluding the Largest Individual Claims
Authors: Ammeter Hans 
Published: July 2016

The theory of extreme values is a special branch of mathematical statistics and was mainly treated by E. J. Gumbel [4]). This theory has only been applied in a few cases to problems in the insurance business. The first practical application to insurance known to the author of the present paper is due to A. Thépaut who has invented a new reinsurance system called ECOMOR [5]. According to this system the reinsurer covers the excess risk for the m largest claims and the ceding company retains an amount equal to the (m + I) largest claim. The credit for having pointed out the importance of the theory of extreme values belongs to R. E. Beard [1]. Recently E. Franckx [3] has found a most remarkable result by disclosing the general form of the distribution for the largest claim occurring in a certain accounting period.The present paper starts from the consideration that not only is the distribution of major claims, which might be eliminated by means of reinsurance, of interest to an insurer but also the distribution of the remaining total loss after excluding the largest claims. The nature of this distribution is important not only in connection with stability and security, but also for statistical investigations of the observed claim ratio. The credibility of such an investigation might be greatly improved if a suitable number of major claims were excluded. To simplify matters, the present paper considers the case where only the largest claim is excluded.
Theorie des valeurs extremes et la tarification de l'excess of loss
Authors: D'Hooge L. 
Published: July 2016

Soit x1 ≤ x2, ≤ … ≤ xn—m + 1 ≤… ≤xn—p+1≤…≤ xn un échantillon ordonné de taille n d'une distribution dont F(x) et T(x) sont respectivement la fonction de répartition et la fonction de densité.Nous appelons xn—m+1 la valeur de rang m de l'échantillon (x1, x2,…xn).Pour simplifier l'écriture nous remplacerons partout l'indice n — m + I par m.Avec cette convention nous notons la valeur caractéristique de rang m et l'intensité de rang m respectivement par um et αm [I].Par définition on a:
Notons enfin par Fm (x) et Tm (x) la fonction de répartition et la fonction de densité de xm pour n → ∞ et m fixe.2.1. Avant de résumer les différents points traités dans cette note, rappelons les particularités qui caractérisent la distribution asymptotique de la plus grande valeur x1 d'un échantillon. On sait que cette distribution existe si et seulement si la fonction de répartition F(x) appartient à un des trois types suivants [2]:a) Type I: F(x) est du type exponentiel.Le domaine de x est illimité vers la droite et F(x) tend vers I pour x → + ∞ au moins aussi rapidement qu'une fonction exponentielle.Tous les moments de F(x) existent.b) Type II: F(x) est du type de Cauchy.Le domaine de x est illiminté vers la droite.
An Outline of a Generalization — started by E. Sparre Andersen — of the classical Ruin Theory
Authors: Thorin Olof 
Published: July 2016

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 inequalitywhere 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.)
Quelques observations statistiques sur la variable „nombre de sinistres” en assurance automobile
Authors: Thyrion P. 
Published: July 2016

RésuméCette note ne risque pas de prêter le flanc à une critique souvent adressée à ASTIN, l'excès de spéculations mathématiques. Elle s'expose plutôt au reproche contraire, car elle a été, autant que possible, délibérément dépouillée de considérations théoriques, pour s'en tenir à quelques constatations arithmétiques fort simples sur la variable „Nombre de sinistres”.Ces constatations sont orientées vers la pratique, particulièrement vers la mise en œuvre d'un système de bonus-malus.
Ruin Probability During A Finite Time Interval
Authors: Beard R. E. 
Published: July 2016

This paper was inspired by comments by H. L. Seal in a series of lectures given to the Actuaries Club in New York and by a paper of his recently published in the Swiss Actuarial Journal (Seal, 1972 [6]). In his lectures he showed that the probability U(w, t) that a risk reserve at every epoch τ, where o < τ ≤ t will be non negative when the initial risk reserve is w is related to , the probability that the aggregate claim outgo through epoch t does not exceed by the relationshipwhere η is the security loading and f(x, t) = (∂/∂x) F(x, t).It is assumed that the d.f. F(x, t) is differentiable eith regard to x with a possible exception at the point x = o.Using an extension of the “ballot theorem” in Chapter III of Feller (1968 [4]) he showed that and observed that if numerical values of F(x, t) were available values of U(w, t) could be computed.
The Story Of 100 Actuarially Guaranteed No-Ruin Casualty Insurance Companies
Authors: Seal Hilary L. 
Published: July 2016

“Most people think that an insurance company's business is to make money out of insuring things. They are wrong. Its business is to take as much money off the public as possible, invest it successfully and hope that the conditions on which it was taken never happen.”The Economist, April 13, 1974 (p. 119)In order to motivate the series of Monte Carlo simulations we have carried out in the following article we would like readers to imagine that a small rural casualty insurance company, the Farm Fire and Flood Damage Ins. Co. (FFFDIC), is to be bought by an entrepreneur (whom we shall designate by EP) provided his consulting actuary (the author of this article) can satisfy his requirements which are as follow:(i) A 15-year investment is foreseen at the end of which time EP wishes to be able to sell, hopefully without loss.(ii) The risk-capital is to be invested and (although some of it must be in easily liquidable securities) should yield a rate of return comparable with that obtainable on the same amount of capital invested in the market.(iii) The premiums will not have risk-loadings, as such, but will be loaded for profit by 15%.(iv) The risk-capital should, on the average, be returnable at the end of the 15-year investment.