Valor de riesgo (VaR) versus la Desviación estándar como concepto de riesgo en la elección de portafolios de inversión

En la actualidad hay una especial preocupación de los inversionistas por realizar sus inversiones de manera más segura, obteniendo una buena rentabilidad y sin poner en riesgo su capital -- En este sentido, la posibilidad de generar nuevas herramientas que permitan tomar mejores decisiones de inversión es cada vez más relevante en el mundo financiero -- Así, uno de los aportes más importantes de los que se dispone para ese propósito es el de Markowitz, que propone la generación de carteras óptimamente diversificadas -- Sin embargo, el problema es cómo escoger entre algunas de estas carteras -- Por ese motivo, este proyecto tuvo como objetivo comparar el modelo de la desviación estándar (Ratio de Sharpe) con el de Value at Risk (VaR) como concepto de riesgo, para la elección de una cartera óptima dentro del entorno de un mercado desarrollado, en este caso, el mercado estadounidense, por medio de un backtesting se analizó también si el ciclo de mercado bajista, estable o alcista tiene incidencia de igual forma en esta elección -- Después de realizar el modelo y aplicarlo se concluyó que bajo situaciones normales, en un mercado desarrollado, elegir una cartera sobre otra tuvo mayores beneficios si se realiza teniendo en cuenta como concepto de riesgo el VaR bajo un modelo de Simulación de Montecarlo, en lugar de la desviación estándar -- Al aplicar este modelo a un entono menos desarrollado y más fluctuante como el colombiano, se determinó que no hay una ventaja significativa entre los dos modelos (desviación estándar y VaR)

Risk management for mathematical optimization under uncertainty

We present a general multistage stochastic mixed 0-1 problem where the uncertainty appears everywhere in the objective function, constraints matrix and right-hand-side. The uncertainty is represented by a scenario tree that can be a symmetric or a nonsymmetric one. The stochastic model is converted in a mixed 0-1 Deterministic Equivalent Model in compact representation. Due to the difficulty of the problem, the solution offered by the stochastic model has been traditionally obtained by optimizing the objective function expected value (i.e., mean) over the scenarios, usually, along a time horizon. This approach (so named risk neutral) has the inconvenience of providing a solution that ignores the variance of the objective value of the scenarios and, so, the occurrence of scenarios with an objective value below the expected one. Alternatively, we present several approaches for risk averse management, namely, a scenario immunization strategy, the optimization of the well known Value-at-Risk (VaR) and several variants of the Conditional Value-at-Risk strategies, the optimization of the expected mean minus the weighted probability of having a "bad" scenario to occur for the given solution provided by the model, the optimization of the objective function expected value subject to stochastic dominance constraints (SDC) for a set of profiles given by the pairs of threshold objective values and either bounds on the probability of not reaching the thresholds or the expected shortfall over them, and the optimization of a mixture of the VaR and SDC strategies.

A gestão do risco associado à negociação de energia eléctrica através de contratos bilaterais e mercado spot

Num mercado de electricidade competitivo onde existe um ambiente de incerteza, as empresas de geração adoptam estratégias que visam a maximização do lucro, e a minimização do risco. Neste contexto, é de extrema importância para desenvolver uma estratégia adequada de gestão de risco ter em conta as diferentes opções de negociação de energia num mercado liberalizado, de forma a suportar a tomada de decisões na gestão de risco. O presente trabalho apresenta um modelo que avalia a melhor estratégia de um produtor de energia eléctrica que comercializa num mercado competitivo, onde existem dois mercados possíveis para a transacção de energia: o mercado organizado (bolsa) e o mercado de contratos bilaterais. O produtor tenta maximizar seus lucros e minimizar os riscos correspondentes, seleccionando o melhor equilíbrio entre os dois mercados possíveis (bolsa e bilateral). O mercado de contratos bilaterais visa gerir adequadamente os riscos inerentes à operação de mercados no curto prazo (mercado organizado) e dar o vendedor / comprador uma capacidade real de escolher o fornecedor com que quer negociar. O modelo apresentado neste trabalho faz uma caracterização explícita do risco no que diz respeito ao agente de mercado na questão da sua atitude face ao risco, medido pelo Value at Risk (VaR), descrito neste trabalho por Lucro-em-Risco (PAR). O preço e os factores de risco de volume são caracterizados por um valor médio e um desvio padrão, e são modelizados por distribuições normais. Os resultados numéricos são obtidos utilizando a simulação de Monte Carlo implementado em Matlab, e que é aplicado a um produtor que mantém uma carteira diversificada de tecnologias de geração, para um horizonte temporal de um ano. Esta dissertação está organizada da seguinte forma: o capítulo 1, 2 e 3 descrevem o estado-da-arte relacionado com a gestão de risco na comercialização de energia eléctrica. O capítulo 4 descreve o modelo desenvolvido e implementado, onde é também apresentado um estudo de caso com uma aplicação do modelo para avaliar o risco de negociação de um produtor. No capítulo 5 são apresentadas as principais conclusões.

Projecto solvência II - Modelação do risco de subscrição numa companhia de seguros não vida

Dissertação apresentada na Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa para obtenção do grau de Mestre em Matemática e Aplicações - Actuariado, Estatística e Investigação Operacional

Risco Operacional nas seguradoras – Estudo do caso Generali

Dissertação apresentada como requisito parcial para obtenção do grau de Mestre em Estatística e Gestão de Informação

Haar wavelets-based approach for quantifying credit portfolio losses

This paper proposes a new methodology to compute Value at Risk (VaR) for quantifying losses in credit portfolios. We approximate the cumulative distribution of the loss function by a finite combination of Haar wavelet basis functions and calculate the coefficients of the approximation by inverting its Laplace transform. The Wavelet Approximation (WA) method is specially suitable for non-smooth distributions, often arising in small or concentrated portfolios, when the hypothesis of the Basel II formulas are violated. To test the methodology we consider the Vasicek one-factor portfolio credit loss model as our model framework. WA is an accurate, robust and fast method, allowing to estimate VaR much more quickly than with a Monte Carlo (MC) method at the same level of accuracy and reliability.

Stochastic approximation methods in stochastic optimization

Stochastic approximation methods for stochastic optimization are considered. Reviewed the main methods of stochastic approximation: stochastic quasi-gradient algorithm, Kiefer-Wolfowitz algorithm and adaptive rules for them, simultaneous perturbation stochastic approximation (SPSA) algorithm. Suggested the model and the solution of the retailer's profit optimization problem and considered an application of the SQG-algorithm for the optimization problems with objective functions given in the form of ordinary differential equation.

Economic capital allocation

For my Licentiate thesis, I conducted research on risk measures. Continuing with this research, I now focus on capital allocation. In the proportional capital allocation principle, the choice of risk measure plays a very important part. In the chapters Introduction and Basic concepts, we introduce three definitions of economic capital, discuss the purpose of capital allocation, give different viewpoints of capital allocation and present an overview of relevant literature. Risk measures are defined and the concept of coherent risk measure is introduced. Examples of important risk measures are given, e. g., Value at Risk (VaR), Tail Value at Risk (TVaR). We also discuss the implications of dependence and review some important distributions. In the following chapter on Capital allocation we introduce different principles for allocating capital. We prefer to work with the proportional allocation method. In the following chapter, Capital allocation based on tails, we focus on insurance business lines with heavy-tailed loss distribution. To emphasize capital allocation based on tails, we define the following risk measures: Conditional Expectation, Upper Tail Covariance and Tail Covariance Premium Adjusted (TCPA). In the final chapter, called Illustrative case study, we simulate two sets of data with five insurance business lines using Normal copulas and Cauchy copulas. The proportional capital allocation is calculated using TCPA as risk measure. It is compared with the result when VaR is used as risk measure and with covariance capital allocation. In this thesis, it is emphasized that no single allocation principle is perfect for all purposes. When focusing on the tail of losses, the allocation based on TCPA is a good one, since TCPA in a sense includes features of TVaR and Tail covariance.

Choix de portefeuille de grande taille et mesures de risque pour preneurs de décision pessimistes

Cette thèse de doctorat consiste en trois chapitres qui traitent des sujets de choix de portefeuilles de grande taille, et de mesure de risque. Le premier chapitre traite du problème d’erreur d’estimation dans les portefeuilles de grande taille, et utilise le cadre d'analyse moyenne-variance. Le second chapitre explore l'importance du risque de devise pour les portefeuilles d'actifs domestiques, et étudie les liens entre la stabilité des poids de portefeuille de grande taille et le risque de devise. Pour finir, sous l'hypothèse que le preneur de décision est pessimiste, le troisième chapitre dérive la prime de risque, une mesure du pessimisme, et propose une méthodologie pour estimer les mesures dérivées. Le premier chapitre améliore le choix optimal de portefeuille dans le cadre du principe moyenne-variance de Markowitz (1952). Ceci est motivé par les résultats très décevants obtenus, lorsque la moyenne et la variance sont remplacées par leurs estimations empiriques. Ce problème est amplifié lorsque le nombre d’actifs est grand et que la matrice de covariance empirique est singulière ou presque singulière. Dans ce chapitre, nous examinons quatre techniques de régularisation pour stabiliser l’inverse de la matrice de covariance: le ridge, spectral cut-off, Landweber-Fridman et LARS Lasso. Ces méthodes font chacune intervenir un paramètre d’ajustement, qui doit être sélectionné. La contribution principale de cette partie, est de dériver une méthode basée uniquement sur les données pour sélectionner le paramètre de régularisation de manière optimale, i.e. pour minimiser la perte espérée d’utilité. Précisément, un critère de validation croisée qui prend une même forme pour les quatre méthodes de régularisation est dérivé. Les règles régularisées obtenues sont alors comparées à la règle utilisant directement les données et à la stratégie naïve 1/N, selon leur perte espérée d’utilité et leur ratio de Sharpe. Ces performances sont mesurée dans l’échantillon (in-sample) et hors-échantillon (out-of-sample) en considérant différentes tailles d’échantillon et nombre d’actifs. Des simulations et de l’illustration empirique menées, il ressort principalement que la régularisation de la matrice de covariance améliore de manière significative la règle de Markowitz basée sur les données, et donne de meilleurs résultats que le portefeuille naïf, surtout dans les cas le problème d’erreur d’estimation est très sévère. Dans le second chapitre, nous investiguons dans quelle mesure, les portefeuilles optimaux et stables d'actifs domestiques, peuvent réduire ou éliminer le risque de devise. Pour cela nous utilisons des rendements mensuelles de 48 industries américaines, au cours de la période 1976-2008. Pour résoudre les problèmes d'instabilité inhérents aux portefeuilles de grandes tailles, nous adoptons la méthode de régularisation spectral cut-off. Ceci aboutit à une famille de portefeuilles optimaux et stables, en permettant aux investisseurs de choisir différents pourcentages des composantes principales (ou dégrées de stabilité). Nos tests empiriques sont basés sur un modèle International d'évaluation d'actifs financiers (IAPM). Dans ce modèle, le risque de devise est décomposé en deux facteurs représentant les devises des pays industrialisés d'une part, et celles des pays émergents d'autres part. Nos résultats indiquent que le risque de devise est primé et varie à travers le temps pour les portefeuilles stables de risque minimum. De plus ces stratégies conduisent à une réduction significative de l'exposition au risque de change, tandis que la contribution de la prime risque de change reste en moyenne inchangée. Les poids de portefeuille optimaux sont une alternative aux poids de capitalisation boursière. Par conséquent ce chapitre complète la littérature selon laquelle la prime de risque est importante au niveau de l'industrie et au niveau national dans la plupart des pays. Dans le dernier chapitre, nous dérivons une mesure de la prime de risque pour des préférences dépendent du rang et proposons une mesure du degré de pessimisme, étant donné une fonction de distorsion. Les mesures introduites généralisent la mesure de prime de risque dérivée dans le cadre de la théorie de l'utilité espérée, qui est fréquemment violée aussi bien dans des situations expérimentales que dans des situations réelles. Dans la grande famille des préférences considérées, une attention particulière est accordée à la CVaR (valeur à risque conditionnelle). Cette dernière mesure de risque est de plus en plus utilisée pour la construction de portefeuilles et est préconisée pour compléter la VaR (valeur à risque) utilisée depuis 1996 par le comité de Bâle. De plus, nous fournissons le cadre statistique nécessaire pour faire de l’inférence sur les mesures proposées. Pour finir, les propriétés des estimateurs proposés sont évaluées à travers une étude Monte-Carlo, et une illustration empirique en utilisant les rendements journaliers du marché boursier américain sur de la période 2000-2011.

Calculo del valor en riesgo del futuro de energia e internacionalizacion del sector energetico en Colombia

El presente trabajo de grado busca definir cuál es el mejor método para determinar el valor en riesgo del contrato de futuro de energía eléctrica que se transa en Colombia, para cumplir con este objetivo se toma como referencia el marco histórico del VaR y de los futuros seguido de las características de la fijación de precios, la estructura del contrato, que políticas y métodos hay para cubrirse del riesgo y como se realiza en otros países, realizando algunos cálculos de los modelos más tradicionales del Var para luego incorporarlo al marco colombiano y al ente supervisor en este caso la Superintendencia Financiera de Colombia.. Además de revisar las diferentes teorías de internacionalización económicas, de proceso y redes aplicadas al sector energético en Colombia., evaluando su proceso, alcance y posibles mercados futuros.

Valor en riesgo del portafolio de TES de los bancos colombianos

En este trabajo se realiza la medición del riesgo de mercado para el portafolio de TES de un banco colombiano determinado, abordando el pronóstico de valor en riesgo (VaR) mediante diferentes modelos multivariados de volatilidad: EWMA, GARCH ortogonal, GARCH robusto, así como distintos modelos de VaR con distribución normal y distribución t-student, evaluando su eficiencia con las metodologías de backtesting propuestas por Candelon et al. (2011) con base en el método generalizado de momentos, junto con los test de independencia y de cobertura condicional planteados por Christoffersen y Pelletier (2004) y por Berkowitz, Christoffersen y Pelletier (2010). Los resultados obtenidos demuestran que la mejor especificación del VaR para la medición del riesgo de mercado del portafolio de TES de los bancos colombianos, es el construido a partir de volatilidades EWMA y basado en la distribución normal, ya que satisface las hipótesis de cobertura no condicional, independencia y cobertura condicional, al igual que los requerimientos estipulados en Basilea II y en la normativa vigente en Colombia.

Implementación de cópulas para la estimación del valor en riesgo

La dependencia entre las series financieras, es un parámetro fundamental para la estimación de modelos de Riesgo. El Valor en Riesgo (VaR) es una de las medidas más importantes utilizadas para la administración y gestión de Riesgos Financieros, en la actualidad existen diferentes métodos para su estimación, como el método por simulación histórica, el cual no asume ninguna distribución sobre los retornos de los factores de riesgo o activos, o los métodos paramétricos que asumen normalidad sobre las distribuciones. En este documento se introduce la teoría de cópulas, como medida de dependencia entre las series, se estima un modelo ARMA-GARCH-Cópula para el cálculo del Valor en Riesgo de un portafolio compuesto por dos series financiera, la tasa de cambio Dólar-Peso y Euro-Peso. Los resultados obtenidos muestran que la estimación del VaR por medio de copulas es más preciso en relación a los métodos tradicionales.

Time-variations in commodity price jumps

In this paper, we study jumps in commodity prices. Unlike assumed in existing models of commodity price dynamics, a simple analysis of the data reveals that the probability of tail events is not constant but depends on the time of the year, i.e. exhibits seasonality. We propose a stochastic volatility jump–diffusion model to capture this seasonal variation. Applying the Markov Chain Monte Carlo (MCMC) methodology, we estimate our model using 20 years of futures data from four different commodity markets. We find strong statistical evidence to suggest that our model with seasonal jump intensity outperforms models featuring a constant jump intensity. To demonstrate the practical relevance of our findings, we show that our model typically improves Value-at-Risk (VaR) forecasts.

A study on time-varying quantile and its applications

This Thesis is the result of my Master Degree studies at the Graduate School of Economics, Getúlio Vargas Foundation, from January 2004 to August 2006. am indebted to my Thesis Advisor, Professor Luiz Renato Lima, who introduced me to the Econometrics' world. In this Thesis, we study time-varying quantile process and we develop two applications, which are presented here as Part and Part II. Each of these parts was transformed in paper. Both papers were submitted. Part shows that asymmetric persistence induces ARCH effects, but the LMARCH test has power against it. On the other hand, the test for asymmetric dynamics proposed by Koenker and Xiao (2004) has correct size under the presence of ARCH errors. These results suggest that the LM-ARCH and the Koenker-Xiao tests may be used in applied research as complementary tools. In the Part II, we compare four different Value-at-Risk (VaR) methodologies through Monte Cario experiments. Our results indicate that the method based on quantile regression with ARCH effect dominates other methods that require distributional assumption. In particular, we show that the non-robust method ologies have higher probability to predict VaRs with too many violations. We illustrate our findings with an empirical exercise in which we estimate VaR for returns of São Paulo stock exchange index, IBOVESPA, during periods of market turmoil. Our results indicate that the robust method based on quantile regression presents the least number of violations.

Ensaios em macroeconometria e finanças

This thesis is composed of three essays referent to the subjects of macroeconometrics and Önance. In each essay, which corresponds to one chapter, the objective is to investigate and analyze advanced econometric techniques, applied to relevant macroeconomic questions, such as the capital mobility hypothesis and the sustainability of public debt. A Önance topic regarding portfolio risk management is also investigated, through an econometric technique used to evaluate Value-at-Risk models. The Örst chapter investigates an intertemporal optimization model to analyze the current account. Based on Campbell & Shillerís (1987) approach, a Wald test is conducted to analyze a set of restrictions imposed to a VAR used to forecast the current account. The estimation is based on three di§erent procedures: OLS, SUR and the two-way error decomposition of Fuller & Battese (1974), due to the presence of global shocks. A note on Granger causality is also provided, which is shown to be a necessary condition to perform the Wald test with serious implications to the validation of the model. An empirical exercise for the G-7 countries is presented, and the results substantially change with the di§erent estimation techniques. A small Monte Carlo simulation is also presented to investigate the size and power of the Wald test based on the considered estimators. The second chapter presents a study about Öscal sustainability based on a quantile autoregression (QAR) model. A novel methodology to separate periods of nonstationarity from stationary ones is proposed, which allows one to identify trajectories of public debt that are not compatible with Öscal sustainability. Moreover, such trajectories are used to construct a debt ceiling, that is, the largest value of public debt that does not jeopardize long-run Öscal sustainability. An out-of-sample forecast of such a ceiling is also constructed, and can be used by policy makers interested in keeping the public debt on a sustainable path. An empirical exercise by using Brazilian data is conducted to show the applicability of the methodology. In the third chapter, an alternative backtest to evaluate the performance of Value-at-Risk (VaR) models is proposed. The econometric methodology allows one to directly test the overall performance of a VaR model, as well as identify periods of an increased risk exposure, which seems to be a novelty in the literature. Quantile regressions provide an appropriate environment to investigate VaR models, since they can naturally be viewed as a conditional quantile function of a given return series. An empirical exercise is conducted for daily S&P500 series, and a Monte Carlo simulation is also presented, revealing that the proposed test might exhibit more power in comparison to other backtests.