# Jean-François Walhin's research while affiliated with Université Catholique de Louvain - UCLouvain and other places

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## Publications (58)

Experience and exposure rating are traditionally considered to be independent but complementary methods for pricing property per risk excess of loss reinsurance. Strengths and limitations of these techniques are well-known. In practice, both methods often lead to quite different prices. In this paper we show that limitations of traditional experien...

This paper describes a particular type of a posteriori rating scheme, namely bonus-malus systems within the general framework of mixed Poisson regression. It is shown that the bonus-malus scale is not independent of the a priori rating applied by the insurer. Considerations about bonus hunger, moral hazard, and signal theory are briefly introduced....

IntroductionProbabilistic ToolsPoisson DistributionMixed Poisson DistributionsStatistical Inference for Discrete DistributionsNumerical IllustrationFurther Reading and Bibliographic Notes

IntroductionCredibility ModelsCredibility Formulas with a Quadratic Loss FunctionCredibility Formulas with an Exponential Loss FunctionDependence in the Mixed Poisson Credibility ModelFurther Reading and Bibliographic Notes

IntroductionDistribution of the Annual Aggregate ClaimsIntroducing a Deductible within a Posteriori RatemakingNumerical IllustrationsFurther Reading and Bibliographic Notes

IntroductionFrench Bonus-Malus SystemPartial LiabilityFurther Reading and Bibliographic Notes

IntroductionMulti-Event Credibility ModelsMulti-Event Bonus-Malus ScalesFurther Reading and Bibliographic Notes

IntroductionTransient Behaviour and Convergence of Bonus-Malus ScalesQuadratic Loss FunctionExponential Loss FunctionNumerical IllustrationsSuper Bonus LevelFurther Reading and Bibliographic Notes

IntroductionModelling Bonus-Malus SystemsTransition ProbabilitiesLong-Term Behaviour of Bonus-Malus SystemsRelativities with a Quadratic Loss FunctionRelativities with an Exponential Loss FunctionSpecial Bonus RuleChange of ScaleDependence in Bonus-Malus ScalesFurther Reading and Bibliographic Notes

IntroductionModelling Claim SeveritiesMeasures of Efficiency for Bonus-Malus ScalesBonus Hunger and Optimal RetentionFurther Reading and Bibliographic Notes

The data of Bissell [A.F. Bissell, A negative binomial model with varying element sizes, Biometrika 59 (1972) 435–441] are counts of the number of flaws in rolls of textile fabric of different lengths. Given that the simple Poisson model is rejected due to overdispersion, we propose an analysis of the data by means of a general family of discrete l...

This article is devoted to the design of bonus-malus scales involving different types of claims. Typically, claims with or without bodily injuries, or claims with full or partial liability of the insured driver, are distinguished and entail different penalties. Under mild assumptions, claim severities can also be taken into account in this way. Num...

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...

We use convex optimization to provide a rigorous proof of de Finetti’s retention result for proportional reinsurance. We then
extend this result to variable quota share reinsurance and surplus reinsurance with table of lines. We demonstrate by a numerical
example that in general neither variable quota share reinsurance nor surplus reinsurance with...

With the deregulation of bonus-malus systems in the EU, it is important to obtain rules in order to transfer policyholders
from one bonus-malus scale to another. The present paper proposes a solution to this problem.
Wegen der Deregulierung der Bonus-Malus Systeme in der EU ist es wichtig, Regeln für den Transfer eines Versicherungsnehmers
von eine...

In this paper we analyze the value creation for an insurance company. We concentrate only on the underwriting risk. We use
a multivariate normal random vector in order to model the underwriting risk of the insurer. Our model accounts for correlations
between risks and between lines of business. We compute return on risk adjusted capital (RORAC) and...

In this pedagogical paper we analyze the diversification benefit. We draw attention to the model and parameter risks which
may have a very important influence on the calculation of the required solvency level of a financial conglomerate. The very
simple numerical applications used in the paper exemplify why discussions between financial conglomerat...

Bonus-malus systems typically lead to high maluses when claims at fault are reported. Such penalties are often difficult to implement in practice. It is shown that this drawback may be avoided by combining a posteriori premium corrections with a deductible varying according to the level occupied in the scale.

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 re...

Bonus-malus systems typically lead to high maluses when claims at fault are reported. Such penalties are often difficult to implement in practice. It is shown in this paper that this drawback may be avoided by combining a posteriori premium corrections with a deductible varying according to the level occupied in the scale.

A Top & Drop cover is a reinsurance cover that may be found on the retrocession world. It oers some capacity that may be used either for a top layer or for a working layer. In the latter case one speaks of a drop. Within the collective risk model we show in this paper how to use the multivariate version of the Panjer's algorithm in order to price t...

We observe on the market that a lot of excess of loss reinsurance treaties have specific clauses such as an annual aggregate
deductible or reinstatements. These clauses imply that the aggregate claims of the ceding company not only depends on the
company’s own retention but also on the aggregate claims of the reinsurer. These risks are obviously de...

We analyze an actuarial approach for the pricing and reserving of minimum death guar-antees in unit-linked life insurance. After summarizing some results on mono-period risk measurement, we explain two possible strategies to deal with multi-period capital alloca-tion problems. The first one uses no future information whereas the second one does. We...

We observe on the market that a lot of excess of loss reinsurance treaties have specific clauses such as an annual aggregate deductible or reinstatements. These clauses imply that the aggregate claims of the ceding company not only depends on the company’s own retention but also on the aggregate claims of the reinsurer. These risks are obviously de...

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.

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 aim of this paper is to develop some bivariate generalizations of the Hofmann distribution. The Hofmann distribution is known to give nice fits for overdispersed data sets. Two bivariate models are proposed. Recursive formulae are given for the evaluation of the probability function. Moments, conditional distributions and marginal distributions...

We show in this paper how to obtain the relativi- ties of the Belgian Bonus-Malus System, including the special bonus rule sending the policyholders in the malus zone to ini- tial level after four claim-free years. The model allows for a priori ratemaking. It is applied to a real-life portfolio.

In this paper we use some concepts related to multivariate stochastic orderings borrowed from reliability theory in order
to find upper and lower bounds on the probability of ruin in discrete time for models where a multivariate dependence exists.
A typical application is the calculation of the ruin probability of a portfolio protected by an excess...

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 fre- quency when the new bonus-malus system will be in use. The future observed claims frequency is used to set up an optimal b...

In 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 rei...

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 Independ...

Premiums 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...

In this paper, we investigate actuarial, financial, economic and commercial aspects of the pricing of an ex-cess of loss treaty. The flexible model we propose allows the calculation of premium rates for all kinds of excess of loss treaties, even with specific clauses. We give a description of the methodology and we illustrate it with various numeri...

The Zero-Inflated Poisson model is extended into a bivariate form. Three new bivariate models are considered. Parameters are estimated by maximum likelihood. Two numerical tramples are discussed.

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. © 2000, Interna...

The 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 adjust...

The aim of this paper if to give some comments on two approximations used to price reinstatements related to excess of loss reinsurance. For the pro rate capita clause, we will study the rate on line method. For the pro rate temporis clause, we will study the use of a trivial approximation. The effect of an aggregate deductible is also looked at.

Sumario: The aim of this note is to extend Bissel's paper to more general compound Poisson distributions. For the particular example given by Bissel we are able to distinguish between a pure Poisson and a compound Poisson part for the number of breaks

For the construction of bonus-malus systems, we propose to show how to apply, thanks to simple mathematics, a parametric method encompassing those encountered in the literature. We also compare this parametric method with a non-parametric one that has not yet been used in the actuarial literature and that however permits a simple formulation of the...

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 Panjer’s formulae for compound distributions. An upper bound is given for the Kolmogorov distance between the true compound distribution and the transformed one. Advantages...

In this paper, we compare the point of view of the regulator and the investors about the required solvency level of an insurance company. We assume that the required solvency level is determined using the Tail Value at Risk and analyze the diversification benefit, both on the required capital and on the residual risk, when merging risks. To describ...

Count data reported over one period of time of- ten show overdispersion and infinite divisibility compared to the Poisson distribution. We propose a methodology which can be extended to pure birth processes derived from Pois- son processes. Our model reveals itself as the most general of this type. Our family encompasses many distributions anal- ys...

Sumario: In this paper we study a class of bivariate mixed Poisson distributions by extending the Hofmann's distribution from the univariate case to the bivariate case. We show how to evaluate the bivariate aggregate claims distribution and we adjust some insurance portfolio's given in the literature

RÉSUMÉ Nous proposons une méthodologie générale pour construire un système bonus-malus équilibré basé sur une fonction de perte exponentielle. Ce travail englobe les travaux sur le sujet et en simplifie les écritures. MOTS-CLÉS Processus de comptage, Poisson mélange, fonction de perte exponentielle, système bonus-malus.

We compare the actuarial and the financial approach for reserving and pricing for guaranteed minimum death benefits in unit-linked life insurance. In the first approach, no hedging strategy is applied, in the second, a dynamic hedging strat-egy based upon the Black and Scholes Model is used. In the financial approach, we incorporate hedging errors...

In this paper we study the impact of distributions with heavy tails and tail dependence on the underwriting pol- icy of an insurer. The aim is to maximize the value creation by varying the number of contracts an insurer sells, under the constraint that investors provide a given amount of economic capital. To model the underwriting risk, we use a mu...

In this paper, we investigate actuarial, nancial, economic and commercial aspects of the pricing of an excess of loss treaty. The exible model we propose allows the calculation of premium rates for all kinds of excess of loss treaties, even with specic clauses. We give a description of the methodology and we illustrate it with various numerical exa...

RESUME Cet article aborde une loi de comptage ignorée dans la littérature actuarielle, la distribution de Poisson-Katz. Celle-ci est obtenue en composant une loi de Katz avec une loi de Poisson. Un lien est fait avec la famille des lois de Panjer (classe (a,b,0)). Des formules récursives sont établies pour l'évaluation des probabilités ainsi que po...

## Citations

... These elements are used to determine how the insured transitions from one class to another, to compute for the right premium, and to apply rewards or penalization. Pitrebois et al. (2003) provided an analytic derivation of the model by Taylor (1997). Analytic formulae for transition probabilities and premium were made, and illustrations were provided. ...

... One of the main topics is the analysis of the problem of optimal coverage and deductible through expected utility (see [3,4] or [5]) and stochastic dominance [6,7]. The interaction between deductibles and bonus-malus systems and their repercussion on the efficiency of the bonusmalus system have been studied in [8][9][10], or [11]. Another topic is the optimal allocation of policy limits and deductibles from the viewpoint of a risk-averse policyholder [12,13] or from the viewpoint of the insurer [14]. ...

... The zibp regression [46] is used to model data that have an excess of counts in the (0, 0) cell. It is obtained as a mixture of a Bivariate Poisson (bp) model and a degenerate distribution at (0, 0). ...

... Therefore, these BMSs are Markovian. For that reason, the academic literature on BMSs uses extensively MCs for their modeling ( [1,[5][6][7][8][9][10][11]). Therefore, a key question in a BMS is fitting the value of the one-step transition probability matrix. ...

... Bonus hunger is known to be a prevalent phenomenon in insurance contracts within any bonus-malus rating system (Lemaire, 2012). It is also known that bonus-malus rating systems induce discrepancies between accident records and claim records (Lemaire, 1977;Walhin and Paris, 2000), which means that ground-up losses and claim amounts generally differ in practice. In this study, we argue that bonus-hunger behavior not only distorts the distributions of claim frequency and severity (Lemaire, 1977;Walhin and Paris, 2000) but can also be a factor in driving the dependence between the claim frequency and severity even when the original accident (ground-up loss) frequency and severity are independent under a given BMS. ...

... Some examples together with the used claim frequency distribution are: Lemaire (1995) (negative binomial), Tremblay (1992) (Poisson inverse Gauß), Walhin and Paris (1999) (Hofmann). Overviews on Bonus Malus systems can be found in the introductions of the following papers: Frangos and Vrontos (2001), Heras et al. (2004), Pitrebois et al. (2005. General Papers on BMS are: Bühlmann (1967), Bühlmann and Gisler (1997), Jewell and Schnieper (1985), Pinquet (1998), Schnieper (1995, Taylor (1997). ...

... Nevertheless, we note that in practice it is common for a BMS to adopt transition rules according to the claim history for the past multiple years in countries such as Belgium, Italy, Korea, and Singapore. In this paper, we revisit a modified BMS which was briefly introduced in Lemaire (1995) and Pitrebois et al. (2003a). Specifically, such a BMS extends the number of Bonus-Malus (BM) levels due to an additional component in the transition rules representing the number of consecutive claim-free years. ...

... Poisson, la Binomial y la Binomial Negativa con parámetros apropiados. Por otra parte, la familia de distribuciones de Hofmann, empleada también para modelar número de reclamos, permite generar la distribución de Poisson, la de Poisson Inversa Gaussianna, la Binomial Negativa y la de Polya-Aeppli (ver [15]). ...

... Split-Atom convolution requires that two pdfs being convolved have the main part of their supports discretized using the common span (grid step size) h s , see step 8 in Algorithm 1. Prior to convolution, either one or both of these pdfs may need arithmetization hereafter referred to as the regriding. In general, this operation takes the discrete pdf p X defined on fine scale support with the span h and determines the new arithmetic pdf p X on coarse scale support with the span h > h with the property ∑ x x m p X (x) = ∑ x x m p X (x ) of equating m moments with p X (see, e.g., Gerber 1982;Vilar 2000;Walhin and Paris 1998). ...

... The expected premium value was calculated based on the BMS principle. Several papers have discussed mixing other distributions to obtain an optimal bonus-malus premium for the claim frequency (Dionne and Vanasse 1992;Lemaire 1995;Walhin and Paris 1999;Tzougas and Frangos 2014;Tzougas et al. 2019;Tzougas 2020). However, premium payments based on the BMS show no difference between a claim made by a policyholder for a small loss and another with a big loss. ...