This project aims to analyze the risk measures Value-at-Risk and Conditional-Value-at-Risk for three stock portfolios with the purpose of evaluating each method's accuracy in modelling Black Swan events. This is achieved by utilizing a parametric approach in the form of a modified (C)VaR with a Cornish-Fisher expansion, a historic approach with a time series spanning ten years and a Markov Monte Carlo simulation modeled with a Brownian motion. From this, it is revealed that the parametric approach at the 99\%-level generates the most favorable results for a 30-day-(C)VaR estimation for each portfolio, followed by the historic approach and, lastly, the Markov Monte Carlo simulation. As such, it is concluded that the parametric approach may serve as a method of evaluating a portfolio's exposure to Black Swan events. / Denna rapport syftar till att analysera riskmåtten Value-at-Risk och Conditional-Value-at-Risk för tre aktieportföljer med målet att utvärdera respektive metods precision i att modellera Black Swan-händelser. Detta uppnås genom att utnyttja en parametrisk metod som tar formen av en modifierad (C)VaR med en Cornish-Fisher-utveckling, en historisk metod med en tidsserie som sträcker sig tio år, och en Markov Monte Carlo-simulering modellerat med en Brownian Motion. Från detta påvisas det att den parametriska metoden vid en 99\%-ig nivå genererar de mest rättvisande resultaten för en 30-dagars-(C)VaR-estimering för respektive portfölj, följt av den historiska metoden och, till sist, Monte Carlo-simulering. På så sätt dras slutsatsen att den parametriska metoden skulle kunna tjäna som en metod för att utvärdera en aktieportföljs exponering till Black Swan-händelser.

Futures contracts represent a suitable instrument for hedging. One conse- quence of their standardized nature is the presence of basis risk. In order to mitigate it an agent might aim to minimize Value at Risk or Expected Shortfall. Among numerous approaches to their modelling, CAViaR models which build upon quantile regression are appealing due to the limited set of assumptions and decent empirical performance. We propose alternative specifications for CAViaR model - power and exponential CAViaR, and an alternative, flexible way of computing Expected Shortfall within CAViaR framework - Implied Expectile Level. Empirical analysis suggests that ex- ponential CAViaR yields competitive results both in Value at Risk and Ex- pected Shortfall modelling and in subsequent Value at Risk and Expected Shortfall hedging. 1

Durante os últimos anos, tem havido muitas mudanças na maneira como as instituições financeiras avaliam o risco. As regulações têm tido um papel muito importante no desenvolvimento das técnicas de medição do risco. Diante das diversidades de técnicas de estimação e análise de risco utilizadas pelas bolsas de valores e de futuros, nacionais e internacionais, bem como as Clearings de controle de risco, este estudo propôs uma análise comparativo de modelos de volatilidade para o cálculo do Value-at-Risk (VaR) aplicados aos principais índices de ações do mercado financeiro internacional. Utilizouse os modelos de volatilidade condicional da família ARCH levando em consideração a presença de longa dependência em seus retornos (memória longa) e assimetria na volatilidade. Em específico, utilizaram-se os modelos GARCH, EGARCH, APARCH, FIGARCH, FIEGARCH, FIAPARCH e HYGARCH estimados a parir de quatro diferentes distribuições, Normal, t-Student, G.E.D. e t-Student Assimétrica. Analisaramse os índices dos principais mercados de ações do mundo, sendo: Dow Jones, S&P 500, Nasdaq, Ibovespa, FTSE e Nikkei 225. Testou-se também a capacidade preditiva do modelo Riskmetrics desenvolvido pelo J.P. Morgan para o calculo do VaR, comparado com os modelos de volatilidade. Os resultados obtidos sugerem que o pacote desenvolvido pelo J.P.Morgan não se aplica adequadamente à realidade do mercado acionário mundial, como ferramenta de gestão e controle do risco das oscilações dos preços das ações de empresas negociadas nas bolsas de Nova Iorque, Nasdaq, BM&FBOVESPA, bolsa de Londres e bolsa de Tóquio. Os modelos que consideram o efeito de memória longa na volatilidade condicional dos retornos dos índices, em especial o modelo FIAPARCH (1,d,1), foram os que obtiveram melhor ajuste e desempenho preditivo do risco de mercado (Value-at-Risk), conforme valores apresentados pelo teste de razão de falha proposto por Kupiec (1995). / In recent years, there have been many changes in how financial institutions assess risk. The regulations have had a very important role in the development of techniques for measuring risk. Considering the diversity of estimation techniques and risk analysis used by stock exchanges and futures, national and international, as well as clearing houses of risk control, this study proposed a comparative analysis of volatility models for calculating Value-at-Risk (VaR) to the major stock indexes of international finance. It used models of conditional volatility of the ARCH family taking into account the presence of long dependence on their returns (long memory) and asymmetry in volatility. Specifically, it used the models GARCH, EGARCH, APARCH, FIGARCH, FIEGARCH, FIAPARCH and HYGARCH estimated the birth of four different distributions, Normal, t-Student, GED and t-Student Asymmetric. It analyzed the contents of the major stock markets of the world, being: Dow Jones, S & P 500, NASDAQ, Bovespa index, FTSE and Nikkei 225. Was also tested the predictive ability of the RiskMetrics model developed by JP Morgan for the calculation of VaR, compared with the models of volatility. The results suggest that the package developed by JPMorgan does not apply adequately to the reality of global stock market as a tool to manage and control the risk of fluctuations in stock prices of companies traded on the New York Stock Exchange, Nasdaq, BM&FBOVESPA, London Stock Exchange and Tokyo Stock Exchange. Models that consider the effect of long memory in conditional volatility of returns of the indices, especially the model FIAPARCH (1, d, 1), were the ones showing better fit and predictive performance of market risk (Value-at-Risk) , according to figures provided by the ratio test proposed by Kupiec (1995).

Hedging has been one of the most important topics in finance. How to effectively hedge the exposed risk draws significant interest from both academicians and practitioners. In a complete financial market, every contingent claim can be hedged perfectly. In an incomplete market, the investor can eliminate his risk exposure by superhedging. However, both perfect hedging and superhedging usually call for a high cost. In some situations, the investor does not have enough capital or is not willing to spend that much to achieve a zero risk position. This brings us to the topic of partial hedging. In this thesis, we establish the risk measure based partial hedging model and study the optimal partial hedging strategies under various criteria. First, we consider two of the most common risk measures known as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We derive the analytical forms of optimal partial hedging strategies under the criterion of minimizing VaR of the investor's total risk exposure. The knock-out call hedging strategy and the bull call spread hedging strategy are shown to be optimal among two admissible sets of hedging strategies. Since VaR risk measure has some undesired properties, we consider the CVaR risk measure and show that bull call spread hedging strategy is optimal under the criterion of minimizing CVaR of the investor's total risk exposure. The comparison between our proposed partial hedging strategies and some other partial hedging strategies, including the well-known quantile hedging strategy, is provided and the advantages of our proposed partial hedging strategies are highlighted. Then we apply the similar approaches in the context of reinsurance. The VaR-based optimal reinsurance strategies are derived under various constraints. Then we study the optimal partial hedging strategies under general risk measures. We provide the necessary and sufficient optimality conditions and use these conditions to study some specific hedging strategies. The robustness of our proposed CVaR-based optimal partial hedging strategy is also discussed in this part. Last but not least, we propose a new method, simulation-based approach, to formulate the optimal partial hedging models. By using the simulation-based approach, we can numerically obtain the optimal partial hedging strategy under various constraints and criteria. The numerical results in the examples in this part coincide with the theoretical results.

As entidades fechadas de previdência complementar (EFPC) possuem significativa relevância na economia brasileira com seus ativos dos fundos de pensão representando 16,8% do PIB em dezembro de 2009. O sistema de gerenciamento de risco dos fundos de pensão ainda não evoluiu na mesma proporção em que evoluiu em outros segmentos do mercado financeiro brasileiro. Para atender suas demandas de gerenciamento de risco, os fundos de pensão têm utilizado os modelos propostos para as instituições financeiras; tais modelos, contudo, não chegam a atender integralmente às suas necessidades. Os órgãos reguladores do setor têm estimulado os fundos de pensão a utilizarem seus próprios modelos para estimar a volatilidade e o Value at Risk (VaR). O objetivo do trabalho é propor uma modelagem de risco a partir da volatilidade estocástica (SV) para o cálculo do Value at Risk (VaR), comparando-a com a volatilidade calculada pela EWMA, proposta pelo Risk Metrics . A aplicação empírica do modelo foi efetuada a partir de uma amostra de uma série de retornos da carteira de uma entidade fechada de previdência complementar (EFPC) - fundo de pensão, a Indusprevi - Sociedade de Previdência Privada do Rio Grande do Sul. A amostra utilizada corresponde às cotas diárias entre o período de 01 de abril de 2004 até 31 de dezembro de 2009, representando 1.439 observações diárias. Os resultados apurados para a amostra demonstraram que a volatilidade estocástica (SV) tende a gerar um Value at Risk (VaR) mais conservador que o calculado a partir da metodologia do EWMA, quando testado pelo Teste de Kupiec (1995) e pela realização de Back testing. Tal fato, no entanto, torna o modelo mais adequado à realidade da Indusprevi e de uma grande maioria de outros fundos, que tendem a adotar políticas de investimentos mais conservadoras. / Pension funds have significant relevance to the Brazilian economy with assets representing, in December 2009, 16.8% of GDP. The pension funds risk management system has not evolved in the same pace as other sectors of the Brazilian financial market. To meet their demands for risk management, pension funds have employed the models proposed for financial institutions. Such models, however, fail to fully satisfy their needs. Government regulators have encouraged pension funds to use their own models so as to estimate volatility and Value at Risk (VaR). The main objective of this thesis is to propose a model of risk based on stochastic volatility (SV) to calculate the Value at Risk (VaR), as well as comparing it with the volatility estimated by EWMA, proposed by Risk MetricsTM. The empirical application of the model was made on a sample of portfolio returns of the pension fund Indusprevi - Sociedade de Previdência Privada do Rio Grande do Sul. The sample comprises 1439 daily quotes during the period April 1, 2004 to December 31, 2009. The results showed that the stochastic volatility (SV) tends to generate a more conservative Value at Risk (VaR) than the EWMA method when applying both the Kupiec (1995) test and back testing. This fact, therefore, makes the model more suitable to the principles of Indusprevi as well as a large majority of other funds, which tend to adopt more conservative investment policies.

Dans cette thèse, nous examinons les apports importants de la modélisation de la dépendance par la théorie des copules dans le cadre des problématiques liées à la gestion des portefeuilles financiers et des produits structurés de crédits.La première partie de cette thèse est consacrée à la gestion des portefeuilles financiers. Nous étudions en premier lieu la relation qui peut s'établir entre le niveau de diversification du portefeuille et le choix de la copule ayant décrit au mieux la structure de dépendance. L'objectif est de pouvoir identifier une caractéristique dans les portefeuilles permettant une sélection plus simple de la copule appropriée. Dans un deuxième chapitre, nous proposons d'étudier l'impact d'une mauvaise spécification du modèle de copule sur les estimations des mesures de risque conventionnelles telles la Value-at-Risk et l'Expected-Shortfall. L'idée est de vérifier l'utilité de développer ces estimations sous l'angle du vrai modèle copule. Dans un troisième chapitre, nous étudions l'impact d'une mauvaise spécification du modèle copule dans le cadre d'un problème d'allocation optimale de portefeuilles. L'objectif principal vise surtout à identifier la sensibilité des investisseurs, en fonction de leurs degrés d'aversion aux risques (pertes), pour l'une au l'autre des composante du modèle copule. Nous proposons ainsi d'établir une passerelle entre les enseignements des théories de la finance comportementale et la modélisation de la dépendance par la théorie des copules.La deuxième partie de la thèse porte sur les produits structurés de crédit. Nous étudions, dans un premier chapitre, l'apport d'un modèle actuariel, utilisant les fonctions copules dans la modélisation de la structure de dépendance entre les temps de défauts, dans le cadre du processus d'estimation des mesures de risque. Enfin, dans un dernier chapitre nous revisitons la notion du « Diversity Score », développée par l'agence de notation Moody's dans l'objectif d'assigner la qualité de produits structurés de crédit en terme de diversification. Nous discutons de l'analogie de cette mesure avec celle de l'approche copule, et nous démontrons son adéquation avec quelques familles de fonctions copules. / In this thesis, we examine one key topic related to copula theory contributions to the financial portfolio management theory and to the study of structured credit products.The first part of this thesis deals with financial portfolio management. We first discuss the relationship between portfolio diversification and the choice of the copula that better describes the dependence structure. The goal is to identify one feature of portfolios making straightforward the process of the appropriate copula selection. In the second chapter, we propose to study the impact of copula model misspecification, on conventional risk measures estimates as for the example of Value-at-Risk and Expected-Shortfall. The idea is to check the validity of developing these estimates under the true copula model. In a third chapter, we apply such approach to optimal portfolio allocation problem. The main objective is to identify investor's sensitivity, depending on their risk (or loss) aversion, to one of the component of the copula model. Thus, we propose one possible linkage between the behavioral finance theory and the copula functions framework.The second part of the thesis focuses on structured credit products. In a first chapter, we study the contribution of an actuarial model, using copulas functions in modeling the dependence structure between times of defaults, in the process of risk measures estimates. Finally, in a last chapter we re-examine the “Diversity Score” concept, developed by the Moody's rating agency to assign the quality of structured credit portfolio in terms of diversification. We discuss the analogy of this measure with that of the copula approach, and demonstrate its adequacy with some copula functions families.

O objetivo principal deste artigo é estimar a contribuição dos bancos no mercado financeiro brasileiro ao risco sistêmico utilizando a metodologia proposta por Adrian e Brunnermeier (2011). Esta aplicação é relevante do ponto de vista de avaliação da regulação vigente, e na verificação dos padrões de risco vigentes mercado nacional. Entre os resultados encontrados, destacam-se três pontos distintos: (a) há uma grande divergência nos patamares de risco entre os períodos de baixa e alta estabilidade monetária; (b) a relação entre tamanho e risco gerado pelas instituições financeiras é não linear; e (c) assim como visto em trabalhos aplicados a outros países, o Value at Risk nem sempre acompanha a contribuição de um banco ao risco sistêmico, colocando em xeque as métricas da regulação vigente. / The main goal this of this paper is estimate the systemic risk contribution of the banks in the Brazilian financial markets, using the CoVaR methodology proposal in Adrian and Brunnermeier (2011). This application is relevant from the point of view of the effective regulation, and the examination of the patterns of the national market risk. Among the obtained results, stand out are three distinctive points: (a) there is a huge difference in levels of risk between poor and high stability environments; (b) the relationship between size and risk generated by financial institutions is not linear; and (c) as seen in previous works applied in others countries, the Value at Risk does not always follow the bank risk contribution to systemic risk, jeopardizing the metrics of the effective regulation.

As entidades fechadas de previdência complementar (EFPC) possuem significativa relevância na economia brasileira com seus ativos dos fundos de pensão representando 16,8% do PIB em dezembro de 2009. O sistema de gerenciamento de risco dos fundos de pensão ainda não evoluiu na mesma proporção em que evoluiu em outros segmentos do mercado financeiro brasileiro. Para atender suas demandas de gerenciamento de risco, os fundos de pensão têm utilizado os modelos propostos para as instituições financeiras; tais modelos, contudo, não chegam a atender integralmente às suas necessidades. Os órgãos reguladores do setor têm estimulado os fundos de pensão a utilizarem seus próprios modelos para estimar a volatilidade e o Value at Risk (VaR). O objetivo do trabalho é propor uma modelagem de risco a partir da volatilidade estocástica (SV) para o cálculo do Value at Risk (VaR), comparando-a com a volatilidade calculada pela EWMA, proposta pelo Risk Metrics . A aplicação empírica do modelo foi efetuada a partir de uma amostra de uma série de retornos da carteira de uma entidade fechada de previdência complementar (EFPC) - fundo de pensão, a Indusprevi - Sociedade de Previdência Privada do Rio Grande do Sul. A amostra utilizada corresponde às cotas diárias entre o período de 01 de abril de 2004 até 31 de dezembro de 2009, representando 1.439 observações diárias. Os resultados apurados para a amostra demonstraram que a volatilidade estocástica (SV) tende a gerar um Value at Risk (VaR) mais conservador que o calculado a partir da metodologia do EWMA, quando testado pelo Teste de Kupiec (1995) e pela realização de Back testing. Tal fato, no entanto, torna o modelo mais adequado à realidade da Indusprevi e de uma grande maioria de outros fundos, que tendem a adotar políticas de investimentos mais conservadoras. / Pension funds have significant relevance to the Brazilian economy with assets representing, in December 2009, 16.8% of GDP. The pension funds risk management system has not evolved in the same pace as other sectors of the Brazilian financial market. To meet their demands for risk management, pension funds have employed the models proposed for financial institutions. Such models, however, fail to fully satisfy their needs. Government regulators have encouraged pension funds to use their own models so as to estimate volatility and Value at Risk (VaR). The main objective of this thesis is to propose a model of risk based on stochastic volatility (SV) to calculate the Value at Risk (VaR), as well as comparing it with the volatility estimated by EWMA, proposed by Risk MetricsTM. The empirical application of the model was made on a sample of portfolio returns of the pension fund Indusprevi - Sociedade de Previdência Privada do Rio Grande do Sul. The sample comprises 1439 daily quotes during the period April 1, 2004 to December 31, 2009. The results showed that the stochastic volatility (SV) tends to generate a more conservative Value at Risk (VaR) than the EWMA method when applying both the Kupiec (1995) test and back testing. This fact, therefore, makes the model more suitable to the principles of Indusprevi as well as a large majority of other funds, which tend to adopt more conservative investment policies.

O objetivo principal deste artigo é estimar a contribuição dos bancos no mercado financeiro brasileiro ao risco sistêmico utilizando a metodologia proposta por Adrian e Brunnermeier (2011). Esta aplicação é relevante do ponto de vista de avaliação da regulação vigente, e na verificação dos padrões de risco vigentes mercado nacional. Entre os resultados encontrados, destacam-se três pontos distintos: (a) há uma grande divergência nos patamares de risco entre os períodos de baixa e alta estabilidade monetária; (b) a relação entre tamanho e risco gerado pelas instituições financeiras é não linear; e (c) assim como visto em trabalhos aplicados a outros países, o Value at Risk nem sempre acompanha a contribuição de um banco ao risco sistêmico, colocando em xeque as métricas da regulação vigente. / The main goal this of this paper is estimate the systemic risk contribution of the banks in the Brazilian financial markets, using the CoVaR methodology proposal in Adrian and Brunnermeier (2011). This application is relevant from the point of view of the effective regulation, and the examination of the patterns of the national market risk. Among the obtained results, stand out are three distinctive points: (a) there is a huge difference in levels of risk between poor and high stability environments; (b) the relationship between size and risk generated by financial institutions is not linear; and (c) as seen in previous works applied in others countries, the Value at Risk does not always follow the bank risk contribution to systemic risk, jeopardizing the metrics of the effective regulation.

nÃo hà / O cenÃrio de incerteza no mercado de capitais brasileiro, durante o perÃodo eleitoral, contribuiu para a formaÃÃo de clusters de volatilidade, que tornou difÃcil a precificaÃÃo dos ativos negociados na bolsa brasileira. Neste contexto, esta pesquisa visa investigar se este cenÃrio de incerteza poderia ter acarretado na quebra de algum dos maiores bancos brasileiros com capital aberto. As aÃÃes dos bancos analisados foram: Banco do Brasil ON, Bradesco PN e ItaÃUnibanco PN. A MÃtrica para essa pesquisa foi a de Value-at-Risk (VaR). Dois desses modelos sÃo ditos incondicionais no que se refere à variÃncia: o VaR Gaussiano incondicional e o VaR Best Fitting incondicional. Os outros dois modelos sÃo chamados de condicionais, assumindo que a volatilidade varia ao longo do tempo. A mÃtrica que melhor representou as sÃries temporais atravÃs do Backtesting foi o VaR Logistic condicional. A partir da estimaÃÃo do melhor modelo, identificou-se que os bancos nÃo quebrariam, mas poderiam ter acumulado expressiva desvalorizaÃÃo. / The scenario of uncertainty in Brazilian capital market, during the election period, contributed to the formation of volatility clusters, that made it difficult to pricing of the securities traded on the Brazilian stock exchange. In this context, this research aims to investigate if this scenario of uncertainty could have led to the breakdown of some of the largest Brazilian banks with open market. The actions of the banks analyzed were: Bank of Brazil ON, Bradesco PN and ItaÃUnibanco PN. The metric for this research was to Value-at-Risk (VaR). Two models that are said unconditional as regards the variance: unconditional Gaussian VaR and VaR Best Fitting unconditionally. The other two are called conditional models, assuming that the volatility varies over time. The best metric that represented the time series by Backtesting was conditional Logistic VaR. From the estimation of the best model, it was identified that the banks would not break, but could have accumulated significant devaluation.