Évaluation et allocation du risque dans le cadre de modèles avancés en actuariat

Dans cette thèse, on s’intéresse à l’évaluation et l’allocation du risque dans le cadre de modèles avancés en actuariat. Dans le premier chapitre, on présente le contexte général de la thèse et on introduit les différents outils et modèles utilisés dans les autres chapitres. Dans le deuxième chapitre, on s’intéresse à un portefeuille d’assurance dont les composantes sont dépendantes. Ces composantes sont distribuées selon une loi mélange d’Erlang multivariée définie à l’aide de la copule Farlie-Gumbel-Morgenstern (FGM). On évalue le risque global de ce portefeuille ainsi que l’allocation du capital. En utilisant certaines propriétés de la copule FGM et la famille de distributions mélange d’Erlang, on obtient des formules explicites de la covariance entre les risques et de la Tail-Value-at-Risk du risque global. On détermine aussi la contribution de chacun des risques au risque global à l’aide de la régle d’allocation de capital basée sur la Tail-Value-at-Risk et celle basée sur la covariance. Dans le troisième chapitre, on évalue le risque pour un portefeuille sur plusieurs périodes en utilisant le modèle de Sparre Andersen. Pour cette fin, on étudie la distribution de la somme escomptée des ladder heights sur un horizon de temps fini ou infini. En particulier, on trouve une expression ferme des moments de cette distribution dans le cas du modèle classique Poisson-composé et le modèle de Sparre Andersen avec des montants de sinistres distribués selon une loi exponentielle. L’élaboration d’une expression exacte de ces moments nous permet d’approximer la distribution de la somme escomptée des ladder heights par une distribution mélange d’Erlang. Pour établir cette approximation, nous utilisons une méthode basée sur les moments. À l’aide de cette approximation, on calcule les mesures de risque VaR et TVaR associées à la somme escomptée des ladder heights. Dans le quatrième chapitre de cette thèse, on étudie la quantification des risques liés aux investissements. On élabore un modèle d’investissement qui est constitué de quatre modules dans le cas de deux économies : l’économie canadienne et l’économie américaine. On applique ce modèle dans le cadre de la quantification et l’allocation des risques. Pour cette fin, on génère des scénarios en utilisant notre modèle d’investissement puis on détermine une allocation du risque à l’aide de la règle d’allocation TVaR. Cette technique est très flexible ce qui nous permet de donner une quantification à la fois du risque d’investissement, risque d’inflation et le risque du taux de change.In this thesis, we are interested in risk evaluation and risk allocation blems using advanced actuarial models. First, we investigate risk aggregation and capital allocation problems for a portfolio of possibly dependent risks whose multivariate distribution is defined with the Farlie-Gumbel-Morgenstern copula and with mixed Erlang distributions for the marginals. In such a context, we first show that the aggregate claim amount has a mixed Erlang distribution. Based on a top-down approach, closed-form expressions for the contribution of each risk are derived using the TVaR and covariance rules. These findings are illustrated with numerical examples. Then, we propose to investigate the distribution of the discounted sum of ascending ladder heights over finite- or infinite-time intervals within the Sparre Andersen risk model. In particular, the moments of the discounted sum of ascending ladder heights over a finite- and an infinite-time intervals are derived in both the classical compound Poisson risk model and the Sparre Andersen risk model with exponential claims. The application of a particular Gerber-Shiu functional is central to the derivation of these results, as is the mixed Erlang distributional assumption. Finally, we define VaR and TVaR risk measures in terms of the discounted sum of ascending ladder heights. We use a moment-matching method to approximate the distribution of the discounted sum of ascending ladder heights allowing the computation of the VaR and TVaR risk measures. In the last chapter, we present a stochastic investment model (SIM) for international investors. We assume that investors are allowed to hold assets in two different economies. This SIM includes four components: interest rates, stocks, inflation and exchange rate models. First, we give a full description of the model and we detail the parameter estimation. The model is estimated using a state-space formulation and an extended Kalman filter. Based on scenarios generated from this SIM, we study the risk allocation to different background risks: asset, inflation and exchange rate risks. The risk allocation is based on the TVaR-based rule

On the moments of the aggregate discounted claims with a general dependence

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Bivariate Sarmanov phase-type distributions for joint lifetimes modeling

Abstract: In this paper, we are interested in the dependence between lifetimes based on a joint survival model. This model is built using the bivariate Sarmanov distribution with Phase-Type marginal distributions. Capitalizing on these two classes of distributions' mathematical properties, we drive some useful closed-form expressions of distributions and quantities of interest in the context of multiple-life insurance contracts. The dependence structure that we consider in this paper is based on a general form of kernel function for the Bivariate Sarmanov distribution. The introduction of this new kernel function allows us to improve the attainable correlation range

Rational distorted beliefs investor; which risk matters?

This paper considers a portfolio strategy in which the investor applies beliefs distortion to allocate different weights to different kinds of risk in the context of Markovitz portfolio. We assume that the investor decomposes the aggregate risk into three different components:fundamental observed factors’ risk, fundamental non-observed factors’ risk and idiosyncratic risk. Our distorted beliefs’ portfolio outperforms the traditional mean–variance and the naive portfolios. We also find that the rational investor prefers to bear more observed fundamental risk and minimize other risks

Systematic extreme potential gain and loss spillover across countries

This paper investigates the existence of systematic extreme risks at a multi-country level that leads to gains and losses spillover. A measure of systematic risk that quantifies both the downside risk and the upside potential in the extreme is introduced. This measure is based on the Conditional-Value-at-Risk (CoVaR) measure and copulas to capture dependencies. Using our approach, we study the contagion effect between different financial markets in the extreme. We show that there is an asymmetric contagion effect from the US stock market to other international markets. The impact is higher when the US market is extremely bear than when it is extremely bull. This paper adds novel findings on the asymmetry between extreme losses and extreme gains and the differences among different countries’ reactions to shocks

Moments of Compound Renewal Sums with Dependent Risks Using Mixing Exponential Models

In this paper, we study the discounted renewal aggregate claims with a full dependence structure. Based on a mixing exponential model, the dependence among the inter-claim times, the claim sizes, as well as the dependence between the inter-claim times and the claim sizes are included. The main contribution of this paper is the derivation of the closed-form expressions for the higher moments of the discounted aggregate renewal claims. Then, explicit expressions of these moments are provided for specific copulas families and some numerical illustrations are given to analyze the impact of dependency on the moments of the discounted aggregate amount of claims