bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the Statistical Estimation of Costs of Claims
Authors: Ajne B. 
Published: July 2016

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.
Windstorm Coverage in Japan
Authors: Wakuri Masao 
Published: July 2016

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.
Largest Claims Reinsurance (LCR). A Quick Method to Calculate LCR-Risk Rates from Excess of Loss Risk Rates
Authors: Benktander G. 
Published: July 2016

Let us denote by E(x) the pure risk premium of an unlimited excess cover with the retention x and by H(x) and m(x) the corresponding expected frequency and severity.We thus have E(x) = H(x) · m(x).H(x) is a non-increasing function of x and for practical purposes we can assume that it is decreasing; H′(x) < o. The equation H(x) = n has then only one solution xn, where n is a fixed integer.Let En denote the risk premium for a reinsurance covering the n largest claims from the bottom.Let us define Intuitively we feel that is a good approximation for En.We shall first show that when the claims size distribution is Pareto and the number of claims is Poisson distributed, is a good approximation for En, being slightly on the safe side. We further include a proof given by G. Ottaviani that the inequality always holds.In the Pareto case we havewhere the Poisson parameter t stands for the expected number of claims in excess of 1 (equal to a suitably chosen monetary unit) and.
A Numerical Illustration of Optimal Semilinear Credibility
Authors: De Vylder Fl. , Ballegeer Y. 
Published: July 2016

The homogeneous (in time) model of credibility theory is defined by a sequence Θ, X1, X2, … of random variables, where for Θ = θ fixed, the variables X1, X2, … are independent and equidistributed. The structure variable Θ may be interpreted as the parameter of a contract chosen at random in a fixed portfolio, the variable Xk as the total cost (or number) of the claims of the kth year of that contract.Bühlmann's linear credibility premium of the year t + 1 may be written in the formwhere f is a linear function. In optimal semilinear credibility, we look for an optimal f, not necessarily linear, such that (1) is closest to Xt+1 in the least squares sense. In the first section we prove that this optimal f, denoted by f*, is solution of an integral equation of Fredholm type, which reduces to a system of linear equations in the case of a finite portfolio. That is a portfolio in which Θ and Xk can assume only a finite number of values.In the second section we see that the structure of such a portfolio is closely connected with the decomposition of a quadratic form in a sum of squares of linear forms.In the last section we calculate numerically the optimal premium for a concrete portfolio in automobile insurance. We limit ourselves to the consideration of the number of claims. The optimal premium is compared with the usual linear premium. The difference is far from negligible.
Optimal Claim Decisions for a Bonus-Malus System: a Continuous Approach
Authors: De Pril Nelson 
Published: July 2016

For the premium calculation the insurer will split up his collectivity of risks into risk groups which are homogeneous with respect to some directly observable risk factors. All risks of such a risk group will be charged the same base premium. But it is clear that by such an a priori classification not all determined factors can be taken into consideration, so that there will still remain accident proneness differentials within a risk group. Since these differentials will be reflected in the course of time by the claim experience of each risk, the insurer can come to a fair tarification by adjusting, each period, the base premium according to the individual claim experience of the risk. Such a system in which earlier neglected risk factors are taken into account a posteriori is an individual experience rating system. Our main interest goes to the following side-effect of experience rating: since an unfavourable claim experience results in a premium increase, an experience rated policyholder is stimulated to self-insure small damages. This phenomenon is well know in connection with bonus-malus systems in motor-car insurance, which explains why it is called “bonus-hunger”.In the present paper a continuous time model for the bonus-malus system is set up which takes into account this hunger for bonus. An insured causing an accident will decide according to a certain decision rule whether to file a claim with his insurance company. The relevant information that he needs to make this decision is: his current risk class, the number of claims he has already filed during that period and the moment at which the decision is to be made.
The Effect of Reinsurance on the Degree of Risk Associated with an Insurer's Portfolio
Authors: Andreadakis M. , Waters H. R. 
Published: July 2016

There are many reasons why an insurer may choose to reinsure a part of his portfolio (see, for example, Carter (1979, p. 5 ff.)) and many ways in which he can assess the effectiveness of the reinsurance arrangements he makes. In this paper we assume the insurer wishes to reinsure a part of his portfolio in order to reduce its “riskiness”. We take as given a portfolio consisting of n independent risks together with the total premium charged to insure these risks and we investigate the effect on the degree of risk associated with the portfolio (see §3 for a definition) of varying the excess of loss or proportional reinsurance limits for each risk.We are given an insurance portfolio consisting of n independent risks. A risk may consist of a single policy or a group of policies: the essential points being that a reinsurance limit, either excess of loss or proportional, is the same for all claims arising from a particular risk, although reinsurance limits may vary from one risk to another. We assume the claims arising from each risk have a compound Poisson distribution. To be more precise, we assume the number of claims arising from the i-th risk is a Poisson process with mean Pi claims each year and the size of each claim has distribution function Fi. As usual, the size of a claim is independent of the time at which it occurs and of all other claims. We also assume that Fi(O) = O for each i, so that we consider only positive claims amounts. We take as given the total annual premium, P, charged by the insurer in respect of these risks. We make no assumption about the way in which P is calculated but we do assume that
The Esscher Premium Principle: A Criticism
Authors: Zehnwirth Benjamin 
Published: July 2016

The Esscher premium principle has recently had some exposure, namely, with the works of Bühlmann (1980) and Gerber (1980).Bühlmann (1980) devised the principle and coined the name for it within the framework of utility theory and risk exchange. Geruber (1980), on the other hand, gives further insight into the principle by studying it within the realm of forecasting in much the same spirit as credibility theorists forecast premiums. However, there is an important distinction: the choice of loss function.The present note sets out to criticize this relatively embryonic principle using decision theoretic arguments and indicates that the Esscher premium is essentially a small perturbation of the well established linearized credibility premium Bühlmann (1970).Let H denote the Esscher premium principle with loading h > o. That is, if X is an observable random variable and Y is a parameter (a risk or a random quantity) to be forecasted then the Esscher premium is given byThat is, H(Y ∣ X) is the Bayes decision rule for estimating Y given the data X relative to the loss functionwhere a is the estimate of Y, and of course the loading h is greater than zero.Now, for the clincher. This loss function is nonsensical from the point of view of estimation. It indicates a loss (or error) to the forecaster that is essentially the antithesis of relative loss.
Bounds on Modified Stop-Loss Premiums in Case of Known Mean and Variance of the Risk Variable
Authors: De Vylder F. , Goovaerts M. , De Pril N. 
Published: July 2016

AbstractIn case of a stop-loss treaty the reinsurer takes over that part of the risk that exceeds a given amount y1. We will deduce bounds on a modified stop-loss treaty where the liability of the reinsurer is limited to y2–y1 in case the claim amount exceeds y2. Upper and lower bounds of this modified stop-loss premium are obtained as a simple application of results obtained earlier by the first author.
Une méthode pour calculer une ristourne adéquate pour années sans sinistres
Authors: Bichsel F. 
Published: July 2016

En Suisse, les sociétés d'assurances bonifient à leurs assurés en responsabilité civile autos une ristourne qui dépend du nombre des années sans sinistres, suivant le barème ci-après:Cette ristourne a pour but de tenir compte des facteurs d'ordre subjectif qui influencent le risque et qui ne peuvent être pris en considération dans les tarifs.Les deux questions suivantes se posent:A) Y a-t-il en réalité des différences dans le risque subjectif justifiant la bonification d'une ristourne pour années sans sinistres?B) Si oui, le barème indiqué ci-haut tient-il compte de ces différences d'une façon adéquate?Le présent travail a pour but de répondre à ces deux questions en appliquant les méthodes modernes des tirages au sort.Question AVoici les résultats de quelques tirages au sort du portefeuille de „La Générale de Berne”S'il n'y avait pas de différences dans le risque subjectif, la variable aléatoire k devrait suivre la distribution de Poisson. Toutefois, on sait que dans ce cas, devrait être égal à s2(k), ce dont il n'est évidemment pas question pour les chiffres ci-dessus. En conséquence, k n'est pas distribué d'après Poisson, et l'existence de différences dans le risque subjectif est prouvée.On pourrait se demander si les différences entre et s2(k) sont de nature aléatoire. Une investigation spéciale a toutefois montré que si k suivait la distribution de Poisson, la probabilité pour des différences aussi élevées que celles des tirages indiqués ci-haut serait pratiquement nulle.
Experience rating and credibility
Authors: Bühlmann Hans 
Published: July 2016

Let me begin with some practical examples of experience rating.a) Swiss Automobile Tariff 1963— Within each tariff-position there are 22 grades:
— The new owner of a car starts at grade 9— The basic premium is determined on the basis of objective characteristics of the risk and essentially depends on the horse-power classification of the car— The 22 grades are experience-rated as follows: For each accident one rises three grades and for each accident-free year one falls one grade. A driver who has I accident in every 4 years hence remains within four adjacent grades.b) Sliding Scale Premiums in ReinsuranceExcess of Loss Contracts often stipulate that:The rate of premium to be applied to the subject premium volume is determined at the end of the cover period as follows:
subject to a minimum of 0,04and a maximum of 0,08c) Participation in Mortality Profit in Group Life InsuranceA group life insurance covers the members of the group on a one year term basis. It is often agreed that at the end of the year mortality profits are given back to the group according to the formularefund = x% gross premiums — y% claims (where x < y)All these examples fall under the heading “Experience Rating”. What do they have in common?Definition: A system by which the premium of the individual risk depends upon the claims experience of this same individual risk.
The optimal reinsurance treaty
Authors: Borch Karl 
Published: July 2016

1. Some years ago I discussed optimal reinsurance treaties, without trying to give a precise definition of this term [1]. I suggested that a reinsurance contract could be called “most efficient” if it, for a given net premium, maximized the reduction of the variance in the claim distribution of the ceding company. I proved under fairly restricted conditions that the Stop Loss contract was most efficient in this respect.I do not consider this a particularly interesting result. I pointed out at the time that there are two parties to a reinsurance contract, and that an arrangement which is very attractive to one party, may be quite unacceptable to the other.2. In spite of my own reservations, it seems that this result —which I did not think deserved to be called a theorem—has caused some interest. Kahn [4] has proved that the result is valid under far more general conditions, and recently Ohlin [5] has proved that the result holds for a much more general class of measures of dispersion.In view of these generalizations it might be useful to state once more, why I think the original result has relatively little interest. In doing so, it is by no means my purpose to reduce the value of the mathematical generalizations of Kahn and Ohlin. Such work has a value in itself, whether the results are immediately useful or not. I merely want to point out that there are other lines of research, which appear more promising, if our purpose is to develop a realistic theory of insurance.
Modern Portfolio Theory: Some Main Results
Authors: Müller Heinz H. 
Published: July 2016

AbstractThis article summarizes some main results in modern portfolio theory. First, the Markowitz approach is presented. Then the capital asset pricing model is derived and its empirical testability is discussed. Afterwards Neumann–Morgenstern utility theory is applied to the portfolio problem. Finally, it is shown how optimal risk allocation in an economy may lead to portfolio insurance.
Experience Rating and Credibility
Authors: Bühlmann Hans 
Published: July 2016

Classical statistics deals with the following standard problem of estimation:Given: random variables X1, X2 … Xn independent, identically distributed, andobservations x1, X2 … xn,Estimate: parameter (or function thereof) of the distribution function common to all Xi.It is not surprising that the “classical actuary” has mostly been involved in solving the actuarial equivalent of this problem in insurance, namelyGiven: risks R1R2 … Rn no contagion, homogeneous group,Find: the proper (common) rate for all risks in the given class.There have, of course, always been actuaries who have questioned the assumptions of independence (no contagion) and/or identical distribution (homogeneity). As long as ratemaking is considered equivalent to the determination of the mean, there seem to be no additional difficulties if the hypothesis of independence is dropped. But is there a way to drop the condition of homogeneity (identical distribution)?
Reciprocal Reinsurance Treaties
Authors: Borch Karl 
Published: July 2016

In this paper we shall study the situation of two insurance companies which are negotiating with the view of concluding a reciprocal reinsurance treaty. We assume that the two companies are under no compulsion to reach an agreement. This means that if the companies conclude a treaty, the treaty must be such that both companies consider themselves better off than without any treaty. We futher assume that no third company can break into the negotiations. This means that the two companies either have to come to terms, or be without any reinsurance.How the two parties reach an agreement in a situation like this, is one of the classical problems of theoretical economics. It is usually referred to as the “Bargaining Problem”. The problem appears very simple, but this is a deception. It has proved extremely difficult to formulate generally acceptable assumptions which give the problem a determinate solution. The “Theory of Games”, developed by von Neumann and Morgenstern (10), does not give a determinate solution, but it has greatly increased our understanding of such problems, and the present paper will draw heavily on that theory.The situation which we propose to study, is very simple, may be too simple to have any bearing on reinsurance negotiations in real life. If there exists a reinsurance market, which also is a perfect market in the sense given to this term in economic theory, bartering between two companies does not make any sense. They could both do equally well or better by dealing in the market at the market price.
Two Studies in Automobile Insurance Ratemaking
Authors: Bailey Robert A. , Simon LeRoy J. 
Published: July 2016

Section A of this paper uses the Canadian experience for private passenger automobiles to show (1) that merit rating is almost as effective as the class plan in separating the better risks from the poorer risks, (2) that both merit rating and class rating leave unanalyzed a considerable amount of variation among risks and (3) that certain available evidence supports the conclusion that annual mileage, which has long been felt to be an important measure of hazard, is a very significant cause of this unanalyzed variation among risks.Section B presents a method for obtaining relativities among groups on which a multiple classification system has been imposed. The customary method of calculating class relativities uses the total experience for each class with all subdivisions within the classes added together. With the customary method it is difficult to make a completely accurate adjustment for different distributions by territory or merit rating, because any change in the class relativities disturbs the other sets of relativities and conversely. It is shown that even if such an adjustment were made, the customary method of calculating relativities one set at a time does not reflect the relative credibility of each subgroup and does not produce the best fit to the actual data. Moreover it produces differences between the actual data and the fitted values which are far too large to be caused by chance. In addition, for private passenger automobile insurance in Canada, it is shown that two sets of relativities which are multiplied together cannot produce the best fit to the actual data, and some of the consequences of trying to do so are explained. Some methods are advanced whereby all sets of relativities for classes, merit ratings, territories, and so forth, can be calculated simultaneously, which will overcome all the deficiencies in the customary method. These improved methods use the technique of minimizing a measure (technically known as the chi-square test) of the differences between the actual data and the fitted values. Some applications to other lines of insurance are mentioned.
The Utility Concept Applied to the Theory of Insurance
Authors: Borch Karl 
Published: July 2016

In some recent papers ((1), (2) and (3)) about reinsurance problems I have made extensive use of utility concepts. It has been shown that if a company follows well defined objectives in its reinsurance policy, these objectives can be represented by a utility function which the company seeks to maximise. This formulation of the problem will in general make it possible to determine a unique reinsurance arrangement which is optimal when the company's objectives and external situation are given.More than 50 years ago Guldberg (4) wrote (about the probability of ruin): “Wie hoch diese Wahrscheinlichkeit gegriffen werden soil, muss dent subjektiven Ermessen oder von Aussen kommenden Bedingungen überlassen bleiben”. This is the traditional approach to reinsurance problems. It does obviously not lead to a determinate solution. Most authors taking this approach conclude their studies by giving a mathematical relation between some measure of “stability”, such as the probability of ruin, and some parameter, for instance maximum retention, to which the company can give any value within a certain range. Such studies do usually not state which particular value the company should select for this parameter, i.e. what degree of stability it should settle for. This question is apparently considered as being outside the field of actuarial mathematics.