The increase in the use of copulas has introduced implementation issues for both practitioners and researchers. One of the issues is to obtain a copula function for a given set of data. The most common approaches for the estimation of the parameters of the copula functions have been the Maximum Likelihood Estimator (MLE) and the Inference Functions for Margins (IFM) methods. Archimedean copulas are one of the most important classes of copulas that are widely used in both finance and insurance for modelling dependent risks. However, simulating multivariate Archimedean copulas has always been a difficult task as the number of dimensions increases. The assessment of capital requirements has always been an important application of stochastic modelling. Capital requirements can vary significantly depending on the model adopted. Several professional bodies have recently discussed the concept of dependencies between insurance risks. They suggest that insurers should use a technique based on copulas to describe the dependence of risks within an insurance company in the context of solvency assessment. The first contribution of this thesis is to provide an insight into the efficiency of parameter estimation methods. This thesis uses numerical experiments to assess the performance of the two common approaches. The second contribution of this thesis is to present a new algorithm to simulate multivariate Exchangeable Archimedean copulas. This algorithm provides a practical solution for simulating one-parameter multivariate Archimedean copulas. Numerical experiments are used to apply this algorithm to determine the "additional" economic capital for an insurance company with multiple lines of business that wants to expand its business by adding another line of business and where the businesses are dependent. The third contribution of this thesis is to quantify the impact of the choice of copulas on the solvency measure of a general insurer within a Dynamic Financial Analysis modelling framework. The results of our experiments provide important guidance for the capital assessment for general insurers.

The increase in the use of copulas has introduced implementation issues for both practitioners and researchers. One of the issues is to obtain a copula function for a given set of data. The most common approaches for the estimation of the parameters of the copula functions have been the Maximum Likelihood Estimator (MLE) and the Inference Functions for Margins (IFM) methods. Archimedean copulas are one of the most important classes of copulas that are widely used in both finance and insurance for modelling dependent risks. However, simulating multivariate Archimedean copulas has always been a difficult task as the number of dimensions increases. The assessment of capital requirements has always been an important application of stochastic modelling. Capital requirements can vary significantly depending on the model adopted. Several professional bodies have recently discussed the concept of dependencies between insurance risks. They suggest that insurers should use a technique based on copulas to describe the dependence of risks within an insurance company in the context of solvency assessment. The first contribution of this thesis is to provide an insight into the efficiency of parameter estimation methods. This thesis uses numerical experiments to assess the performance of the two common approaches. The second contribution of this thesis is to present a new algorithm to simulate multivariate Exchangeable Archimedean copulas. This algorithm provides a practical solution for simulating one-parameter multivariate Archimedean copulas. Numerical experiments are used to apply this algorithm to determine the "additional" economic capital for an insurance company with multiple lines of business that wants to expand its business by adding another line of business and where the businesses are dependent. The third contribution of this thesis is to quantify the impact of the choice of copulas on the solvency measure of a general insurer within a Dynamic Financial Analysis modelling framework. The results of our experiments provide important guidance for the capital assessment for general insurers.

Credibility theory is widely used in insurance. It is included in the examination of the Society of Actuaries and in the construction and evaluation of actuarial models. In particular, the Buhlmann credibility model has played a fundamental role in both actuarial theory and practice. It provides a mathematical rigorous procedure for deciding how much credibility should be given to the actual experience rating of an individual risk relative to the manual rating common to a particular class of risks. However, for any selected risk, the Buhlmann model assumes that the outcome random variables in both experience periods and future periods are independent and identically distributed. In addition, the Buhlmann method uses sample mean-based estimators to insure the selected risk, which may be a poor estimator of future costs if only a few observations of past events (costs) are available. We present an extension of the Buhlmann model and propose a general method based on a linear combination of both robust and efficient estimators in a dependence framework. The performance of the proposed procedure is demonstrated by Monte Carlo simulations.

Depuis longtemps les modèles de type phase sont utilisés dans plusieurs domaines scientifiques pour décrire des systèmes qui peuvent être caractérisés par différents états. Les modèles sont bien connus en théorie des files d’attentes, en économie et en assurance.<p><p>La thèse est focalisée sur différentes applications des modèles de type phase en assurance et montre leurs avantages. En particulier, le modèle de Lin et Liu en 2007 est intéressant, parce qu’il décrit le processus de vieillissement de l’organisme humain. La durée de vie d’un individu suit une loi de type phase et les états de ce modèle représentent des états de santé. Le fait que le modèle prévoit la connexion entre les états de santé et l’âge de l’individu le rend très utile en assurance.<p><p>Les résultats principaux de la thèse sont des nouveaux modèles et méthodes en assurance pension et en assurance santé qui utilisent l’hypothèse de la loi de type phase pour décrire la durée de vie d’un individu.<p><p>En assurance pension le but d’estimer la profitabilité d’un fonds de pension. Pour cette raison, on construit un modèle « profit-test » qui demande la modélisation de plusieurs caractéristiques. On décrit l’évolution des participants du fonds en adaptant le modèle du vieillissement aux causes multiples de sortie. L’estimation des profits futurs exige qu’on détermine les valeurs des cotisations pour chaque état de santé, ainsi que l’ancienneté et l’état de santé initial pour chaque participant. Cela nous permet d’obtenir la distribution de profits futurs et de développer des méthodes pour estimer les risques de longevité et de changements de marché. De plus, on suppose que la diminution des taux de mortalité pour les pensionnés influence les profits futurs plus que pour les participants actifs. C’est pourquoi, pour évaluer l’impact de changement de santé sur la profitabilité, on modélise séparément les profits venant des pensionnés.<p><p>En assurance santé, on utilise le modèle de type phase pour calculer la distribution de la valeur actualisée des coûts futurs de santé. On développe des algorithmes récursifs qui permettent d’évaluer la distribution au cours d’une période courte, en utilisant des modèles fluides en temps continu, et pendant toute la durée de vie de l’individu, en construisant des modèles en temps discret. Les trois modèles en temps discret correspondent à des hypothèses différentes qu’on fait pour les coûts: dans le premier modèle on suppose que les coûts de santé sont indépendants et identiquement distribués et ne dépendent pas du vieillissement de l’individu; dans les deux autres modèles on suppose que les coûts dépendent de son état de santé.<p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

In this work we study two problems motivated by Risk Management: the optimal design of financial products from an investor's point of view and the calculation of bounds and approximations for sums involving non-independent random variables. The element that interconnects these two topics is the notion of conditioning, a fundamental concept in probability and statistics which appears to be a useful device in finance. In the first part of the dissertation, we analyse structured products that are now widespread in the banking and insurance industry. These products typically protect the investor against bearish stock markets while offering upside participation when the markets are bullish. Examples of these products include capital guaranteed funds commercialised by banks, and equity linked contracts sold by insurers. The design of these products is complex in general and it is vital to examine to which extent they are actually interesting from the investor's point of view and whether they cannot be dominated by other strategies. In the academic literature on structured products the focus has been almost exclusively on the pricing and hedging of these instruments and less on their performance from an investor's point of view. In this work we analyse the attractiveness of these products. We assess the theoretical cost of inefficiency when buying a structured product and describe the optimal strategy explicitly if possible. Moreover we examine the cost of the inefficiency in practice. We extend the results of Dybvig (1988a, 1988b) and Cox & Leland (1982, 2000) who in the context of a complete, one-dimensional market investigated the inefficiency of path-dependent pay-offs. In the dissertation we consider this problem in one-dimensional Levy and multidimensional Black-Scholes financial markets and we provide evidence that path-dependent pay-offs should not be preferred by decision makers with a fixed investment horizon, and they should buy path-independent structures instead. In these market settings we also demonstrate the optimal contract that provides the given distribution to the consumer, and in the case of risk- averse investors we are able to propose two ways of improving the design of financial products. Finally we illustrate the theory with a few well-known securities and strategies e.g. dollar cost averaging, buy-and-hold investments and widely used portfolio insurance strategies. The second part of the dissertation considers the problem of finding the distribution of a sum of non- independent random variables. Such dependent sums appear quite often in insurance and finance, for instance in case of the aggregate claim distribution or loss distribution of an investment portfolio. An interesting avenue to cope with this problem consists in using so-called convex bounds, studied by Dhaene et al. (2002a, 2002b), who applied these to sums of log-normal random variables. In their papers they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In the dissertation we prove that unlike the log-normal case the construction of a convex lower bound in explicit form appears to be out of reach for general sums of log-elliptical risks and we show how we can construct stop-loss bounds and we use these to construct mean preserving approximations for general sums of log-elliptical distributions in explicit form. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Cette thèse traite de processus de détérioration en théorie du risque et en biomathématique.<p><p>En théorie du risque, le processus de détérioration étudié est celui des sinistres supportés par une compagnie d'assurance.<p><p>Le premier chapitre examine la distribution de Markov-Polya comme loi possible pour modéliser le nombre de sinistres et établit certains liens avec la famille de lois de Katz/Panjer. Nous construisons la loi de Markov-Polya sur base d'un modèle de survenance des sinistres et nous montrons qu'elle satisfait une récurrence élégante. Celle-ci permet notamment de déduire un algorithme efficace pour la loi composée correspondante. Nous déduisons la famille de Katz/Panjer comme famille limite de la loi de Markov-Polya.<p><p>Le second chapitre traite de la famille dite "Lagrangian Katz" qui étend celle de Katz/Panjer. Nous motivons par un problème de premier passage son utilisation comme loi du nombre de sinistres. Nous caractérisons toutes les lois qui en font partie et nous déduisons un algorithme efficace pour la loi composée. Nous examinons également son indice de dispersion ainsi que son comportement asymptotique. <p><p>Dans le troisième chapitre, nous étudions la probabilité de ruine sur horizon fini dans un modèle discret avec taux d'intérêt positifs. Nous déterminons un algorithme ainsi que différentes bornes pour cette probabilité. Une borne particulière nous permet de construire deux mesures de risque. Nous examinons également la possibilité de faire appel à de la réassurance proportionelle avec des niveaux de rétention égaux ou différents sur les périodes successives.<p><p>Dans le cadre de processus épidémiques, la détérioration étudiée consiste en la propagation d'une maladie de type SIE (susceptible - infecté - éliminé). La manière dont un infecté contamine les susceptibles est décrite par des distributions de survie particulières. Nous en déduisons la distribution du nombre total de personnes infectées à la fin de l'épidémie. Nous examinons en détails les épidémies dites de type Markov-Polya et hypergéométrique. Nous approximons ensuite cette loi par un processus de branchement. Nous étudions également un processus de détérioration similaire en théorie de la fiabilité où le processus de détérioration consiste en la propagation de pannes en cascade dans un système de composantes interconnectées. <p><p><p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished