José Paris

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

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.

The True Claim Amount and Frequency Distributions within a Bonus-Malus System

We apply Lemaires algorithm and a non-parametric mixed Poisson fit to a motor insurance portfolio in order to find the true claim frequency and claim amount distributions. The algorithm we develop accounts for the fact that observed distributions are distorted by bonus hunger, when a bonus-malus system is used by the insurer.

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.

Some comments on the individual risk model and multivariate extension

SummaryIn this paper we first concentrate on the univariate individual risk model and make some comments on the computing time of the related algorithms. We then obtain the multivariate approximate De Pril’s algorithm. The main point of the paper is the derivation of a multivariate extension of the exact Dhaene and Vandebroek algorithm. Excess of loss reinsurance with reinstatements is used as an application.ZusammenfassungIn dieser Arbeit konzentrieren wir uns zunächst auf das univariate Risikomodell und machen einige Bemerkungen zur Rechenzeit des verwendeten Algorithmus. Wir erhalten dann den multivariaten approximierten De-Pril-Algorithmus. Der Schwerpunkt dieser Arbeit liegt in der Herleitung einer multivariaten Erweiterung des exakten Dhaene/Vandebroek-Algorithmus. Die Schadenexzedentenrückversicherung mit Wiederauffüllung wird als Beispiel einer Anwendung herangezogen.

The practical replacement of a bonus-malus system

In this paper we will show how to set up a practical bonus-malus system with a finite number of classes. We will use the actual claim amount and claims frequency distributions in order to predict the future observed claims frequency when the new bonus-malus system will be in use. The future observed claims frequency is used to set up an optimal bonus-malus system as well as the transient and stationary distributions of the drivers in the new bonusmalus system. When the number of classes as well as the transition rules of the new bonus-malus system have been adopted, the premium levels are obtained by minimizing a certain distance between the levels of the practical bonus-malus system and the corresponding optimal bonus-malus system. Some iterations are necessary in order to reach stabilization of the future observed claims frequency and the levels of the practical bonus-malus system.

The Mixed Bivariate Hofmann Distribution

In this paper we study a class of Mixed Bivariate Poisson Distributions by extending the Hofmann Distribution from the univariate case to the bivariate case. We show how to evaluate the bivariate aggregate claims distribution and we fit some insurance portfolios given in the literature. This study typically extends the use of the Bivariate Independent Poisson Distribution, the Mixed Bivariate Negative Binomial and the Mixed Bivariate Poisson Inverse Gaussian Distribution.