La théorie américaine de la crédibilité est considérée par nos collégues américains comme une pièce-maîtresse de la théorie actuarielle „non life” (Longley Cook: An Introduction to Credibility Theory). Elle a re¸u la consécration de l'expérience sur le marché le plus large du monde. Dirigeants d'entreprises et autorités de contrôle se sont accordéd sur son emploi. Ce sont la des résultats positifs qui ne peuvent que faire réfléchir tous les actuaires. ASTIN a mis le probléme á l'ordre du jour du colloque de Lucerne en 1965. Cette année, la réunion d'Arnhem reprend le même sujet, á juste titre.

Mais il faut le constater, si les méthodes d'adaptation américaines donnent des résultats pratiques indiscutables, elles ont été établies par des méthodes pragmatiques, d'abord basées sur certains cas particuliers et admises, ex cathedra, en toute généralité. Cette situation laisse un sentiment de malaise et souléve de nombreuses critiques. (Derron: Kredibilität und Versicherungwesen) (Mayerson: A Bayesan view of Credibility). La théorie de la crédibilité impose avant tout la recherche d'une base théorique plus valable.

La question est délicate. Elle touche directement au probléme de l'inférence statistique qui constitue entre probabilistes et statisticiens un domaine trés controversé. La présente note apporte une contribution trés subjective á la théorie américaine. Elle tend á élargir le débat et conduit á la conclusion que l'établissement d'une théorie du comportement de l'assureur serait souhaitable. Dans une telle théorie, les mathématiques ont leur importance, mais elles présupposent l'admission á priori de certaines régies d'inférence.

The purpose of this paper is to describe a technical procedure, which enables one to compute values of the generalised Poisson distribution function, with an accuracy considered sufficient for insurance companies and with satisfactory speed. The procedure requires a fast medium sized computer.

The computation of values of the generalised Poisson distribution function has become a timely problem in Finland, because of the introduction by the Supervisory Service of more stringent requirements in determining limits of the so called equalisation reserves, which have their theoretical basis in the random fluctuations of claims amounts. The question has also been discussed in papers submitted by Dr. Pentikainen [3] and Dr. Pesonen [4] to this Colloquium. Because the practical computation is a further problem, the Federation of Finnish Insurance Companies set up a committee in 1962 to gather and work up the necessary statistics from various branches of insurance and to develop the computational methods ready for use in practice. The committee has almost completed its work, and one of the results, a procedure to compute values of the generalised Poisson function with a mixed method, is described below. The method is referred to briefly in [4]. The programming and further planning work has been done by Dr. Loimaranta and M. Sc. Porn.

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].

Mortgage insurance indemnifies a mortage lender against loss on default by the borrower. The sequence of events leading to a claim under this type of insurance is relatively complex, depending not only on the credit worthiness of the borrower but also on a number of external economic factors.

Prominent among these external factors are the loan to valuation ratio of the insured loan, the disposable income of the borrower, and movements in property values. A broad theoretical model of the functional dependencies of claim frequency and average claim size on these variables is established in Sections 6 and 7. Section 8 fits these models, extended by other “internal” variables such as the geographic location of the mortgaged property, to a real data set.

Section 9 compares the fitted model with the data, and finds an acceptable fit despite extreme fluctuations in the claims experience recorded in the data set.

In the context of the Solvency II directive, the operation of an internal risk model is a possible way for risk assessment and for the determination of the solvency capital requirement of an insurance company in the European Union. A Monte Carlo procedure is customary to generate a model output. To be compliant with the directive, validation of the internal risk model is conducted on the basis of the model output. For this purpose, we suggest a new test for checking whether there is a significant change in the modeled solvency capital requirement. Asymptotic properties of the test statistic are investigated and a bootstrap approximation is justified. A simulation study investigates the performance of the test in the finite sample case and confirms the theoretical results. The internal risk model and the application of the test is illustrated in a simplified example. The method has more general usage for inference of a broad class of law-invariant and coherent risk measures on the basis of a paired sample.

In pricing insurance contracts based on the individual policyholder’s aggregate losses for non-life insurers, the literature has mainly focused on using detailed information from policies and closed claims. However, the information on open claims can reflect shifts in the distribution of the expected claim payments better than closed claims. Such shifts may be needed to be reflected in the ratemaking process earlier rather than later, especially when insurers are experiencing environmental changes. In practice, actuaries use ad hoc techniques to adjust data to current levels to determine premiums. This paper presents an intuitive ratemaking model, employing a marked Poisson process framework, which ensures that the multivariate risk analysis is done more routinely using all reported claims and makes an adjustment for Incurred But Not Reported claims. Utilizing data from the Wisconsin Local Government Property Insurance Fund, we find that by determining rates based on current data, the proposed ratemaking model leads to better alignment of premiums and provides insurers with a more financially sound portfolio.

An insurance company aims to increase the volume of its business.

The volume is measured by the premium income during one year.

The company makes use of N tariff groups.

pn = premium charged by the company per unit of insurance in tariff group n.

qn = premium per unit of insurance in tariff group n corresponding to the statistical experience in the said group.

Ln = The size of the market in tariff group n. If all insurances in the group pay the same premium and if the company is able to cover the whole of its potential market then Ln is equal to the number of policies. If the premium per policy is equal to pn times the sum insured and if the company is able to cover the whole of its potential market then Ln is equal to the total sum insured.

Fn(p) = the proportion of Ln that actually buys insurance in tariff group n if the company charges the premium p per unit of insurance.

In his report to the XVIth International Congress of Actuaries in Brussels [5] in 1960 the author has given a survey of the German Motor Insurance Business and in this it was mentioned that a change might be expected from the flat-tariff prescribed by law into individual tariffs for every insurance company. The Federal Ministry of Economics decided this question at the end of 1959 by signing the price regulation PR 15/59 [I] The federal price-fixing treaty had thus come to an end after several decades.

Three steps are planned for the transitional period: For the years i960 and 1961 the flat-tariff remained in existence; free competition was introduced by providing for a return of premiums from technical surplus. At the beginning of 1962 the second phase of liberalisation began: Between 1962–1965 each insurance company has to compute its own tariff which has to be approved by the Federal Ministry of Economics. No deviations are permitted from the approved tariffs. Nobody can tell yet what the third phase in 1966 will bring.

At the time the price-fixing agreements were cancelled it was unanimously agreed that proper competition should be maintained and that ruinous price competition which would harm the interests of victims of traffic accidents must be avoided [4]. Trusts or tariff rings were not taken into consideration. However, some market regulations were set up by co-operation between the different parties.

The benefits (insurance conditions) are the same for all insurance enterprises. Also there are only few alternatives in the tariff conditions.

The paper develops a methodology to enable microscopic models of transportation systems to be accessible for a statistical study of traffic accidents. Our approach is intended to permit an understanding not only of historical losses but also of incidents that may occur in altered, potential future systems. Through such a counterfactual analysis, it is possible, from an insurance, but also from an engineering perspective, to assess the impact of changes in the design of vehicles and transport systems in terms of their impact on road safety and functionality.

Structurally, we characterize the total loss distribution approximatively as a mean-variance mixture. This also yields valuation procedures that can be used instead of Monte Carlo simulation. Specifically, we construct an implementation based on the open-source traffic simulator SUMO and illustrate the potential of the approach in counterfactual case studies.

Different rating methods which allow for exceptional large claims are discussed. A robust Bayesian statistical model is proposed which can cope with non negative, skewed data. An example from fire insurance is analyzed. The performance of the posterior mean is compared to the performance of a robust credibility estimator.

“War is the father of all things”. This sentence of Heraklit (between 540 and 535 BC) has a lot of truth in general. I am quoting it in connection with World War II (1939-1945) and inventions originating in this period.

The best known examples of inventions (or first practical use of such invention) are

• Nuclear Fission leading to the construction of the atomic bomb and nuclear reactors,

• The Programmable Electronic Computer. Best known is the machine of John von Neumann as Los Alamos: Mathematical Analyser Numerical Integrator and Computer (MANIAC),

• Radar to guide the airplanes particularly at night and in bad weather conditions,

• Penicillin to fight bacteriological diseases.

Some of these inventions relied on original discoveries already made before the war, but during the war they were for the first time used on a large scale.

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

This paper describes a rather simple application of credibility which was used to develop experience rating plans for Blue Cross and Blue Shield organizations in the United States. These are non-profit corporations which provide prepaid coverage for the cost of hospitalization and medical-surgical care. The benefits are in many instances full service benefits so that there is no additional cost to the subscribing member.

While coverage is being granted to individual applicants, the bulk of the unterwriting is composed of groups, that is employees of a given commercial entity. Since the plans are voluntary and, in many instances, the employer either acts only as collecting agent for his employees or else pays only a fraction of the premium, certain minimum participation percentages have been established to prevent anti-selection.

Because of the competion of Insurance Companies which have concentrated their efforts on larger and more profitable groups (especially those where the employer pays all of the premiums) it was early recognized that a uniform or “community” rate will result in a gradual loss of groups with good experience thereby requiring substantial increases in the average rates. The original program of experience rating which the author has developed for the Massachusetts Blue Cross in 1949 has been since adopted “mutatis mutandis” by a number of other Blue Cross and Blue Shield organizations.