Value at Risk (VaR) and Expected Shortfall (ES) are methods often used to measure market risk. Inaccurate and unreliable Value at Risk and Expected Shortfall models can lead to underestimation of the market risk that a firm or financial institution is exposed to, and therefore may jeopardize the well-being or survival of the firm or financial institution during adverse markets. The objective of this study is therefore to examine various Value at Risk and Expected Shortfall models, including fatter tail models, in order to analyze the accuracy and reliability of these models. Thirteen VaR and ES models under three main approaches (Parametric, Non-Parametric and Semi-Parametric) are examined in this study. The results of this study show that the proposed model (ARMA(1,1)-GJR-GARCH(1,1)-SGED) gives the most balanced Value at Risk results. The semi-parametric model (Extreme Value Theory, EVT) is the most accurate Value at Risk model in this study for S&P 500. / October 2014

La libéralisation des échanges, le développement des moyens de transport de marchandises à faible coût et l’essor économique des pays émergents font de la globalisation (mondialisation) des chaînes logistiques un phénomène irréversible. Si ces chaines globalisées permettent de réduire les coûts, en contrepartie, elles multiplient les risques de rupture depuis la phase d’approvisionnement jusqu’à la phase finale de distribution. Dans cette thèse, nous nous focalisons sur la phase amont. Nous traitons plus spécifiquement le cas d’une centrale d’achat devant sélectionner des fournisseurs et allouer les commandes aux fournisseurs retenus. Chacun des fournisseurs risque de ne pas livrer ses commandes pour des raisons qui lui sont propres (problèmes internes, mauvaise qualité) ou externes (catastrophe naturelle, problèmes de transport). Selon que les fournisseurs sélectionnés livrent ou non leurs commandes, l’opération dégagera un profit ou sera déficitaire. L’objectif de cette thèse, est de fournir des outils d’aide à la décision à un décideur confronté à ce problème tout en prenant en compte le comportement du dit décideur face au risque. Des programmes stochastiques en nombre entiers mixtes ont été proposés pour modéliser ce problème. La première partie du travail porte sur l’élaboration d’un outil visuel d’aide à la décision permettant à un décideur de trouver une solution maximisant le profit espéré pour un risque de perte fixé. La deuxième partie applique les techniques d’estimation et de quantification du risque VAR et CVaR à ce problème. L’objectif est d’aider un décideur qui vise à minimiser la valeur de l’espérance du coût (utilisation de VaR) ou à minimiser la valeur de l’espérance du coût dans le pire des cas (utilisation de VAR et CVaR). Selon nos résultats, il apparaît que le décideur doit prendre en compte les différents scénarios possibles quelque soit leurs probabilités de réalisation, pour que la décision soit efficace. / Trade liberalization, the development of mean of transport and the development economic of emerging countries which lead to globalization of supply chain is irreversible phenomen. They can reduce costs, in return, they multiply the risk of disruption from upstream stage to downstream stage. In this thesis, we focus on the inbound supply chain stage. We treat more specifically the case of a purchasing central to select suppliers and allocate the orders. Each of the suppliers cannot deliver its orders due to internal reasons (poor quality problems) or external reasons (natural disasters, transport problems). According to the selected suppliers deliver their orders or not, the transaction operation will generate a profit or loss. The objective of this thesis is to provide decision support tools to a decision maker faced with this problem by taking into account the behavior of decision maker toward risk. We proposed stochastic mixed integer linear programs to model this problem. In the first part, we focuses on the development of a decision support visual tool that allows a decision maker to find a compromise between maximizing the expected profit and minimize the risk of loss. In the second part, we integrated the techniques of estimation of risk VaR and CVaR in this problem. The objective is to help decision maker to minimize the expected cost and minimize the conditional value at risk simultanously via calculating of VaR. Result shows that the decision maker must tack into account the different scenarios of disruption regardless their probability of realisation.

Made available in DSpace on 2010-04-20T21:00:12Z (GMT). No. of bitstreams: 4 Lucas Machado Braga de Araujo.pdf.jpg: 18740 bytes, checksum: e39415b36953ced37a6d87a19c381b38 (MD5) Lucas Machado Braga de Araujo.pdf.txt: 95713 bytes, checksum: a95ad7b44ec22b2f5ba2dbfe4f6798bb (MD5) Lucas Machado Braga de Araujo.pdf: 758972 bytes, checksum: c20d926964cc37d2ee7bdfc1bdf6aafc (MD5) license.txt: 4886 bytes, checksum: f5ef202e66d94f51fd6c9a09505731a6 (MD5) Previous issue date: 2009-02-03T00:00:00Z / The aim of this work is to show that the optimization of a portfolio composed of Brazilian hedge funds presents better results when the risk measure considered is Conditional Value-at-Risk. Portfolio optimization models aim to select assets that maximize the investor‘s return for a given level of risk. Therefore the definition of an appropriate measure of risk is of fundamental importance to the allocation process. The traditional methodology of portfolio optimization, developed by Markowitz, uses the variance of assets returns as risk measure. However variance is a measure appropriate only for cases where the returns are normally distributed or that the investor utility function is quadratic. Nevertheless it will be shown that the returns of Brazilian hedge funds usually do not have a Normal distribution. Consequently, to perform the optimization of a portfolio composed by Brazilian hedge funds is necessary to use an alternative risk measure. / O objetivo do trabalho é demonstrar que a otimização de uma carteira composta por fundos multimercados brasileiros gera melhores resultados quando a medida de risco utilizada é o Conditional Value-at-Risk. Modelos de otimização de carteira têm como objetivo selecionar ativos que maximizem o retorno do investidor para um determinado nível de risco. Assim, a definição de uma medida apropriada de risco é de fundamental importância para o processo de alocação. A metodologia tradicional de otimização de carteiras, desenvolvida por Markowitz, utiliza como medida de risco a variância dos retornos. Entretanto, a variância é uma medida apenas apropriada para casos em que os retornos são normalmente distribuídos ou em que os investidores possuem funções de utilidade quadrática. Porém, o trabalho mostra que os retornos dos fundos multimercados brasileiros tendem a não apresentar distribuição normal. Logo, para efetuar a otimização de uma carteira composta por fundos multimercados brasileiros é necessário utilizar uma medida de risco alternativa.

Orientador: Roberto Andreani / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T15:35:32Z (GMT). No. of bitstreams: 1 Magro_RogerioCorrea_M.pdf: 1309841 bytes, checksum: 3935050b45cf1bf5bbba46ac64603d72 (MD5) Previous issue date: 2008 / Resumo: Dado um capital C e n opções de investimento (ativos), o problema de seleção de portfolio consiste em aplicar C da melhor forma possivel para um determinado perfil de investidor. Visto que, em geral, os valores futuros destes ativos não são conhecidos, a questão fundamental a ser respondida e: Como mensurar a incerteza? No presente trabalho são apresentadas tres medidas de risco: O modelo de Markowitz, o Value-at-Risk (VaR) e o Conditional Value-At-Risk (CVaR). Defendemos que, sob o ponto de vista teorico, o Valor em Risco (VaR) e a melhor dentre as tres medidas. O motivo de tal escolha deve-se ao fato de que, para o VaR, podemos controlar a influencia que os cenários catastroficos possuem sobre nossas decisões. Em contrapartida, o processo computacional envolvido na escolha de um portfolio ótimo sob a metodologia VaR apresenta-se notadamente mais custoso do que aqueles envolvidos nos calculos das demais medidas consideradas. Dessa forma, nosso objetivo e tentar explorar essa vantagem computacional do Modelo de Markowitz e do CVaR no sentido de tentar aproximar suas decisões aquelas apontadas pela medida eleita. Para tal, consideraremos soluções VaR em seu sentido original (utilizando apenas o parametro de confiabilidade ao buscar portfolios otimos) e soluções com controle de perda (impondo uma cota superior para a perda esperada) / Abstract: Given a capital C and n investment options (assets), the problem of portfolio selection consists of applying C in the best possible way for a certain investor profile. Because, in general, the future values of these assets are unknown, the fundamental question to be answered is: How to measure the uncertainty? In the present work three risk measures are presented: The Markowitz¿s model, the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR). We defended that, under the theoretical point of view, the Value in Risk (VaR) is the best amongst the three measures. The reason of such a choice is due to the fact that, for VaR, we can control the influence that the catastrophic sceneries possess about our decisions. In the other hand, the computational process involved in the choice of a optimal portfolio under the VaR methodology comes notedly more expensive than those involved in the calculations of the other considered measures. In that way, our objective is to try to explore that computational advantage of the Markowitz¿s Model and of CVaR in the sense of trying to approach its decisions the those pointed by the elect measure. For such, we will consider VaR solutions in its original sense (just using the confidence level parameter when looking for optimal portfolios) and solutions with loss control (imposing a superior quota for the expected loss) / Mestrado / Otimização / Mestre em Matemática Aplicada

Cyber risk or data breach risk can be estimated similarly as other types of operational risk. First we identify problems of cyber risk models in existing literature. A large dataset consisting of 5,713 loss events enables us to apply extreme value theory. We adopt goodness of fit tests adjusted for distribution functions with estimated parameters. These tests are often overlooked in the literature even though they are essential for correct results. We model aggregate losses in three different industries separately and then we combine them using a copula. A t-test reveals that potential one-year global losses due to data breach risk are larger than the GDP of the Czech Republic. Moreover, one-year global cyber risk measured with a 99% CVaR amounts to 2.5% of the global GDP. Unlike others we compare risk measures with other quantities which allows wider audience to understand the magnitude of the cyber risk. An estimate of global data breach risk is a useful indicator not only for insurers, but also for any organization processing sensitive data.

<p>Optimization under uncertainty is the field of optimization where the data or the optimization model itself has uncertainties associated with it. Such problems are more commonly referred to as stochastic optimization problems. These problems capture the broad idea of making optimal decisions under uncertainty. An important class of these stochastic optimization problems is chance-constrained optimization problems, where the decision maker seeks to choose the best decision such that the probability of violating a set of uncertainty constraints is within a predefined probabilistic threshold risk level. Such stochastic optimization problems have found a lot of interest in the service industry as the service providers need to satisfy a minimum service level agreement (SLA) with their customers. Satisfying SLA in the presence of uncertainty in the form of probabilistic failure of infrastructure poses many interesting and challenging questions. In this thesis, we answer a few of these questions.</p> <p>We first explore the problem of quantifying uncertainties that adversely impact the service provider's infrastructure, thereby hurting the service level agreements. In particular we address the probability quantification problem, where given an uncertainty set, the goal is to quantify the probability of an event, on which the optimal value of an optimization problem exceeds a predefined threshold value. The novel techniques we propose, use and develop ideas from diverse literatures such as mixed integer nonlinear program, chance-constrained programming, approximate sampling and counting techniques, and large deviation bounds. Our approach yields the first polynomial time approximation scheme for the specific probability quantification problem we consider. </p> <p>Our next work is inspired by the ideas of risk averse decision making. Here, we focus on studying the problem of minimizing risk functions. As a special case we also explore the problem of minimizing the Value at Risk (VaR), which is a well know non-convex problem. For more than a decade, the well-known, best convex approximation to this problem has been obtained by minimizing the Conditional Value at Risk (CVaR). We proposed a new two-stage model which formulates these risk functions, which eventually leads to a bilinear optimization problem, a special case of which is the VaR minimization problem. We come up with enhancements to this bilinear formulation and use convexification techniques to obtain tighter lower and upper convex approximations to the problem. We also find that the approximation obtained by CVaR minimization is a special case of our method. The overestimates we construct help us to develop tighter convex inner approximations for the chance constraint optimization problems.</p>

Cette thèse étudie la gestion et la couverture du risque en s’appuyant sur la Value-at-Risk (VaR) et la Value-at-Risk Conditionnelle (CVaR), comme mesures de risque. La première partie propose un modèle d’évolution de prix que nous confrontons à des données réelles issues de la bourse de Paris (Euronext PARIS). Notre modèle prend en compte les probabilités d’occurrence des pertes extrêmes et les changements de régimes observés sur les données. Notre approche consiste à détecter les différentes périodes de chaque régime par la construction d’une chaîne de Markov cachée et à estimer la queue de distribution de chaque régime par des lois puissances. Nous montrons empiriquement que ces dernières sont plus adaptées que les lois normales et les lois stables. L’estimation de la VaR est validée par plusieurs backtests et comparée aux résultats d’autres modèles classiques sur une base de 56 actifs boursiers. Dans la deuxième partie, nous supposons que les prix boursiers sont modélisés par des exponentielles de processus de Lévy. Dans un premier temps, nous développons une méthode numérique pour le calcul de la VaR et la CVaR cumulatives. Ce problème est résolu en utilisant la formalisation de Rockafellar et Uryasev, que nous évaluons numériquement par inversion de Fourier. Dans un deuxième temps, nous nous intéressons à la minimisation du risque de couverture des options européennes, sous une contrainte budgétaire sur le capital initial. En mesurant ce risque par la CVaR, nous établissons une équivalence entre ce problème et un problème de type Neyman-Pearson, pour lequel nous proposons une approximation numérique s’appuyant sur la relaxation de la contrainte / This thesis studies the risk management and hedging, based on the Value-at-Risk (VaR) and the Conditional Value-at-Risk (CVaR) as risk measures. The first part offers a stocks return model that we test in real data from NSYE Euronext. Our model takes into account the probability of occurrence of extreme losses and the regime switching observed in the data. Our approach is to detect the different periods of each regime by constructing a hidden Markov chain and estimate the tail of each regime distribution by power laws. We empirically show that powers laws are more suitable than Gaussian law and stable laws. The estimated VaR is validated by several backtests and compared to other conventional models results on a basis of 56 stock market assets. In the second part, we assume that stock prices are modeled by exponentials of a Lévy process. First, we develop a numerical method to compute the cumulative VaR and CVaR. This problem is solved by using the formalization of Rockafellar and Uryasev, which we numerically evaluate by Fourier inversion techniques. Secondly, we are interested in minimizing the hedging risk of European options under a budget constraint on the initial capital. By measuring this risk by CVaR, we establish an equivalence between this problem and a problem of Neyman-Pearson type, for which we propose a numerical approximation based on the constraint relaxation

Shorter product life cycles and rapid product obsolescence provide increasing incentives to introduce newproducts to markets more quickly. As a consequence of rapidly changing market conditions, firms focus onimproving their new product development processes to reap the benefits of early market entry. Researchershave analyzed market entry, but have seldom provided quantitative approaches for the product rolloverproblem. This research builds upon the literature by using established optimization methods to examine howfirms can minimize their net loss during the rollover process. Specifically, our work explicitly optimizes thetiming of removal of old products and introduction of new products, the optimal strategy, and the magnitudeof net losses when the market entry approval date of a new product is unknown. In the first paper, we use theconditional value at risk to optimize the net loss and investigate the effect of risk perception of the manageron the rollover process. We compare it to the minimization of the classical expected net loss. We deriveconditions for optimality and unique closed-form solutions for single and dual rollover cases. In the secondpaper, we investigate the rollover problem, but for a time-dependent demand rate for the second producttrying to approximate the Bass Model. Finally, in the third paper, we apply the data-driven optimizationapproach to the product rollover problem where the probability distribution of the approval date is unknown.We rather have historical observations of approval dates. We develop the optimal times of rollover and showthe superiority of the data-driven method over the conditional value at risk in case where it is difficult to guessthe real probability distribution

Modern financial asset pricing theory is a broad, and at times, complex field. The literature review in this study covers many of the asset pricing techniques including factor models, random walk models, correlation models, Bayesian methods, autoregressive models, moment-matching models, stochastic jumps and mean reversion models. An important topic in finance is portfolio opti-misation with respect to risk and reward such as the mean variance optimisation introduced by Markowitz (1952). This study covers optimisation techniques such as single period mean variance optimisation, optimisation with risk aversion, multi-period stochastic programs, two-fund separa- tion theory, downside optimisation techniques and multi-period optimisation such as the Bellman dynamic programming model. The question asked in this study is, in the context of investing for South African individuals in a multi-asset portfolio, whether an active investment strategy is signi cantly di erent from a passive investment strategy. The passive strategy is built using stochastic programming with moment matching methods for non-Gaussian asset class distributions. The strategy is optimised in a framework using a downside risk metric, the conditional variance at risk. The active strategy is built with forward forecasts for asset classes using the time-varying transitional-probability Markov regime switching model. The active portfolio is finalised by a dynamic optimisation using a two-stage stochastic programme with recourse, which is solved as a large linear program. A hypothesis test is used to establish whether the results of two strategies are statistically different. The performance of the strategies are also reviewed relative to multi-asset peer rankings. Lastly, we consider whether the findings reveal information on the degree of effi ciency in the market place for multi-asset investments for the South African investor. / Operations Management / M. Sc. (Operations Research)

Les consommateurs d’aujourd’hui ont des goûts très variés et cherchent les produits les plus récents. Avec l’accélération technologique, les cycles de vie des produits se sont raccourcis et donc, de nouveaux produits doivent être introduits au marché plus souvent et progressivement, les anciens doivent y être retirés. L’introduction d’un nouveau produit est une source de croissance et d’avantage concurrentiel. Les directeurs du Marketing et Supply Chain se sont confrontés à la question de savoir comment gérer avec succès le remplacement de leurs produits et d’optimiser les coûts de la chaîne d’approvisionnement associée. Dans une situation idéale, la procédure de rollover est efficace et claire: l’ancien produit est vendu jusqu’à une date prévue où un nouveau produit est introduit. Dans la vie réelle, la situation est moins favorable. Le but de notre travail est d’analyser et de caractériser la politique optimale du rollover avec une date de disponibilitéstochastique pour l’introduction du nouveau produit sur le marché. Pour résoudre le problème d’optimisation,nous utilisons dans notre premier article deux mesures de minimisation: le coût moyen et le coût de la valeurconditionnelle à risque. On obtient des solutions en forme explicite pour les politiques optimales. En outre, nous caractérisons l’influence des paramètres de coûts sur la structure de la politique optimale. Dans cet esprit, nous analysons aussi le comportement de la politique de rollover optimale dans des contextes différents. Dans notre deuxième article, nous examinons le même problème mais avec une demande constante pour le premier produit et une demande linéaire au début puis constante pour le deuxième. Ce modèle est inspiré par la demande de Bass. Dans notre troisième article, la date de disponibilité du nouveau produit existe mais elle est inconnue. La seule information disponible est un ensemble historique d’échantillons qui sont tirés de la vraie distribution. Nous résoudrons le problème avec l’approche data drivenet nous obtenons des formulations tractables. Nous développons aussi des bornes sur le nombre d’échantillons nécessaires pour garantir qu’avec une forte probabilité, le coût n’est pas très loin du vrai coût optimal. / Shorter product life cycles and rapid product obsolescence provide increasing incentives to introduce newproducts to markets more quickly. As a consequence of rapidly changing market conditions, firms focus onimproving their new product development processes to reap the benefits of early market entry. Researchershave analyzed market entry, but have seldom provided quantitative approaches for the product rolloverproblem. This research builds upon the literature by using established optimization methods to examine howfirms can minimize their net loss during the rollover process. Specifically, our work explicitly optimizes thetiming of removal of old products and introduction of new products, the optimal strategy, and the magnitudeof net losses when the market entry approval date of a new product is unknown. In the first paper, we use theconditional value at risk to optimize the net loss and investigate the effect of risk perception of the manageron the rollover process. We compare it to the minimization of the classical expected net loss. We deriveconditions for optimality and unique closed-form solutions for single and dual rollover cases. In the secondpaper, we investigate the rollover problem, but for a time-dependent demand rate for the second producttrying to approximate the Bass Model. Finally, in the third paper, we apply the data-driven optimizationapproach to the product rollover problem where the probability distribution of the approval date is unknown.We rather have historical observations of approval dates. We develop the optimal times of rollover and showthe superiority of the data-driven method over the conditional value at risk in case where it is difficult to guessthe real probability distribution