One of the major problem faced by banks is how to manage the risk exposure in large portfolios. According to Basel II regulation banks has to measure the risk using Value-at-Risk with confidence level 99%. However, this regulation does not specify the way to calculate Valueat- Risk. The easiest way to calculate Value-at-Risk is to assume that portfolio returns are normally distributed. Altough, this is the most common way to calculate Value-at-Risk, there exists also other methods. The previous crisis shows that the regular methods are unfortunately not always enough to prevent bankruptcy. This paper is devoted to compare the classical methods of estimating risk with other methods such as Cornish-Fisher Expansion (CFVaR) and assuming generalized hyperbolic distribution. To be able to do this study, we estimate the risk in a large portfolio consisting of ten stocks. These stocks are chosen from the NASDAQ 100-list in order to have highly liquid stocks (bluechips). The stocks are chosen from different sectors to make the portfolio welldiversified. To investigate the impact of dependence between the stocks in the portfolio we remove the two most correlated stocks and consider the resulting eight stock portfolio as well. In both portfolios we put equal weight to the included stocks. The results show that for a well-diversified large portfolio none of the risk measures are violated. However, for a portfolio consisting of only one highly volatile stock we prove that we have a violation in the classical methods but not when we use the modern methods mentioned above.

This paper links two applications of Extreme Value Theory (EVT) to analyze Taiwanese financial markets: 1. computation of Value at Risk (VaR) and Expected Shortfall (ES) 2. estimates of cross-market dependence under extreme events. Daily data from the Taiwan Stock Exchange Capitalization Weight Stock Index (TAIEX) and the foreign exchange rate, USD/NTD, are employed to analyze the behavior of each return and the dependence structure between the foreign exchange market and the equity market. In the univariate case, when computing risk measures, EVT provides us a more accurate way to estimate VaR. In bivariate case, when measuring extremal dependence, the results of whole period data show the extremal dependence between two markets is asymptotically independent, and the analyses of subperiods illustrate that the relation is slightly dependent in specific periods. Therefore, there is no significant evidence that extreme events appeared in one market (the equity market or the foreign exchange market) will affect another in Taiwan.

Made available in DSpace on 2010-04-20T21:00:11Z (GMT). No. of bitstreams: 4 Mauricio Mussashi Irie.pdf.jpg: 16022 bytes, checksum: d3dcf1b8020749a12b3baae53334cda5 (MD5) Mauricio Mussashi Irie.pdf.txt: 96735 bytes, checksum: 944036c3b3d1ae823db79daedb65dd6e (MD5) Mauricio Mussashi Irie.pdf: 7355110 bytes, checksum: 880246debd0d864c44768ebd4eaf2e6e (MD5) license.txt: 4886 bytes, checksum: 8fa2d810f5b64e058d76fb4986924cf0 (MD5) Previous issue date: 2009-02-06T00:00:00Z / Este trabalho explora com cuidado o lado específico da implementação de um modelo de alocação de ativos em que o risco é tratado de maneira integrada, não somente através do desvio padrão do portfólio, mas também considerando outras métricas de risco como, por exemplo, o Expected Shortfall. Além disso, utilizamos algumas técnicas de como trabalhar com as variáveis de modo a extrair do mercado os chamados "invariantes de mercado", fenômenos que se repetem e podem ser modelados como variáveis aleatórias independentes e identicamente distribuídas. Utilizamos as distribuições empíricas dos invariantes, juntamente com o método de Cópulas para gerar um conjunto de cenários multivariados simulados de preços. Esses cenários são independentes de distribuição, portanto são não paramétricos. Através dos mesmos, avaliamos a distribuição de retornos simulados de um portfólio através de um índice de satisfação que é baseado em uma função de utilidade quadrática e utiliza o Expected Shortfall como métrica de risco. O índice de satisfação incorpora o trade-off do investidor entre risco e retorno. Finalmente, escolhemos como alocação ótima aquela que maximiza o índice de satisfação ajustado a um parâmetro de aversão ao risco. Perseguindo esses passos, é possível obter um portfólio no qual a alocação em cada ativo, ou classe de ativos, reflete o prêmio esperado ao risco incorrido. / The present work carefully explores the implementation of an asset allocation model in which the risk measure considered is fully integrated, not only through the standard deviation for the portfolio, but also considering other risk metrics, for instance, the Expected Shortfall. Moreover, some statistical tools are used to extract from the market the so called “market invariants”, which are phenomena that tend to repeat themselves and can be modeled as i.i.d. random variables. We use the empirical distribution of the invariants, along with the Method of Copula to generate a set of simulated multivariate price scenarios. These scenarios are independent of distribution, therefore they are non-parametric. With these scenarios we evaluate the simulated return distribution of a portfolio through a satisfaction index which is based on a quadratic utility function and the risk measure considered is the Expected Shortfall. The satisfaction index summarizes the investor trade-off between risk and return. Finally, we choose the optimal allocation that maximizes the satisfaction index adjusted to a risk aversion parameter. In pursuing these steps, it is possible to obtain a portfolio in which the allocation of each asset class or security fully reflects the expected premium to the risk assumed.

Made available in DSpace on 2010-04-20T21:00:37Z (GMT). No. of bitstreams: 3 marceloferreirasantos.pdf.jpg: 18994 bytes, checksum: 6ed4621da38fff28ddcbe493a614c144 (MD5) marceloferreirasantos.pdf: 594325 bytes, checksum: 5a8a7414557d37e5a6d7f21f9a190d47 (MD5) marceloferreirasantos.pdf.txt: 84863 bytes, checksum: 64ea5d2052e93075b4d3e1a6a2ff8b86 (MD5) Previous issue date: 2008-02-08T00:00:00Z / The goal of this work was to show alternatives models to the traditional way of measuring market risk for Brazilian financial assets. In order to cover the maximum possible risk factors in Brazil, we have used the main proxies for Fixed Income products. In times of volatility, market risk management is highly criticized for working in models based on normal distribution. Here it is the best contribution of the VaR and also the greatest criticism of it. In addition, our financial market is characterized by extreme illiquidity in the secondary market in spite of certain governmental bonds. The first stage was to research academic production about the theme in Brazil or worldwide. To our surprise, little has been said in country about stable distribution applied to financial market, whether in risk management, options pricing, or portfolio management. After this step, we selected a set of variables to be used aiming to cover a large part of Brazilian financial assets. Thus, we were able to identify or not a presence of normality condition so that we could model risk measure, VaR and ES, for chosen assets. The theoretical and practical conditions were created: market demand (heavy criticisms of Gausian approach), ample selection of assets (in spite of eventual doubts about liquidity), academic experience, and international knowledge (by means of detailed and meticulous study of the production about the theme in the main circles). In this way, four principal approaches have been analyzed in order to calculate risk measures whether they be coherent or not. It is important to mention that this work might be useful for large initiatives, for example, those which incorporate several assets within linear risk portfolios or, even, for non-linear portfolios. / O objetivo deste trabalho foi mostrar modelagens alternativas à tradicional maneira de se apurar o risco de mercado para ativos financeiros brasileiros. Procurou-se cobrir o máximo possível de fatores de risco existentes no Brasil; para tanto utilizamos as principais proxies para instrumentos de Renda Fixa. Em momentos de volatilidade, o gerenciamento de risco de mercado é bastante criticado por trabalhar dentro de modelagens fundamentadas na distribuição normal. Aqui reside a maior contribuição do VaR e também a maior crítica a ele. Adicionado a isso, temos um mercado caracterizado pela extrema iliquidez no mercado secundário até mesmo em certos tipos de títulos públicos federais. O primeiro passo foi fazer um levantamento da produção acadêmica sobre o tema, seja no Brasil ou no mundo. Para a nossa surpresa, pouco, no nosso país, tem se falado em distribuições estáveis aplicadas ao mercado financeiro, seja em gerenciamento de risco, precificação de opções ou administração de carteiras. Após essa etapa, passamos a seleção das variáveis a serem utilizadas buscando cobrir uma grande parte dos ativos financeiros brasileiros. Assim, deveríamos identificar a presença ou não da condição de normalidade para, aí sim, realizarmos as modelagens das medidas de risco, VaR e ES, para os ativos escolhidos, As condições teóricas e práticas estavam criadas: demanda de mercado (crítica ao método gausiano bastante difundido), ampla cobertura de ativos (apesar do eventual questionamento da liquidez), experiência acadêmica e conhecimento internacional (por meio de detalhado e criterioso estudo da produção sobre o tema nos principais meios). Analisou-se, desta forma, quatro principais abordagens para o cálculo de medidas de risco sendo elas coerentes (ES) ou não (VaR). É importante mencionar que se trata de um trabalho que poderá servir de insumo inicial para trabalhos mais grandiosos, por exemplo, aqueles que incorporarem vários ativos dentro de uma carteira de riscos lineares ou, até mesmo, para ativos que apresentem risco não-direcionais.

Submitted by Camila Corrêa Moraes (camila.cmoraes@gmail.com) on 2013-02-24T03:00:19Z No. of bitstreams: 1 DISSERTAÇÃO CAMILA MORAES.pdf: 4708711 bytes, checksum: 3c2acb024f3dbcde7627bb8afea462fd (MD5) / Rejected by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br), reason: Prezada Camila, Seu titulo não confere com a Ata, não podemos aprovar o trabalho, pois não temos informação do orientador (verso da Ata) da mudança do título. Aguardo email do seu orientador informando a alteração e posteriormente o professor deve assinar o verso da ata. Att. Suzi 3799-7876 on 2013-02-25T15:26:27Z (GMT) / Submitted by Camila Corrêa Moraes (camila.cmoraes@gmail.com) on 2013-02-26T17:46:32Z No. of bitstreams: 1 DISSERTAÇÃO CAMILA MORAES.pdf: 4708711 bytes, checksum: 3c2acb024f3dbcde7627bb8afea462fd (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2013-02-26T17:50:59Z (GMT) No. of bitstreams: 1 DISSERTAÇÃO CAMILA MORAES.pdf: 4708711 bytes, checksum: 3c2acb024f3dbcde7627bb8afea462fd (MD5) / Made available in DSpace on 2013-02-26T18:41:00Z (GMT). No. of bitstreams: 1 DISSERTAÇÃO CAMILA MORAES.pdf: 4708711 bytes, checksum: 3c2acb024f3dbcde7627bb8afea462fd (MD5) Previous issue date: 2013-01-29 / Value at Risk (VaR) and Expected Shortfall (ES) are quantitative models to measure market risk of financial assets portfolios. The purpose of this study is to evaluate the results of these models for a portfolio traded in the Brazilian market through four backtesting methods - Basel Traffic Light Test, Kupiec Test, Christoffersen Test and McNeil and Frey Test - covering periods of domestic (2002) and international (2008) financial crisis. The VaR model described here presents two approaches - Parametric, where it is assumed that the distribution of asset returns follow a Normal, and Historical Simulation, where there are no assumption about the distribution of asset returns, but it is assumed that they are independent and identically distributed. The results of VaR were also evaluated with the Cornish-Fisher expansion, which tries to approximate the empirical distribution to a Normal distribution using the values of skewness and kurtosis. Another feature observed was the property of coherence, which evaluates if the risk measure follows four basic axioms - monotonicity, translation invariance, homogeneity and subadditivity. VaR is not considered a coherent risk measure because it doesn´t follow the subadditivity feature in all cases. On the other hand the ES follows the four axioms, thus considered a coherent risk measure. The ES model was evaluated according to the Parametric Normal approach. This work also verified through backtests, if the property of coherency improves the accuracy of the analyzed risk measures / Value at Risk (VaR) e Expected Shortfall (ES) são modelos quantitativos para mensuração do risco de mercado em carteiras de ativos financeiros. O propósito deste trabalho é avaliar os resultados de tais modelos para ativos negociados no mercado brasileiro através de quatro metodologias de backtesting - Basel Traffic Light Test, Teste de Kupiec, Teste de Christoffersen e Teste de McNeil e Frey – abrangendo períodos de crise financeira doméstica (2002) e internacional (2008). O modelo de VaR aqui apresentado utilizou duas abordagens – Paramétrica Normal, onde se assume que a distribuição dos retornos dos ativos segue uma Normal, e Simulação Histórica, onde não há hipótese a respeito da distribuição dos retornos dos ativos, porém assume-se que os mesmos são independentes e identicamente distribuídos. Também foram avaliados os resultados do VaR com a expansão de Cornish-Fisher, a qual visa aproximar a distribuição empírica a uma distribuição Normal utilizando os valores de curtose e assimetria para tal. Outra característica observada foi a propriedade de coerência, a qual avalia se a medida de risco obedece a quatro axiomas básicos – monotonicidade, invariância sob translações, homogeneidade e subaditividade. O VaR não é considerado uma medida de risco coerente, pois não apresenta a característica de subaditividade em todos os casos. Por outro lado o ES obedece aos quatro axiomas, considerado assim uma medida coerente. O modelo de ES foi avaliado segundo a abordagem Paramétrica Normal. Neste trabalho também se verificou através dos backtests, o quanto a propriedade de coerência de uma medida de risco melhora sua precisão.

Quantifier et mesurer le risque dans un environnement partiellement ou totalement incertain est probablement l'un des enjeux majeurs de la recherche appliquée en mathématiques financières. Cela concerne l'économie, la finance, mais d'autres domaines comme la santé via les assurances par exemple. L'une des difficultés fondamentales de ce processus de gestion des risques est de modéliser les actifs sous-jacents, puis d'approcher le risque à partir des observations ou des simulations. Comme dans ce domaine, l'aléa ou l'incertitude joue un rôle fondamental dans l'évolution des actifs, le recours aux processus stochastiques et aux méthodes statistiques devient crucial. Dans la pratique l'approche paramétrique est largement utilisée. Elle consiste à choisir le modèle dans une famille paramétrique, de quantifier le risque en fonction des paramètres, et d'estimer le risque en remplaçant les paramètres par leurs estimations. Cette approche présente un risque majeur, celui de mal spécifier le modèle, et donc de sous-estimer ou sur-estimer le risque. Partant de ce constat et dans une perspective de minimiser le risque de modèle, nous avons choisi d'aborder la question de la quantification du risque avec une approche non-paramétrique qui s'applique à des modèles aussi généraux que possible. Nous nous sommes concentrés sur deux mesures de risque largement utilisées dans la pratique et qui sont parfois imposées par les réglementations nationales ou internationales. Il s'agit de la Value at Risk (VaR) qui quantifie le niveau de perte maximum avec un niveau de confiance élevé (95% ou 99%). La seconde mesure est l'Expected Shortfall (ES) qui nous renseigne sur la perte moyenne au delà de la VaR. / To quantify and measure the risk in an environment partially or completely uncertain is probably one of the major issues of the applied research in financial mathematics. That relates to the economy, finance, but many other fields like health via the insurances for example. One of the fundamental difficulties of this process of management of risks is to model the under lying credits, then approach the risk from observations or simulations. As in this field, the risk or uncertainty plays a fundamental role in the evolution of the credits; the recourse to the stochastic processes and with the statistical methods becomes crucial. In practice the parametric approach is largely used.It consists in choosing the model in a parametric family, to quantify the risk according to the parameters, and to estimate its risk by replacing the parameters by their estimates. This approach presents a main risk, that badly to specify the model, and thus to underestimate or over-estimate the risk. Based within and with a view to minimizing the risk model, we choose to tackle the question of the quantification of the risk with a nonparametric approach which applies to models as general as possible. We concentrate to two measures of risk largely used in practice and which are sometimes imposed by the national or international regulations. They are the Value at Risk (VaR) which quantifies the maximum level of loss with a high degree of confidence (95% or 99%). The second measure is the Expected Shortfall (ES) which informs about the average loss beyond the VaR.

Thesis (MComm)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Accurate estimation of Value at Risk (VaR) and Expected Shortfall (ES) is critical in the management of extreme market risks. These risks occur with small probability, but the financial impacts could be large. Traditional models to estimate VaR and ES are investigated. Following usual practice, 99% 10 day VaR and ES measures are calculated. A comprehensive theoretical background is first provided and then the models are applied to the Africa Financials Index from 29/01/1996 to 30/04/2013. The models considered include independent, identically distributed (i.i.d.) models and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) stochastic volatility models. Extreme Value Theory (EVT) models that focus especially on extreme market returns are also investigated. For this, the Peaks Over Threshold (POT) approach to EVT is followed. For the calculation of VaR, various scaling methods from one day to ten days are considered and their performance evaluated. The GARCH models fail to converge during periods of extreme returns. During these periods, EVT forecast results may be used. As a novel approach, this study considers the augmentation of the GARCH models with EVT forecasts. The two-step procedure of pre-filtering with a GARCH model and then applying EVT, as suggested by McNeil (1999), is also investigated. This study identifies some of the practical issues in model fitting. It is shown that no single forecasting model is universally optimal and the choice will depend on the nature of the data. For this data series, the best approach was to augment the GARCH stochastic volatility models with EVT forecasts during periods where the first do not converge. Model performance is judged by the actual number of VaR and ES violations compared to the expected number. The expected number is taken as the number of return observations over the entire sample period, multiplied by 0.01 for 99% VaR and ES calculations. / AFRIKAANSE OPSOMMING: Akkurate beraming van Waarde op Risiko (Value at Risk) en Verwagte Tekort (Expected Shortfall) is krities vir die bestuur van ekstreme mark risiko’s. Hierdie risiko’s kom met klein waarskynlikheid voor, maar die finansiële impakte is potensieel groot. Tradisionele modelle om Waarde op Risiko en Verwagte Tekort te beraam, word ondersoek. In ooreenstemming met die algemene praktyk, word 99% 10 dag maatstawwe bereken. ‘n Omvattende teoretiese agtergrond word eers gegee en daarna word die modelle toegepas op die Africa Financials Index vanaf 29/01/1996 tot 30/04/2013. Die modelle wat oorweeg word sluit onafhanklike, identies verdeelde modelle en Veralgemeende Auto-regressiewe Voorwaardelike Heteroskedastiese (GARCH) stogastiese volatiliteitsmodelle in. Ekstreemwaarde Teorie modelle, wat spesifiek op ekstreme mark opbrengste fokus, word ook ondersoek. In hierdie verband word die Peaks Over Threshold (POT) benadering tot Ekstreemwaarde Teorie gevolg. Vir die berekening van Waarde op Risiko word verskillende skaleringsmetodes van een dag na tien dae oorweeg en die prestasie van elk word ge-evalueer. Die GARCH modelle konvergeer nie gedurende tydperke van ekstreme opbrengste nie. Gedurende hierdie tydperke, kan Ekstreemwaarde Teorie modelle gebruik word. As ‘n nuwe benadering oorweeg hierdie studie die aanvulling van die GARCH modelle met Ekstreemwaarde Teorie vooruitskattings. Die sogenaamde twee-stap prosedure wat voor-af filtrering met ‘n GARCH model behels, gevolg deur die toepassing van Ekstreemwaarde Teorie (soos voorgestel deur McNeil, 1999), word ook ondersoek. Hierdie studie identifiseer sommige van die praktiese probleme in model passing. Daar word gewys dat geen enkele vooruistkattingsmodel universeel optimaal is nie en die keuse van die model hang af van die aard van die data. Die beste benadering vir die data reeks wat in hierdie studie gebruik word, was om die GARCH stogastiese volatiliteitsmodelle met Ekstreemwaarde Teorie vooruitskattings aan te vul waar die voorafgenoemde nie konvergeer nie. Die prestasie van die modelle word beoordeel deur die werklike aantal Waarde op Risiko en Verwagte Tekort oortredings met die verwagte aantal te vergelyk. Die verwagte aantal word geneem as die aantal obrengste waargeneem oor die hele steekproefperiode, vermenigvuldig met 0.01 vir die 99% Waarde op Risiko en Verwagte Tekort berekeninge.

The thesis describes commonly used measures of risk, such as volatility, Value at Risk (VaR) and Expected Shortfall (ES), and is tasked with creating models for measuring market risk. It is concerned with the risk over daily and over monthly horizons and shows the shortcomings of a square-root-of-time approach for converting VaR and ES between horizons. Parametric models, geometric Brownian motion (GBM) and GARCH process, and non-parametric models, historical simulation (HS) and some its possible improvements, are presented. The application of these mentioned models is demonstrated using real data. The accuracy of VaR models is proved through backtesting and the results are discussed. Part of this thesis is also a simulation study, which reveals the precision of VaR and ES estimates.

The debate on quantitative operational risk modeling has only started at the beginning of the last decade and the best-practices are still far from being established. Estimation of capital requirements for operational risk under Advanced Measurement Approaches of Basel II is critically dependent on the choice of risk measure, which quantifies the risk exposure based on the underlying simulated distribution of losses. Despite its well-known caveats Value-at-Risk remains a predominant risk measure used in the context of operational risk management. We describe several serious drawbacks of Value-at-Risk and explain why it can possibly lead to misleading conclusions. As a remedy we suggest the use of coherent risk measures - and namely the statistic known as Expected Shortfall - as a suitable alternative or complement for quantification of operational risk exposure. We demonstrate that application of Expected Shortfall in operational loss modeling is feasible and produces reasonable and consistent results. We also consider a variety of statistical techniques for modeling of underlying loss distribution and evaluate extreme value theory framework as the most suitable for this purpose. Using stress tests we further compare the robustness and consistency of selected models and their implied risk capital estimates...

Dans cette thèse, on propose des méthodes analytiques ou numériques pour l'estimation de la VaR ou l'Expected Shortfall des portefeuilles linéaires, quadratiques, lorsque le vecteur des facteurs de risques suit un mélange convexe de distributions elliptiques. Aussi, on introduit pour la prémière fois la notion de "portefeuille quadratique" d'actifs de bases (ie. actions).