[pt] A preocupação com o aquecimento global e a poluição tem aumentado significativamente o interesse no desenvolvimento de fontes renováveis de energia. Este estudo tem como eixo principal a energia eólica, o uso dessa energia elimina resíduos indesejados e prejudiciais à saúde e ao meio ambiente causados por outras fontes de energia, como carvão e usinas nucleares. Este trabalho objetiva analisar a relação de dependência entre a velocidade do vento e a geração de energia eólica, esta é uma relação bastante complexa, por isso busca-se entender a natureza estocástica de ambas as variáveis. Como ferramenta metodológica foi utilizada a teoria da cópula. O estudo baseia-se na análise e modelagem da dependência entre dados de velocidade do vento e geração de energia eólica, para um banco de dados horário de um parque eólico do estado da Bahia, no período de janeiro a dezembro de 2017, após encontrar a cópula correspondente a estrutura de dependência para o ano completo e para cada mês individualmente, foram geradas simulações e apresentadas as probabilidades de ocorrência dos cenários em intervalos pré-definidos, os resultados obtidos foram significativos, testes estatísticos adequados foram realizados, evidenciando a qualidade do ajuste. / [en] Concern about global warming and pollution has significantly increased interest in developing renewable energy sources. This study has wind energy as its main axis, the use of this energy eliminates unwanted and harmful waste to health and the environment caused by other energy sources, such as coal and nuclear power plants. This work aims to analyze the dependence relationship between wind speed and wind energy generation, this is a very complex relationship, so we seek to understand the stochastic nature of both variables. As a methodological tool, the copula theory was used. The study is based on the analysis and modeling of the dependence between wind speed data and wind energy generation, for an hourly database of a wind farm in the state of Bahia, from January to December 2017, after finding the copula corresponding to the dependency structure for the entire year and for each month individually, simulations were generated and the probabilities of occurrence of the scenarios were presented at pre-defined intervals, the results obtained were significant, adequate statistical tests were performed, evidencing the quality of the fit .

Quantification of risks is one of the pillars of the contemporary insurance industry. Natural catastrophes and their modelling represents one of the most important areas of non-life insurance in the Czech Republic. One of the key inputs of catastrophe models is a spatial dependence structure in the portfolio of an insurance company. Copulas represents a more general view on dependence structures and broaden the classical approach, which is implicitly using the dependence structure of a multivariate normal distribution. The goal of this work, with respect to absence of comprehensive monographs in the Czech Republic, is to provide a theoretical basis for use of copulas. It focuses on general properties of copulas and specifics of two most commonly used families of copulas -- Archimedean and elliptical. The other goal is to quantify difference between the given copula and the classical approach, which uses dependency structure of a multivariate normal distribution, in modelled flood losses in the Czech Republic. Results are largely dependent on scale of losses in individual areas. If the areas have approximately a "tower" structure (i.e., one area significantly outweighs others), the effect of a change in the dependency structure compared to the classical approach is between 5-10% (up and down depending on a copula) at 99.5 percentile of original losses (a return period of once in 200 years). In case that all areas are approximately similarly distributed the difference, owing to the dependency structure, can be up to 30%, which means rather an important difference when buying the most common form of reinsurance -- an excess of loss treaty. The classical approach has an indisputable advantage in its simplicity with which data can be generated. In spite of having a simple form, it is not so simple to generate Archimedean copulas for a growing number of dimensions. For a higher number of dimensions the complexity of data generation greatly increases. For above mentioned reasons it is worth considering whether conditions of 2 similarly distributed variables and not too high dimensionality are fulfilled, before general forms of dependence are applied.

This thesis investigates correlations and linkages in credit risk that widely exist in all sectors of the financial markets. The main body of this thesis is constructed around four empirical chapters. I started with extending two main issues focused by earlier empirical studies on credit derivatives markets: the determinants of CDS spreads and the relationship between CDS spreads and bond yield spreads, with a special focus on the effect of the subprime crisis. By having observed that the linear relationship can not fully explain the variation in CDS spreads, the third empirical chapter investigated the dependence structure between CDS spread changes and market variables using a nonlinear copula method. The last chapter investigated the relationship between the CDS spread and another credit spread - the TED spread, in that a MVGARCH model and twelve copulas are set forth including three time varying copulas. The results of this thesis greatly enhanced our understanding about the effect of the subprime crisis on the credit default swap market, upon which a set of useful practical suggestions are made to policy makers and market participants.

Initialement, la théorie du risque supposait l’indépendance entre les différentes variables aléatoires et autres paramètres intervenant dans la modélisation actuarielle. De nos jours, cette hypothèse d’indépendance est souvent relâchée afin de tenir compte de possibles interactions entre les différents éléments des modèles. Dans cette thèse, nous proposons d’introduire des modèles de dépendance pour différents aspects de la théorie du risque. Dans un premier temps, nous suggérons l’emploi des copules comme structure de dépendance. Nous abordons tout d’abord un problème d’allocation de capital basée sur la Tail-Value-at-Risk pour lequel nous supposons un lien introduit par une copule entre les différents risques. Nous obtenons des formules explicites pour le capital à allouer à l’ensemble du portefeuille ainsi que la contribution de chacun des risques lorsque nous utilisons la copule Farlie-Gumbel-Morgenstern. Pour les autres copules, nous fournissons une méthode d’approximation. Au deuxième chapitre, nous considérons le processus aléatoire de la somme des valeurs présentes des sinistres pour lequel les variables aléatoires du montant d’un sinistre et de temps écoulé depuis le sinistre précédent sont liées par une copule Farlie-Gumbel-Morgenstern. Nous montrons comment obtenir des formes explicites pour les deux premiers moments puis le moment d’ordre m de ce processus. Le troisième chapitre suppose un autre type de dépendance causée par un environnement extérieur. Dans le contexte de l’étude de la probabilité de ruine d’une compagnie de réassurance, nous utilisons un environnement markovien pour modéliser les cycles de souscription. Nous supposons en premier lieu des temps de changement de phases de cycle déterministes puis nous les considérons ensuite influencés en retour par les montants des sinistres. Nous obtenons, à l’aide de la méthode d’erlangisation, une approximation de la probabilité de ruine en temps fini. / Initially, it was supposed in risk theory that the random variables and other parameters of actuarial models were independent. Nowadays, this hypothesis is often relaxed to take into account possible interactions. In this thesis, we propose to introduce some dependence models for different aspects of risk theory. In a first part, we use copulas as dependence structure. We first tackle a problem of capital allocation based on the Tail-Value-at-Risk where the risks are supposed to be dependent according to a copula. We obtain explicit formulas for the capital to be allocated to the overall portfolio but also for the contribution of each risk when we use a Farlie-Gumbel-Morenstern copula. For the other copulas, we give an approximation method. In the second chapter, we consider the stochastic process of the discounted aggregate claims where the random variables for the claim amount and the time since the last claim are linked by a Farlie-Gumbel-Morgenstern copula. We show how to obtain exact expressions for the first two moments and for the moment of order m of the process. The third chapter assumes another type of dependence that is caused by an external environment. In the context of the study of the ruin probability for a reinsurance company, we use a Markovian environment to model the underwriting cycles. We suppose first deterministic cycle phase changes and then that these changes can also be influenced by the claim amounts. We use the erlangization method to obtain an approximation for the finite time ruin probability.

Neste trabalho são apresentados modelos de regressão com respostas bivariadas obtidos através de funções cópulas. O objetivo de utilizar estes modelos bivariados é modelar a correlação entre eventos e captar nos modelos de regressão a influência da associação entre as variáveis resposta na presença de censura nos dados. Os parâmetros dos modelos, são estimados por meio dos métodos de máxima verossimilhança e jackknife. Alguns métodos de análise de sensibilidade como influência global, local e local total de um indivíduo, são introduzidos e calculados considerando diferentes esquemas de perturbação. Uma análise de resíduos foi proposta para verificar a qualidade do ajuste dos modelos utilizados e também foi proposta novas medidas de resíduos para respostas bivariadas. Métodos de simulação de Monte Carlo foram conduzidos para estudar a distribuição empírica dos resíduos marginais e bivariados propostos. Finalmente, os resultados são aplicados à dois conjuntos de dados dsponíveis na literatura. / In this work bivariate response regression models are presented with the use of copulas. The objective of this approach is to model the correlation between events and capture the influence of this correlation in the regression parameters. The models are used in the context of survival analysis and are ¯tted to two data sets available in the literature. Inferences are obtained using maximum likelihood and Jackknife methods. Sensitivity techniques such as local and global in°uence are proposed and calculated. A residual analysis is proposed to check the adequacy of the models and simulation methods are used to asses the empirical distribution of the marginal univariate and bivariate residual measures proposed.

The aim of this thesis is to provide an overview of methods used in cause-of-death mortality analysis and to demonstrate the application on real data. In Chapter 1 we present the continuous model based on the force of mortality and review the approach using copula functions. In Chapter 2 we focus on the multinomial logit model formulated for cause-specific mortality data, discuss life tables construction and derive life expectancy. In Chapter 3 we apply the multinomial logit model on the data from Czech Statistical Office. We identify the regression model, check its assumptions, present the outputs including the fitted life expectancy, and predicted mortality rates. Later in Chapter 3 we consider several stress scenarios in order to demonstrate the impact of shocked mortality rates on the life expectancy.

[pt] Esta dissertação tem por objetivo investigar como a crise do subprime impactou a estrutura de dependência entre os mercados emergentes e desenvolvidos, utilizando como proxy para esses mercados os índices MSCI (Morgan Stanley Capital International). A metodologia proposta é baseada na construção de distribuições bivariadas através de cópulas condicionais. A distribuição marginal de cada um dos índices é obtida via ajuste de modelos GARCH univariados e a modelagem de dependência é realizada através das cópulas normal, normal GAS (Generalised Autoregressive Score) e Joe- Clayton simétrica, considerando os parâmetros fixos (forma estática) ou variantes no tempo (forma dinâmica). Diante dos resultados obtidos, a cópula normal GAS (variantes no tempo) com quebra estrutural se mostrou a mais adequada para capturar a dependência entre os retornos dos mercados emergentes e desenvolvidos. Através do arcabouço utilizado pode-se verificar que as medidas de correlação e de dependência de cauda entre os mercados emergentes e desenvolvidos aumentaram significativamente no período da crise do suprime. Por fim, avaliou-se o ajuste das diversas cópulas aqui propostas via VaR (Value at Risk), verificando-se que a cópula normal GAS apresentou o melhor ajuste. / [en] The aim of this dissertation is to analyze how the subprime crisis impacted the dependence structure among the emerging and developed markets by using the MSCI (Morgan Stanley Capital International) market index as proxy for each of these markets. The proposed methodology is based on the construction of bivariate distributions via conditional copulas. The marginal distribution for each of the indexes makes use of univariate GARCH models and model dependence is captured via the following copulas: normal, normal GAS (Generalised Autoregressive Score) and Joe-Clayton symmetric considering both fixed parameters (static framework) and time varying parameters (dynamic framework). Our results show that the normal GAS copula with structural break was the most adequate to capture dependence between the returns of emerging and developed markets. Throughout the proposed framework it was possible to infer that correlation and tail dependence measures between these markets sharply increased during the subprime crisis. Finally VaR (Value at Risk) coverage was used as goodness of fit measure, and on this metric the normal GAS copula has also outperformed the others.

A copula is the representation of a multivariate distribution. Copulas are used to model multivariate data in many fields. Recent developments include copula models for spatial data and for discrete marginals. We will present a new methodological approach for modeling discrete spatial processes and for predicting the process at unobserved locations. We employ Bayesian methodology for both estimation and prediction. Comparisons between the new method and Generalized Additive Model (GAM) are done to test the performance of the prediction. Although there exists a large variety of copula functions, only a few are practically manageable and in certain problems one would like to choose the Gaussian copula to model the dependence. Furthermore, most copulas are exchangeable, thus implying symmetric dependence. However, none of them is flexible enough to catch the tailed (upper tailed or lower tailed) distribution as well as elliptical distributions. An elliptical copula is the copula corresponding to an elliptical distribution by Sklar's theorem, so it can be used appropriately and effectively only to fit elliptical distributions. While in reality, data may be better described by a "fat-tailed" or "tailed" copula than by an elliptical copula. This dissertation proposes a novel pseudo-copula (the modified Gaussian pseudo-copula) based on the Gaussian copula to model dependencies in multivariate data. Our modified Gaussian pseudo-copula differs from the standard Gaussian copula in that it can model the tail dependence. The modified Gaussian pseudo-copula captures properties from both elliptical copulas and Archimedean copulas. The modified Gaussian pseudo-copula and its properties are described. We focus on issues related to the dependence of extreme values. We give our pseudo-copula characteristics in the bivariate case, which can be extended to multivariate cases easily. The proposed pseudo-copula is assessed by estimating the measure of association from two real data sets, one from finance and one from insurance. A simulation study is done to test the goodness-of-fit of this new model. / Graduation date: 2012

Graphical models use Markov properties to establish associations among dependent variables. To estimate spatial correlation and other parameters in graphical models, the conditional independences and joint probability distribution of the graph need to be specified. We can rely on Gaussian multivariate models to derive the joint distribution when all the nodes of the graph are assumed to be normally distributed. However, when some of the nodes are discrete, the Gaussian model no longer affords an appropriate joint distribution function. We develop methods specifying the joint distribution of a chain graph with both discrete and continuous components, with spatial dependencies assumed among all variables on the graph. We propose a new group of chain graphs known as the generalized tree networks. Constructing the chain graph as a generalized tree network, we partition its joint distributions according to the maximal cliques. Copula models help us to model correlation among discrete variables in the cliques. We examine the method by analyzing datasets with simulated Gaussian and Bernoulli Markov random fields, as well as with a real dataset involving household income and election results. Estimates from the graphical models are compared with those from spatial random effects models and multivariate regression models. / Graduation date: 2013

This thesis addresses three topics in the area of statistics and probability, with applications in risk management. First, for the testing problems in the high-dimensional (HD) data analysis, we present a novel method to formulate empirical likelihood tests and jackknife empirical likelihood tests by splitting the sample into subgroups. New tests are constructed to test the equality of two HD means, the coefficient in the HD linear models and the HD covariance matrices. Second, we propose jackknife empirical likelihood methods to formulate interval estimations for important quantities in actuarial science and risk management, such as the risk-distortion measures, Spearman's rho and parametric copulas. Lastly, we introduce the theory of completely mixable (CM) distributions. We give properties of the CM distributions, show that a few classes of distributions are CM and use the new technique to find the bounds for the sum of individual risks with given marginal distributions but unspecific dependence structure. The result partially solves a problem that had been a challenge for decades, and directly leads to the bounds on quantities of interest in risk management, such as the variance, the stop-loss premium, the price of the European options and the Value-at-Risk associated with a joint portfolio.