In an earlier note the present author deduced bounds for the approximation error of stop loss premiums when the aggregate claims distribution is calculated by a method introduced by Bertram. From the error bounds of the stop loss premiums we deduced bounds for the approximation error of the cumulative distribution and the discrete density of the aggregate claims. In the present note we shall improve the bounds for the cumulative distribution and the discrete density.

Nous donnerons ci-après un aperçu de quelques domaines d'application de la théorie du risque dans l'assurance de choses et en particulier dans l'assurance incendie en Autriche.

Il convient cependant de signaler que ce domaine n'a pas encore reçu beaucoup d'applications pratiques en Autriche, mais, ainsi que le montrera la suite de ce travail, l'application pratique de la théorie du risque a déjà conduit à quelques résultats positifs.

Du point de vue de la pratique actuarielle dans l'assurance de choses, il faut distinguer deux problèmes principaux:

a) la détermination du risque individuel;

b) la détermination du risque collectif.

Ces deux problèmes feront l'objet des deux prochains chapitres de ce travail. Un dernier chapitre mettra en lumière les relations de ces problèmes avec d'autres disciplines et fera état, pour terminer, de quelques perspectives pratiques.

Dans l'assurance de choses, il faut distinguer trois méthodes de détermination du coût de la couverture du risque individuel:

a) estimation empirique du coût du risque individuel;

b) détermination du coût à l'aide de séries d'observations par la comparaison des sinistres et des primes ou des sinistres et des sommes assurées;

c) détermination statistique exacte des primes à l'aide de la théorie individuelle du risque.

In his paper “Actuarial Activity in General Insurance in the Northern Countries of Europe” L. Wilhelmsen gives amongst other things an account of the work carried on by Centralstället för nordisk ömsesidig Brandförsäkringsstatistik (CNÖB, the Northern Central Office for Fire Insurance Statistics from Mutual Companies). In this he states that one part of the organisation's work is the carrying out of special investigations of current problems with material collected on each occasion for the purpose.

The object of this paper is to report investigations carried out in CNÖB on the connection between temperature and risk premium in fire insurance. The material used is made up exclusively of civil risks (buildings) and the material has been taken from Sweden and Norway. The background to the investigation consists in the fact that in Scandinavia the risk premiums for fire insurance show an apparent geographical variation, in that the amount clearly increases in the most northerly provinces.

As we know that the fire damage directly or indirectly caused by heating systems (chimney fires, cracked building blocks, embers from fireplaces or chimneys etc.) represents a large proportion of the damage in civil risks (40-50%, in some materials 60% or more), that the proportion is greatest in the northern parts and that in the Swedish material about 50% of the damage in any one area falls in the four months December to March, it is quite reasonable to trace the influence of the temperature factor behind the geographical variation; the colder the climate, the more lighting of fires and the more damage. This line of argument is connected exclusively to the frequency of damage; we shall return later to the mean degree of damage.

The response to the invitation for the 5th Colloquium has been, especially for subject 1, very gratifying; a large number of meritorious papers have been sent in. Fortunately, it took not much time in this case to find a good guide for all this work. R. E. Beard presents in his article “Some Actuarial Aspects of Non-Life Insurance Company Management”) a broad cross-section of the many problems that confront the non-life insurer. It was therefore regarded as not quite unwise to follow his steps and to discuss the other papers at suitable points.

In the first half of his work R. E. Beard deals with a practical example from motor insurance, which is fully calculated with the help of numerical data. The question relates to the estimation of the value of outstanding claims, a quantity which is of decisive importance for the evaluation of the results of a portfolio. Apart from chance fluctuations which might affect the statistics from year to year, there are three major factors that render the task of estimation difficult:

(a) the natural growth of the portfolio

(b) the fact that bigger claims require longer time for settlement than the smaller ones

(c) the change in the value of money with time.

In Industrial Fire insurance an aggregate limit for the amount retained by the policyholder under a deductible policy has been agreed upon more frequently in recent times. This agreement is equivalent to a stop-loss cover on the retained loss amount. For the Poisson-lognormal model the corresponding stop-loss net premium is calculated using various methods (normal power, translated gamma, various discretisations) and the methods are compared. Finally, the influence of the model parameters is examined and it is demonstrated how a variety of parameter value combinations can be reduced to only a few rating curves.

We develop a semi-parametric predictor of the IBNR reserve in a macro-model when the claim amount for a certain accident and development year can be expressed in a loglinear form composed of a deterministic part and a random error. We need to make assumptions only on the first two moments of the error, without any specified parametric assumption on its distribution. We give its properties, present its advantages and compare the estimates obtained with various predictors of the IBNR reserve, parametric and non-parametric, using a data set.

The unbayesed credibility procedure proposed by Gerber is revisited. Its performance is discussed, connections are drawn to earlier literature, and some possible ideas of generalizations are investigated (and found fruitless).

This paper studies optimal defined-contribution (DC) pension management under stochastic interest rates and expected inflation. In addition to financial risk, we consider the risk of pre-retirement death and introduce life insurance to the pension account as an option to manage this risk. We formulate this pension management problem as a random horizon utility maximization problem and derive its explicit solution under the assumption of constant relative risk aversion utility. We calibrate our model to the U.S. data and demonstrate that the pension member’s demand for life insurance has a hump-shaped pattern with age and a U-shaped pattern with the real interest rate and expected inflation. The optimal pension account balance in our model resembles a variable annuity, wherein the death benefits are endogenously determined and depend on various factors including age, mortality, account balance, future contributions, preferences, and market conditions. Our study suggests that offering variable annuities with more flexible death benefits within the DC account could better cater to the bequest demands of its members.

By experience rating the main problem is to estimate the credibilities. We have for the credibility αk the famous formula)

but it is often troublesome to find suitable estimates for the variances and . In the present paper a general method to estimate them from the actual statistics is given.

A disadvantage of the method is that good estimates require relatively extensive statistical material. If one of the variances is known, the method can be easily modified to give the other variance from statistics of moderate size.

The method is based on the Maximum Likelihood principle and leads to a system of non-linear equations. The equations can be solved by an iterative process, easily programmable for computers.

The mathematical model underlying the experience rating problem differs in our case lightly from the usual one.

A generalized estimating equations (GEE) approach is developed to estimate structural parameters of a regression credibility model with independent or moving average errors. A comprehensive account is given to illustrate how GEE estimators are worked out within an extended Hachemeister (1975) framework. Evidenced by results of simulation studies, the proposed GEE estimators appear to outperform those given by Hachemeister, and have led to a remarkable improvement in accuracy of the credibility estimators so constructed.

Le dépouillement statistique d'une classe de tarif hétérogène permet d'observer les fréquences relatives qi des contrats qui pendant une période de référence ont été frappés de (i — 1) sinistres.

On a:

et

Ces fréquences déterminent dans l'espace Rm un vecteur à m composantes, que nous notons:

Une base de décomposition est constituée par un éventail fini de η classes homogènes. Chaque classe homogène Ck est définie par une variable aléatoire, dont la loi de probabilité (λk) dépend du paramètre λk. Théoriquement ce paramètre peut être tout à fait général (un triplet réel p.e.); pour les applications pratiques traitées au paragraphe 4, il suffit pourtant de considérer λk comme un nombre réel.

Si Pi(λk) représente la probabilité à priori d'avoir exactement (i — 1) sinistres dans la classe Ck, nous pouvons noter (λk) par:

(p1(λk), p2(λk), … Pm(λk))

On a:

et

Nous supposons que les vecteurs (λk) sont linéairement indépendants.

Résoudre l'hétérogénéité d'une classe de tarif consiste tout d'abord à choisir une base de décomposition, à représentation analytique connue, et puis à décomposer la classe de tarif par l'éventail des classes homogènes Ck de cette base.