This thesis project studies robust portfolio optimization with Expected Short-fall applied to a reference portfolio consisting of Swedish linear assets with stocks and a bond index. Specifically, the classical robust optimization definition, focusing on uncertainties in parameters, is extended to also include uncertainties in log-return distribution. My contribution to the robust optimization community is to study portfolio optimization with Expected Shortfall with log-returns modeled by either elliptical distributions or by a normal copula with asymmetric marginal distributions. The robust optimization problem is solved with worst-case parameters from box and ellipsoidal un-certainty sets constructed from historical data and may be used when an investor has a more conservative view on the market than history suggests. With elliptically distributed log-returns, the optimization problem is equivalent to Markowitz mean-variance optimization, connected through the risk aversion coefficient. The results show that the optimal holding vector is almost independent of elliptical distribution used to model log-returns, while Expected Shortfall is strongly dependent on elliptical distribution with higher Expected Shortfall as a result of fatter distribution tails. To model the tails of the log-returns asymmetrically, generalized Pareto distributions are used together with a normal copula to capture multivariate dependence. In this case, the optimization problem is not equivalent to Markowitz mean-variance optimization and the advantages of using Expected Shortfall as risk measure are utilized. With the asymmetric log-return model there is a noticeable difference in optimal holding vector compared to the elliptical distributed model. Furthermore the Expected Shortfall in-creases, which follows from better modeled distribution tails. The general conclusions in this thesis project is that portfolio optimization with Expected Shortfall is an important problem being advantageous over Markowitz mean-variance optimization problem when log-returns are modeled with asymmetric distributions. The major drawback of portfolio optimization with Expected Shortfall is that it is a simulation based optimization problem introducing statistical uncertainty, and if the log-returns are drawn from a copula the simulation process involves more steps which potentially can make the program slower than drawing from an elliptical distribution. Thus, portfolio optimization with Expected Shortfall is appropriate to employ when trades are made on daily basis. / Examensarbetet behandlar robust portföljoptimering med Expected Shortfall tillämpad på en referensportfölj bestående av svenska linjära tillgångar med aktier och ett obligationsindex. Specifikt så utvidgas den klassiska definitionen av robust optimering som fokuserar på parameterosäkerhet till att även inkludera osäkerhet i log-avkastningsfördelning. Mitt bidrag till den robusta optimeringslitteraturen är att studera portföljoptimering med Expected Shortfall med log-avkastningar modellerade med antingen elliptiska fördelningar eller med en norma-copul med asymmetriska marginalfördelningar. Det robusta optimeringsproblemet löses med värsta tänkbara scenario parametrar från box och ellipsoid osäkerhetsset konstruerade från historiska data och kan användas när investeraren har en mer konservativ syn på marknaden än vad den historiska datan föreslår. Med elliptiskt fördelade log-avkastningar är optimeringsproblemet ekvivalent med Markowitz väntevärde-varians optimering, kopplade med riskaversionskoefficienten. Resultaten visar att den optimala viktvektorn är nästan oberoende av vilken elliptisk fördelning som används för att modellera log-avkastningar, medan Expected Shortfall är starkt beroende av elliptisk fördelning med högre Expected Shortfall som resultat av fetare fördelningssvansar. För att modellera svansarna till log-avkastningsfördelningen asymmetriskt används generaliserade Paretofördelningar tillsammans med en normal-copula för att fånga det multivariata beroendet. I det här fallet är optimeringsproblemet inte ekvivalent till Markowitz väntevärde-varians optimering och fördelarna med att använda Expected Shortfall som riskmått används. Med asymmetrisk log-avkastningsmodell uppstår märkbara skillnader i optimala viktvektorn jämfört med elliptiska fördelningsmodeller. Därutöver ökar Expected Shortfall, vilket följer av bättre modellerade fördelningssvansar. De generella slutsatserna i examensarbetet är att portföljoptimering med Expected Shortfall är ett viktigt problem som är fördelaktigt över Markowitz väntevärde-varians optimering när log-avkastningar är modellerade med asymmetriska fördelningar. Den största nackdelen med portföljoptimering med Expected Shortfall är att det är ett simuleringsbaserat optimeringsproblem som introducerar statistisk osäkerhet, och om log-avkastningar dras från en copula så involverar simuleringsprocessen flera steg som potentiellt kan göra programmet långsammare än att dra från en elliptisk fördelning. Därför är portföljoptimering med Expected Shortfall lämpligt att använda när handel sker på daglig basis.

Capital Asset Pricing Model (CAPM) is an equilibrium model to test relationship between expected return and market risk (Sharpe, 1964). The model research on pricing and return when the securities market reaches equilibrium and investors are rational and investing by diversification based on Markovitz portfolio theory (Markovitz, 1952). Fama and MacBeth (1973) proposed a cross-sectional testing methodology on CAPM and this regression method has been widely used in testing CAPM in developed markets since then. While CAPM is hard to explain more and more market anomalies (excessive return in smaller market value company) in cross section regression, Fama and French (1992) added two more factors (SMB and HML) and proposed three factor model. The empirical results show that three factor model is superior to CAPM in developed markets. Relevant studies have been conducted by Manjuunatha (2006) and Trimech et al. (2015) but show different results. This dissertation will use Fama-MacBeth cross section approach to test CAPM and Fama-French's three factor model in Chinese and Polish stock market respectively. Following Fama and MacBeth (1972) and Shweta and Anil (2015), three sub periods of Polish and Chinese stock market returns ranging from 2007 to 2018 are examined. The empirical results in this thesis...

本文係針對臺灣50指數期貨與基金於2003年6月30日上市之後，臺灣期貨及現貨市場報酬率間領先落後關係與波動性的變化來進行探討。研究分為兩部份，第一部份是觀察臺灣50指數期貨與現貨之間的關聯性，並探討臺灣加權股價指數、金融保險類股股價指數及電子類股股價指數期貨與現貨市場間的變化；第二部份是採用可模擬現貨走勢的臺灣50指數基金、國泰金及臺積電的股價來做為現貨的替代變數，觀察其與期貨之間的關連性是否與第一部份的結果類似，若是實證結果極為相同，則相關機構與一般投資人將可運用各期貨與其標的指數中市值最大的股票來進行套利操作。此外，本文在進行模型估計時，首度採用一階段估計法，來聯合估計雙變量GARCH模型中的條件平均數方程式與條件變異數方程式，以避免過去相關文獻將兩條方程式個別估計時所造成的估計誤差。 實證結果所獲得的重要結論如下：首先，臺灣期貨市場的發展仍未趨成熟，並不具有價格發現的功能，在考慮風險溢酬方面，僅有臺灣50指數期貨與現貨的投資人會在報酬率之外，額外要求用以補償的風險溢酬，再者，臺灣50指數期貨與基金的上市，並沒有對臺灣現有的期貨與現貨市場造成顯著的影響，然而，替代變數並不能完全取代現貨指數，但相較之下，國泰金在臺灣50指數期貨與基金上市之後的那段期間模擬成效最好。 / This paper investigates the change of lead-lag relationship in returns and volatilities in Taiwan futures and spot markets after the introduction of Taiwan 50 Futures and Taiwan Top50 Tracker Fund (TTT) on June 30, 2003. The study divides into two parts. The first part examines the relationship between Taiwan 50 Futures and spot markets, and also discusses the change of Taiwan Stock Exchange Capitalization Weighted Stock Index, Taiwan Stock Exchange Banking and Insurance Sector Index, and Taiwan Stock Exchange Electronic Sector Index in futures and spot markets. Another part uses the stock price of TTT, Cathay Financial Holding Company and Taiwan Semiconductor Manufacturing Company as the substitutive variables of spot index and goes a step further to examine the relationships between them and futures individually. Additionally, this research used One-Pass Method for first time to estimate jointly the conditional mean equation and conditional variance equation of Bivariate GARCH Model to avoid estimating error in previous relative studies with Two-Pass Method. The major empirical results are as follows: first, the development of Taiwan futures market is incomplete. The futures market does not play the price discovery role to the spot market. Second, under the consideration of risk premium, only investors in Taiwan 50 Futures and spot markets would ask for compensated risk premium excepting returns. Third, the opening of Taiwan 50 Futures and TTT does not influence significantly Taiwan futures and spot markets. Last but not least, these substitutive variables can not replace spot index perfectly. However, comparing with others, the stock price of Cathay Financial Holding Company is the very model of Taiwan Stock Exchange Banking and Insurance Sector Index after the introduction of Taiwan 50 Futures and TTT.

附最低保證投資型保險商品的特色在於無論投資者的投資績效好壞，保險金額皆享有一最低投資保證，過去關於此類商品的研究皆假設標的資產為單一資產，或依固定比例之投資組合，並沒有考慮到投資人自行配置投資組合的效果，但大部分市售商品中，投資人可以自行配置投資標，此情況之下，保險公司如何衡量適當的保證成本即為一相當重要之課題。 本研究假設投資人風險偏好服從冪次效用函數，並假設與保單所連結之投資標的有兩種資產，一為具有高風險高報酬的資產，另一為具有低風險低報酬之資產，在每個保單年度之初，投資人可以選擇配置在兩種資產之比例，我們運用黃迪揚(2009)所提出的動態規劃數值解之方法，計算出在考慮投資人自行配置資產之下，保證成本將會比固定比例之投資高出12個百分點。 此外，為了瞭解在不同資產報酬率的模型之下，保證成本是否會有不一樣的結論，除了對數常態模型之外，我們假設高風險資產與低風險資產服從ARIMA-GARCH(Autoregressive Integrated Moving Average-Generalized Autoregressive Conditional Heteroscedastic )模型，並得到較高的保證成本。 / The main characteristic of variable annuities (VA) with minimum benefits is that the benefit will be guaranteed. Previous literatures assume a specific underling asset return process when considering the guaranteed cost of VA; but they do not consider the portfolio choice opportunity of the policyholders. However, it is common for policyholders to rebalance his portfolio in many types of VA products. Therefore it’s important for insurance companies to apply an approximate method to measure the guaranteed cost. In this research, we assume that there are two potential assets in policyholders’ portfolio; one with high risk and high return and the other one with low risk and low return. The utility function of the policyholder is assumed to follow a power utility. We consider the asset allocation effect on the guaranteed cost for a VA with guaranteed minimum withdrawal benefits, finding that the guaranteed cost will increase 12% compared with a specific underling asset. The model effect of the asset return process is also examined by considering two different asset processes, the lognormal model and ARIMA-GARCH model. The solution of dynamic programming problem is solved by the numerical approach proposed by Huang (2009). Finally we get the conclusion which the guaranteed cost given by the ARIMA-GARCH model is greater than the lognormal model.

Made available in DSpace on 2016-06-02T20:06:01Z (GMT). No. of bitstreams: 1 2117.pdf: 990773 bytes, checksum: a7b62936541ab91d8ae3424f62aa0f40 (MD5) Previous issue date: 2008-08-22 / In the financial market usually notices are taken of the shares sequentially over the time in order to characterize them a time series. However, the major interest is to forecast the behavior of these shares. Motivated by this fact, a lot of models were created based on the past information considering constant averages and variance over time. Although, in financial series a feature often presented is called volatility, which can be noticed by the variance to vary in time. In order to catch this characteristic were developed the models of the family GARCH, that model the conditional variance through known information. These models were well used and have passed by many formulation modifications to be able to catch different effects, such as the effect leverage EGARCH. Thus, the goal is to estimate volatility patterns obeying the specifications of the family GARCH verifying which ones of them describe better the data inside and outside the sample. / No mercado financeiro costuma-se fazer observações sobre as carteiras sequencialmente ao longo do tempo, caracterizando uma série temporal. Contudo, o maior interesse está em prever o comportamento destas carteiras. Motivado por este fato, foram criados muitos modelos de previsão baseando-se em observações passadas considerando a média e variância constantes no tempo. Porém, nas séries financeiras uma característica muito presente é a chamada volatilidade, que pode ser observada pela variância não constante no tempo. A fim de captar esta característica, desenvolveram-se os modelos da família GARCH, que modelam a variância condicional através de informações passadas. Estes modelos foram muito utilizados e sofreram muitas modificações nas formulações para poderem captar diferentes efeitos, como o efeito de leverage (EGARCH). Assim, deseja-se estimar modelos de volatilidade obedecendo às especificações da família GARCH, verificando quais deles descrevem melhor os dados dentro e fora da amostra.

This master's thesis studies portfolio optimization using linear programming algorithms. The contribution of this thesis is an extension of the convex framework for portfolio optimization with Conditional Value-at-Risk, introduced by Rockafeller and Uryasev. The extended framework considers risk measures in this thesis belonging to the intersecting classes of coherent risk measures and distortion risk measures, which are known as coherent distortion risk measures. The considered risk measures belonging to this class are the Conditional Value-at-Risk, the Wang Transform, the Block Maxima and the Dual Block Maxima measures. The extended portfolio optimization framework is applied to a reference portfolio consisting of stocks, options and a bond index. All assets are from the Swedish market. The returns of the assets in the reference portfolio are modelled with elliptical distribution and normal copulas with asymmetric marginal return distributions. The portfolio optimization framework is a simulation-based framework that measures the risk using the simulated scenarios from the assumed portfolio distribution model. To model the return data with asymmetric distributions, the tails of the marginal distributions are fitted with generalized Pareto distributions, and the dependence structure between the assets are captured using a normal copula. The result obtained from the optimizations is compared to different distributional return assumptions of the portfolio and the four risk measures. A Markowitz solution to the problem is computed using the mean average deviation as the risk measure. The solution is the benchmark solution which optimal solutions using the coherent distortion risk measures are compared to. The coherent distortion risk measures have the tractable property of being able to assign user-defined weights to different parts of the loss distribution and hence value increasing loss severities as greater risks. The user-defined loss weighting property and the asymmetric return distribution models are used to find optimal portfolios that account for extreme losses. An important finding of this project is that optimal solutions for asset returns simulated from asymmetric distributions are associated with greater risks, which is a consequence of more accurate modelling of distribution tails. Furthermore, weighting larger losses with increasingly larger weights show that the portfolio risk is greater, and a safer position is taken. / Denna masteruppsats behandlar portföljoptimering med linjära programmeringsalgoritmer. Bidraget av uppsatsen är en utvidgning av det konvexa ramverket för portföljoptimering med Conditional Value-at-Risk, som introducerades av Rockafeller och Uryasev. Det utvidgade ramverket behandlar riskmått som tillhör en sammansättning av den koherenta riskmåttklassen och distortions riksmåttklassen. Denna klass benämns som koherenta distortionsriskmått. De riskmått som tillhör denna klass och behandlas i uppsatsen och är Conditional Value-at-Risk, Wang Transformen, Block Maxima och Dual Block Maxima måtten. Det utvidgade portföljoptimeringsramverket appliceras på en referensportfölj bestående av aktier, optioner och ett obligationsindex från den Svenska aktiemarknaden. Tillgångarnas avkastningar, i referens portföljen, modelleras med både elliptiska fördelningar och normal-copula med asymmetriska marginalfördelningar. Portföljoptimeringsramverket är ett simuleringsbaserat ramverk som mäter risk baserat på scenarion simulerade från fördelningsmodellen som antagits för portföljen. För att modellera tillgångarnas avkastningar med asymmetriska fördelningar modelleras marginalfördelningarnas svansar med generaliserade Paretofördelningar och en normal-copula modellerar det ömsesidiga beroendet mellan tillgångarna. Resultatet av portföljoptimeringarna jämförs sinsemellan för de olika portföljernas avkastningsantaganden och de fyra riskmåtten. Problemet löses även med Markowitz optimering där "mean average deviation" används som riskmått. Denna lösning kommer vara den "benchmarklösning" som kommer jämföras mot de optimala lösningarna vilka beräknas i optimeringen med de koherenta distortionsriskmåtten. Den speciella egenskapen hos de koherenta distortionsriskmåtten som gör det möjligt att ange användarspecificerade vikter vid olika delar av förlustfördelningen och kan därför värdera mer extrema förluster som större risker. Den användardefinerade viktningsegenskapen hos riskmåtten studeras i kombination med den asymmetriska fördelningsmodellen för att utforska portföljer som tar extrema förluster i beaktande. En viktig upptäckt är att optimala lösningar till avkastningar som är modellerade med asymmetriska fördelningar är associerade med ökad risk, vilket är en konsekvens av mer exakt modellering av tillgångarnas fördelningssvansar. En annan upptäckt är, om större vikter läggs på högre förluster så ökar portföljrisken och en säkrare portföljstrategi antas.