Jean-François Walhin

Statistical procedure for improving the precision of the measurement of the essential work of fracture of thin sheets

The precision of the measurement of the essential work of fracture requires an accurate determination of the critical ligament length below which the ligament is in a mixed mode stress state and the failure mechanisms considerably change. A statistical procedure is proposed which allows to determine easily and accurately this critical ligament length. After rejection of the specimens having too small ligament lengths, we is obtained by extrapolation of the remaining data for zero ligament length. The number of data points from which we is calculated is also shown to strongly influence the precision of the measurement. The procedure is applied to the measurement of the toughness of sheets of Al and Zn alloys and of a low density polyethylene.

Using Mixed Poisson Processes in Connection with Bonus-Malus Systems

For the constructmn of bonus-malus systems, we propose to show how to apply, thanks to sunple mathematms, a parametric method encompassing those encountered m the hterature. We also compare this parametric method with a non-paramemc one that has not yet been used m the actuarial hterature and that however permits a simple formulatmn of the stationary and transmon probabflmes m a portfoho whenever we have the mtent~on to construct a bonus-malus system with fimte number of classes

Setting a bonus-malus scale in the presence of other rating factors: Taylor's work revisited

In this paper, we propose an analytic analogue to the simulation procedure described in Taylor (1997). We apply the formulas to a Belgian data set and discuss the interaction between a priori and a posteriori ratemakings.

The Effect of Excess-of-Loss Reinsurance with Reinstatements of the Cedent's Portfolio

ZusammenfassungDer Anpassungskoeffizient für das Risiko im Eigenbehalt nach Schadenexzedenten-Rückversicherung mit Wiederauffüllung wird berechnet.Dafür brauchen wir eine multivariable Aggregat-Schadenverteilung. Diese Verteilung wird uns durch eine multivariable Ausdehnung von Panjers Rekursion in einfacher Weise geliefert.Numerische Beispiele zeigen den Vorteil für den Zedenten, beim Kauf einer Schadenexzedenten-Rückversicherung mit Wiederauffüllung, den Anpassungskoeffizient seines Portefeuilles zu berechnen.Es folgt eine Diskussion über die optimale Berechnungsmethode.SummaryThe adjustment coefficient for the cedent’s retained risk after excess of loss reinsurance with reinstatements is calculated.Therefore we need a multivariate aggregate claims distribution. This distribution is easily given by a multivariate extension of Panjer’s recursion.Numerical examples show the interest for the cedent to calculate the adjustment coefficient for its portfolio when buying excess of loss reinsurance with reinstatements.An optimal organization of the calculations is discussed.

On the Use of Equispaced Discrete Distributions

The Kolmogorov distance is used to transform arithmetic severities into equispaced arithmetic severities in order to reduce the number of calculations when using algorithms like Panjers formulae for compound distributions. An upper bound is given for the Kolmogorov distance between the true compound distribution and the transformed one. Advantages of the Kolmogorov distance and disadvantages of the total variation distance are discussed. When the bounds are too big, a BerryEsseen result can be used. Then almost every case can be handled by the techniques described in this paper. Numerical examples show the interest of the methods.

Stochastic projection for large individual losses

In this paper we investigate how to estimate ultimate values of large losses. The method is based on the development of individual losses and therefore allows to compute the netting impact of excess of loss reinsurance. In particular the index clause is properly accounted for. A numerical example based on real-life data is provided.

On the optimality of proportional reinsurance

Proportional reinsurance is often thought to be a very simple method of covering the portfolio of an insurer. Theoreticians are not really interested in analysing the optimality properties of these types of reinsurance covers. In this paper, we will use a real-life insurance portfolio in order to compare four proportional structures: quota share reinsurance, variable quota share reinsurance, surplus reinsurance and surplus reinsurance with a table of lines.

Recursive Formulae for Some Bivariate Counting Distributions Obtained by the Trivariate Reduction Method

In this paper we study some bivariate counting distributions that are obtained by the trivariate reduction method. We work with Poisson compound distributions and we use their good properties in order to derive recursive algorithms for the bivariate distribution and bivariate aggregate claims distribution. A data set is also fitted.

Excess of loss reinsurance with reinstatements: premium calculation and ruin probability of the cedent

ZusammenfassungDie Prämien für Schadenexzedentenverträge mit Wiederauffüllung werden auf Basis des Standardabweichung-Prämienprinzips und der PH-Transformation kalkuliert. Die praktische Anwendung dieser Prämienprinzipien wird besprochen.Der bivariate Panjers Algorithmus wird benutzt, um die zeitendliche Ruinwahrscheinlichkeit des Zedenten zu schätzen, wenn dieser Schadenexzedentenverträge mit Wiederauffüllung kauft.SummaryPremiums for excess of loss treaties with reinstatements are calculated according to the standard deviation and PH transform premium principles. Some comments are given regarding the practical use of these premium principles.The bivariate Panjer’s algorithm is used in order to find finite time ruin probabilities of the Ceding Company when it buys excess of loss treaties with reinstatements.

An actuarial analysis of the French bonus-malus system

The bonus-malus system in force in France differs from most of those used in industrialized countries around the world. Policyholders do not move inside a scale but their premium is obtained with the help of multiplicative CRM coefficients (CRM stands for the acronym of the French coefficient de réduction-majoration). The French bonus-malus system has been the topic of very few scientific investigations in the actuarial literature. This paper purposes to analyze this bonus-malus system in details. Despite its apparent simplicity, it will be seen that it leads to nontrivial mathematical problems. The financial equilibrium of the bonus-malus system is also investigated thanks to the multivariate De Prils algorithm for the convolution of independent and identically distributed random vectors.