<p>In this thesis we investigate portfolio optimization under Value at Risk, Average Value at Risk and Limited expected loss constraints in a framework, where stocks follow a geometric Brownian motion. We solve the problem of minimizing Value at Risk and Average Value at Risk, and the problem of finding maximal expected wealth with Value at Risk, Average Value at Risk, Limited expected loss and Variance constraints. Furthermore, in a model where the stocks follow an exponential Ornstein-Uhlenbeck process, we examine portfolio selection under Value at Risk and Average Value at Risk constraints. In both geometric Brownian motion (GBM) and exponential Ornstein-Uhlenbeck (O.U) models, the risk-reward criterion is employed and the optimal strategy is found. Secondly, the Value at Risk, Average Value at Risk and Variance is minimized subject to an expected return constraint. By running numerical experiments we illustrate the effect of Value at Risk, Average Value at Risk, Limited expected loss and Variance on the optimal portfolios. Furthermore, in the exponential O.U model we study the effect of mean-reversion on the optimal strategies. Lastly we compare the leverage in a portfolio where the stocks follow a GBM model to that of a portfolio where the stocks follow the exponential O.U model.</p> / Master of Science (MSc)

由於國民財富增加，對於不動產投資一事越來越熱衷，房屋不再只是供人居住使用，而成為重要的投資工具之一，但一般購屋投資者只考量投資『報酬』，卻忽略其『風險』，且由於傳統上對於投資不動產之風險只能以報酬率的標準差或變異數作計算，僅能知道其風險高或低，並不能夠確實知道其『風險值』，此外，投資者必須有分散風險之觀念，選擇適合的投資工具，以建立最佳的投資組合來分散風險。 本文以『市場風險』為主，並以『購屋者投資』角度，探討國內外衡量不動產投資風險之估計方法、模型，找出風險因子以建立一套衡量不動產投資風險因子之模式，並估計風險值，以評估投資之可行性。以1975第1季年至2003年第4季之預售屋平均房價季資料為主軸之時間範圍，並以台北市為研究的地理範圍，以預售屋住宅為研究標的，並以購屋消費者角色作分析，運用各種風險衡量方法，包括樣本變異數法、指數加權移動平均法、GRACH模型、歷史模擬方法、蒙地卡羅結構法、拔靴法、GRACH－拔靴法及VAR－拔靴法等估計風險值。 本文之實證結果顯示： 一、以考量風險因子之VAR模型Ⅰ－拔靴法及VAR模型Ⅱ－拔靴法所估計之風險值最小，表示投資淨值一千萬元，有5﹪的機率可能的最大損失會大於591,218元或577,564元。 二、以未考量風險因子之歷史模擬法及GARCH－拔靴法所估計風險值較大，表示投資淨值一千萬，有5﹪的機率可能的最大損失會大於2,816,827元或2,344,946元，因此，考量風險因子之VAR模型－拔靴法為較適當之模型，因有考量影響風險之因子，較能準確估計出實際之風險值。 三、假設個案中估計調整後報酬率，在95﹪的信賴水準之下，未考慮風險因子模型估計之調整後報酬率為1.80﹪及2.32﹪，即持有一季後，調整後報酬約18及23萬元左右，而以考量風險因子之模型估計之調整後報酬率為2.37﹪及2.38﹪，即持有一季後報酬約24萬元左右。 四、顯示投資組合於三種不同之投資工具時，當投資預售屋比例較大時，風險值是較小，而投資營建股價比例較大時，其風險值是較大。 / As a result of national wealth increased, regarded real estate investment more and more desires, houses not only supply to live but also become one of investment tool, but general purchase investors only considered invest return but ignored risk at invest, as a result of traditional just estimated standard or variance of return represented risk, just to know the high or low of risk, but should not indeed to know the value at risk, investors must had concept of diversification, choose a appropriate investment tool and built the better portfolio to decrease risk. The current thesis was considered market risk and designed to examine the method or model of measure real estate risk, and looked for risk factors to build a set of model of real estate investment risk factors, and estimated value at risk to evaluate the feasible of investment. The current thesis used dates of time range are from 1975Q1 to 2003Q4, geography range is Taipei, pre-sales residential housing, and role of purchase consumer, apply many kinds of methods of measure risk, including Sample Variance, Exponentially Weighted Moving Average, Generalized Autoregressive Conditional Heteroskedasticity（GRACH）, Historical Simulation Method, Monte Carlo Simulation, Classical Bootstrap, GARCH－Bootstrap and VAR－Bootstrap, to estimate value at risk. The empirical result showed that the first, there had minimum value at risk by considering VAR－Bootstrap of risk factors, represented investitive net value are NT 10,000,000, maximum loss of 5﹪probability will greater than NT 591,218 or NT 577,564. Secondly, there are bigger value at risk by Historical Simulation Method of risk-factors free, represented investitive net value are NT 10,000,000, maximum loss of 5﹪probability will greater than NT 2,816,827 or NT 2,344,946. So used considering VAR－Bootstrap of risk factors were more appropriated model, because model of considering risk factors were able to accurate estimate reality value at risk. The third, case study estimated adjusted return, at 95﹪confidence level, risk-factors estimated rate of adjusted return were 2.37﹪and 2.38﹪, hold one quarterly period the return about two hundred and forty thousand dollars, If we have not consider risk-factors, estimated rate of adjusted return were 1.80﹪and 2.32﹪, hold one quarterly period the return about one hundred and eighty thousand dollars or two hundred and thirty thousand dollars. The last, invest portfolio three kinds of investment tool, if invest ratio of pre-sales residential housing were bigger, then value at risk were smaller, and if invest ratio of construct stock were bigger, then value at risk were bigger.

在本篇文章中，我們採用長期記憶模型來估計S&P500、Nasdaq100和Dow Jones Industrial Index三個股票指數期貨的日收盤價的風險值。為了更準確地計算風險值，本文採用常態分配、t分配以及偏斜t分配來做模型估計以及風險值之計算。有鑒於大多數探討風險值的文獻只考慮買入部位的風險，本研究除了估計買入部位的風險值，也估計放空部位的風險值，以期更能全面性地估算風險。實證結果顯示，ARFIMA-FIGARCH模型配合偏斜t分配較其他兩種分配更能精確地估算樣本內的風險值。基於ARFIMA-FIGARCH模型配合偏斜t分配在樣本內風險值計算的優異表現，我們利用此模型搭配來實際求算樣本外風險值。結果如同樣本內風險值一般，ARFIMA-FIGARCH模型配合偏斜t分配在樣本外也有相當好的風險預測能力。 / In this thesis, we estimate Value-at-Risk (VaR) for daily closing price of three stock index futures contracts, S&P500, Nasdaq100, and Dow Jones, using the double long memory models. Due to the existence of a long-term persistence characterized in our data, the ARFIMA-FIGARCH models are used to compute the VaR. In order to investigate better, three kinds of density distributions, normal, Student-t, and skewed Student-t distributions, are used for estimating models and computing the VaR. In addition to the VaR for the long trading positions which most researches focus on to date, the VaR for the short trading positions are calculated as well in this study. From the empirical results we show that for the three stock index futures, the ARFIMA-FIGARCH models with skewed Student-t distribution perform better in computing in-sample VaR both in long and short trading positions than symmetric models and has a quite excellent performance in forecasting out-of-sample VaR as well.

金融自由化後，金融商品交易的多樣性在活絡金融市場方面佔有很重的份量，也使得投資者有更多樣化的投資管道及標地。投資者購買金融商品除了追求較高的報酬外，對於投資風險的管理也是不容乎視。2007年，美國的次級房貸subprimemortgage風爆使得雷曼兄弟和AIG集團爆發財務危機，正是投資者追求高報酬之後，在風險管理上並未妥善管理所造成。 　　 　　衡量風險時，通常會使用變異數或標準差當做衡量指標，即在衡量其波動性，因此波動性裏含有許多訊息。在本論文中，我們將探討波動性所透露出來的兩個訊息，一個是風險值(VaR)，文中將分別使用二種衡量可解釋長期記憶的GARCH模型探討台股指數期貨及新加坡的摩台股指數期貨這兩個期貨市場的VaR。另外則是試圖尋找出這兩個期貨市場殘差值的波動性之間的長期共整合關係。 本論文主要由三篇文章組成，第一篇是利用Baillie, Bollerslev, and Millelsen (1996) 所提出的長期記憶模型FIGARCH來計算台指期貨的風險值（VaR）；第二篇也是利用長期記憶模型來計算新加坡的摩台指期貨的風險值，但這次的長期記憶模型增加一個由Tse (1998)　提出的可以考慮不對稱性波動的FIAPARCH模型。 　　這兩個模型都搭配三種不同的分配來計算VaR，分別是Normal, Student-t和skewed Student-t分配；實證結果顯示，這兩個期貨市場報酬的波動皆具有長期記憶，表示之前影響指數期貨報酬率的因素對未來指數期貨報酬率會有較長時間的影響力。而在傳統認為差殘值服從常態分配的假定下所計算出的VaR的配適情況較以Student-t分配計算出的VaR的配適情況不具效率，這除了說明傳統的常態分配假說在計算此兩個指數期貨報酬率是不適用之外，亦得出他們是具有肥尾（厚尾）的現象。 　　第三篇則是結合前兩篇的結果來探討此兩個指數期貨報酬率之間的波動性是否具有長期關係。因為台指期貨報酬率與摩台指期貨報酬率的波動性皆具有長期記憶，故在此部分，利用Engle-Granger (1987) 的兩階段共整合模型來求此兩個指數期貨報酬率之間的波動性是否存在長期關係。實證結果顯示，他們確實存在長期共整合關係，且摩台指期貨報酬率的波動性較台指期貨報酬率的波動性強，因此我們可以在台指期貨市場買入期指，而在新加坡的摩台指期貨市場避險 / The finance commodity exchange's multiplicity holds the very heavy component in the detachable money market aspect, after the financial liberalization. It also enables the investor to have many chances and commodities of investment. The investor purchases the financial commodity besides the higher reward, and does not allow regarding investment risk's management to regard. In 2007, the securitization commodity violation of US's subprimemortgage explodes causes Lehman Brothers and the AIG group erupts the financial crisis. This is precisely the investor pursues the high reward, and their administration centers have not created properly in the risk management. When we measure risks, we usually adopt the variance or the standard deviation. That is to weight its property of volatilities. There is much information in the volatilities. In this thesis, we discussed two kinds of information which the property of volatilities discloses. One is the value at risk (VaR hereafter). In this article, we use long-term memory's GARCH model to explain that the VaR of Taiwan stock index futures returns and Singapore's MSCI Taiwan index futures returns. Moreover, we attempts to seek for whether there are long relationship of the residuals volatilities between these two futures markets. This thesis was combined by three essays. The first essay employed the FIGARCH model of Baillie, Bollerslev, and Millelsen (1996) to calculated the VaR of Taiwan stock index futures returns. The second essay employed the FIGARCH model and FIAPARCH model of Tse (1998) to calculated the VaR of Singapore's MSCI Taiwan index futures returns. We calculated the VaRs of the different two futures markets by using the FIGARCH and FIAPARCH models with three different distributions-normal, student-t and skewed student-t. The empirical results showed the two futures markets both has long memory. It is not efficient to calculated the VaRs by using the traditional normal distribution. The Student-t distribution fitted the model better than the normal distribution. The third essay, we employed the Engle-Granger (1987) two-step cointegration model to test whether there are long relationship of the residuals volatilities between the Taiwan stock index futures returns and Singapore's MSCI Taiwan index futures returns. The empirical results showed that there was fractional cointegration between the two futures markets and the volatility in Taiwan stock index futures market is about 83% of that in MSCI Taiwan Index Futures market.

We examined the influence of news, related to the main central banks, on the conditional volatility of the stock returns of eighteen major European banks. We model their conditional volatility with GARCH, EGARCH and TGARCH models plugging in variables representing news. As a practical application we evaluate whether applying the news into the volatility modeling improves the performance of the Value-at-Risk (VaR) measure for given banks. The two types of news variables we use are constructed from the press releases of main central banks and from the search query at Factiva Dow Jones news database. The information contained in news is proxied by daily news counts. Using the EGARCH setup we are able to model individual volatility reaction functions of the banks' stock returns to different news variables. We show that the content, origin of the news and also the amount of news (news count) matter to the conditional volatility behavior. The results confirm that increase in the amount of media coverage causes increase in volatility. Certain news types have calming effect (speeches of the central banks' representatives) on volatility while others stir it (monetary news). Finally, we conclude that adding the news into the modeling only slightly improves the VaR out-of-sample performance.

[pt] A partir da década de 90, a metodologia de Valor em Risco (VaR) se difundiu pelo mundo, tanto em instituições financeiras quanto em não financeiras, como uma boa prática de mensuração de riscos. Um dos fatos estilizados mais pronunciados acerca das distribuições de retornos financeiros diz respeito à presença de caudas pesadas. Isso torna os modelos paramétricos tradicionais de cálculo de Valor em Risco (VaR) inadequados para a estimação de VaR de baixas probabilidades, dado que estes se baseiam na hipótese de normalidade para as distribuições dos retornos. Sendo assim, o objetivo do presente trabalho é investigar o desempenho de modelos baseados na Teoria dos Valores Extremos para o cálculo do VaR. Os resultados indicam que os modelos baseados na Teoria dos Valores Extremos são adequados para a modelagem das caudas, e consequentemente para a estimação de Valor em Risco quando os níveis de probabilidade de interesse são baixos. / [en] Since the 90 decade, the use of Value at Risk (VaR) methodology has been disseminated among both financial and non-financial institutions around the world, as a good practice in terms of risks management. The existence of fat tails is one of the striking stylized facts of financial returns distributions. This fact makes the use of traditional parametric models for Value at Risk (VaR) estimation unsuitable for the estimation of low probability events. This is because traditional models are based on the conditional normality assumption for financial returns distributions. The main purpose of this dissertation is to investigate the performance of VaR models based on Extreme Value Theory. The results indicates that Extreme Value Theory based models are suitable for low probability VaR estimation.

[pt] O objetivo principal do trabalho é o estudo dos resultados obtidos com a aplicação de diferentes modelos para estimar a volatilidade das ações brasileiras. Foram analisadas as séries de retornos diários de seis ações, num período de 1200 dias de pregão. Inicialmente, as séries foram estudadas quanto a suas propriedades estatísticas: estacionariedade, distribuição incondicional e independência. Concluiu-se que as séries são estacionárias na média, mas não houve conclusão quanto à variância, nesta análise inicial. A distribuição dos retornos não é normal, por apresentar leptocurtose. Os retornos mostraram dependência no tempo, linear e, principalmente, não linear. Modelada a dependência linear, foram aplicados dez modelos diferentes para tentar capturar a dependência não linear através da modelagem da volatilidade: os modelos foram avaliados, dentro e fora da amostra, pelos seus resíduos e pelos erros de previsão. Os resultados indicaram que os modelos menos elaborados tendem a representar pior o processo gerador dos dados, mas que os modelos pouco parcimoniosos são de difícil estimação e seus resultados não correspondem ao que seria esperado em função de sua sofisticação. As volatilidades estimadas pelos dez modelos foram utilizadas para prever valor em risco (VaR), usando- se dois processos para determinar os quantis das distribuições dos resíduos: distribuição empírica e teoria de valores extremos. Os resultados indicaram que os modelos menos elaborados prevêem melhor o VaR. Isto se deve à não estacionariedade das séries na variância, que fica evidente ao longo do trabalho. / [en] This thesis aims to study the results of applying different models to estimate Brazilian stock volatilities. The models are applied to six series of daily returns, and each series has 1200 days. We studied first the series` main statistical features: Stationarity, unconditional distribution and independence. We concluded that the series are mean stationary, but there was no conclusion on variance stationarity, in this first analysis. Return distribution is not normal, because of the high kurtosis. Returns showed time dependence, linear and, mainly, not linear. We modeled the linear dependence, and then applied ten different volatility models, in order to try to capture the non linear dependence. We evaluated the different models, in sample and out of sample, by analyzing their residuals and their forecast errors. The results showed that the less sophisticated models tend to give a worst representation of the data generating process; they also showed that the less parsimonious models are difficult to estimate, and their results are not as good as we could expect from their sophistication. We used the ten models` volatility forecasts to estimate value-at-risk (VaR) and two methods to estimate the residual distribution quantiles: empirical distribution and extreme value theory. The results showed that the less sophisticated models give better VaR estimates. This is a consequence of the variance non stationarity, that became apparent along the thesis. / [es] EL objetivo principal del trabajo es el estudio de los resultados obtenidos con la aplicación dediferentes modelos para estimar la volatilidad de las acciones brasileras. Fueron analizadas series de retornos diários de seis acciones, en un período de 1200 días de pregón. Inicialmente, las series fueron estudiadas con respecto a sus propriedades estadísticas: estacionalidad, distribucción incondicional e independencia. Se concluye que las series son estacionarias en la media, pero no se llega a ninguna conclusión respecto a la varianza, en este análisis inicial. La distribucción de los retornos no es normal, ya que presenta leptocurtosis. Los retornos muestran dependencia en el tempo, lineal y, principalmente, no lineal. Después de modelar la dependencia lineal, se aplicaron diez modelos diferentes para intentar capturar la dependencia no lineal modelando la volatilidad: los modelos fueron evaluados, dentro y fuera de la amostra, por sus residuos y por los errores de previsión. Los resultados indicaran que los modelos menos elaborados tienden a representar peor el proceso generador de los datos, mientras que los modelos poco parcimoniosos son de difícil estimación y sus resultados no corresponden al que sería esperado en función de su sofisticación. Las volatilidades estimadas por los diez modelos se utilizaron para prever valor en riesgo (VaR), usando dos procesos para determinar los quantis de las distribuciones de los residuos: distribucción empírica y teoría de valores extremos. Los resultados indicaran que los modelos menos elaborados preveen mejor el VaR. Esto se debe a la no estacionalidad de las series en la varianza, que resulta evidente a lo largo del trabajo.

This thesis addresses some practical issues that are similar to what a risk manager would be facing. To protect portfolio against unexpected turbulent drop, risk managers might use options to hedge the portfolio. Since the price of an option is not a linear function of the price of the underlying security or index, consequently option hedged portfolio's value is a not linear combination of the market prices of the underlying securities. Three Value-at-Risk (VaR) models, traditional estimate based Monte Carlo model, GARCH based Monte Carlo model, and resampling model, are developed to estimate risk of non-linear portfolios. The results from the models by setting different levels of hedging strategies are useful to evaluate and compare these strategies, and therefore may assist risk managers in making practical decisions in risk management.

This work presents an empirical analysis of popular scenario generation methods for stochastic optimization, including quasi-Monte Carlo, moment matching, and methods based on probability metrics, as well as a new method referred to as Voronoi cell sampling. Solution quality is assessed by measuring the error that arises from using scenarios to solve a multi-dimensional newsvendor problem, for which analytical solutions are available. In addition to the expected value, the work also studies scenario quality when minimizing the expected shortfall using the conditional value-at-risk. To quickly solve problems with millions of random parameters, a reformulation of the risk-averse newsvendor problem is proposed which can be solved via Benders decomposition. The empirical analysis identifies Voronoi cell sampling as the method that provides the lowest errors, with particularly good results for heavy-tailed distributions. A controversial finding concerns evidence for the ineffectiveness of widely used methods based on minimizing probability metrics under high-dimensional randomness.

A análise de risco de mercado, o risco associado a perdas financeiras resultantes de utilizações de preços de mercado, é fundamental para instituições financeiras e gestores de carteiras. A alocação dos ativos nas carteiras envolve decisões risco/retorno eficientes, frequentemente limitadas por uma política de risco. Muitos modelos tradicionais simplificam a estimação do risco de mercado impondo muitas suposições, como distribuições simétricas, correlações lineares, normalidade, entre outras. A utilização de cópulas exibiliza a estimação da estrutura de dependência dessas séries de tempo, possibilitando a modelagem de séries de tempo multivariadas em dois passos: estimações marginais e da dependência entre as séries. Neste trabalho, utilizou-se um modelo de cópulas com dinâmica semiparamétrica para medição de risco de mercado. A estrutura dinâmica das cópulas conta com um parâmetro de dependência que varia ao longo do tempo, em que a proposta semiparamétrica possibilita a modelagem de qualquer tipo de forma funcional que a estrutura dinâmica venha a apresentar. O modelo proposto por Hafner e Reznikova (2010), de dinâmica semiparamétrica, é comparado com o modelo sugerido por Patton (2006), que apresenta dinâmica paramétrica. Todas as cópulas no trabalho são bivariadas. Os dados consistem em quatro séries de tempo do mercado brasileiro de ações. Para cada um desses pares, utilizou-se modelos ARMA-GARCH para a estimação das marginais, enquanto a dependência entre as séries foi estimada utilizando os dois modelos de cópulas dinâmicas mencionados. Para comparar as metodologias estimaram-se duas medidas de risco de mercado: Valor em Risco e Expected Shortfall. Testes de hipóteses foram implementados para verificar a qualidade das estimativas de risco. / Market risk management, i.e. managing the risk associated with nancial loss resulting from market price uctuations, is fundamental to nancial institutions and portfolio managers. Allocations involve e cient risk/return decisions, often restricted by an investment policy statement. Many traditional models simplify risk estimation imposing several assumptions, like symmetrical distributions, the existence of only linear correlations, normality, among others. The modelling of the dependence structure of these time series can be exibly achieved by using copulas. This approach can model a complex multivariate time series structure by analyzing the problem in two blocks: marginal distributions estimation and dependence estimation. The dynamic structure of these copulas can account for a dependence parameter that changes over time, whereas the semiparametric option makes it possible to model any kind of functional form in the dynamic structure. We compare the model suggested by Hafner and Reznikova (2010), which is a dynamic semiparametric one, with the model suggested by Patton (2006), which is also dynamic but fully parametric. The copulas in this work are all bivariate. The data consists of four Brazilian stock market time series. For each of these pairs, ARMA-GARCH models have been used to model the marginals, while the dependences between the series are modeled by using the two methods mentioned above. For the comparison between these methodologies, we estimate Value at Risk and Expected Shortfall of the portfolios built for each pair of assets. Hypothesis tests are implemented to verify the quality of the risk estimates.