This thesis is an implementation project of a portfolio optimization model, with the purpose of creating a decision support tool. It aims to provide quantitative input to the portfolio construction process at Handelsbanken Fonder, by applying Konno & Yamazaki’s Mean Absolute Deviation method, with a Feinstein & Thapa modification. Additionally, the Black-Litterman model is implemented to approximate the input of expected return. The linear optimization problem was then solved by the Simplex algorithm. The main deliverable is a model that can assist portfolio managers in making investment decisions. Back-testing of the model showed that it did not outperform the benchmark portfolios, which is likely a result of only allowing long positions in the model. Nevertheless, the model provides value by giving the user a second opinion on the efficient frontier, for any given investment decision. / Den här uppsatsen är ettimplementationsprojekt av enportföljoptimerings-modell, med syftet att skapaett beslutsstödjande verktyg. Den strävar efter att ge ett kvantitativt bidragtill portföljallokerings-processen på Handelsbanken Fonder, genom att användaKonno & Yamazaki’s Mean Absolute Deviation-metod med en Feinstein &Thapa-modifiering. Vidare har Black-Littermanmodellen implementerats för attapproximera den förväntade avkastningen. Det linjära optimeringsproblemetlöstes sedan med Simplex-algorithmen. Det huvudsakliga resultatet är en modellsom kan assisterafondförvaltare i investeringsbeslut. Utförda utfallstestvisade att modellen inte överträffade de använda benchmark-fonderna, vilketsannolikt är ett resultat av att modellen enbart tillåterlånga positioner.Likväl, kan modellen vara värdefull genom att erbjuda användaren ett alternativpå den effektiva fronten, för ett givet investeringsbeslut.

Portfolio optimization is a crucial practice in finance aimed at maximizing the return while minimizing the risk through strategic asset allocation. This paper explores two distinct approaches to modeling robust portfolio optimization, comparing their efficacy in balancing the return and the risk. The first approach focuses on diversifying the portfolio by varying the number of stocks and sector allocation, while the second approach emphasizes minimizing risk by selecting stocks with low correlation. Theoretical foundations and mathematical formulations underpinning these approaches are discussed, incorporating concepts from Modern Portfolio Theory and Mixed Integer Linear Programming. Practical implementation involves data collection from Yahoo Finance API and computational analysis using Python and the optimization tool Gurobi. The results of these methodologies are evaluated, considering factors such as budget constraints, maximum and minimum investment limits, binary constraints, and correlation thresholds. The study concludes by discussing the implications of these findings and their relevance in contemporary financial decision-making processes.

本篇論文主要在外匯市場建立市場投資組合、利差交易投資組合與動能交易投資組合，探討透過不同情境適當改變投資組合比重配置，是否能夠顯著提升交易策略的報酬表現。 以1999年1月至2015年10月為樣本期間，根據28個國家外匯市場資料建構市場投資組合、利差交易投資組合與動能交易投資組合等，之後根據三種投資組合報酬情況透過馬可夫情境轉換模型區分成三種情境。按三種情境的各種投資組合超額報酬表現，再利用馬可維茲的平均數-變異數投資組合模型配置各情境下各項交易的比重，再依據計算出的預期情境與相對應比重進行投資。其結果顯示在樣本期間內，本篇論文的交易策略相較於外匯市場投資組合、利差交易投資組合與動能交易組合有較佳的投資表現。 在樣本外測試部分，採用自2012年中開始的連續情境二資料進行分析。報酬方面，在其他交易型態呈現負報酬較多情況下，就本文交易策略而言，投資者隨時根據其各種交易平均報酬與共變異數進行交易比重配置，適時放空交易策略或投資無風險資產，產生正報酬。但從標準差可以推斷投資者面對未來的不確定，在整個樣本外期間歷時的34個月當中標準差亦無法有效降低，說明了投資者面對下一期總體環境的高不確定性。 / In this thesis, we mainly investigate whether it could improve the performance of currency portfolio by adjusting weights among carry trade, momentum and market return in foreign exchange market under different kinds of regimes. Based on a sample of 28 market currencies, we form three kinds of transactions in our portfolio, including carry trade, momentum, and market return. Under Markov switching model, we divide the sample period into three regimes, and then determine weights among carry trade, momentum and market return by parameters of each re-gime using Markowitz mean-variance analysis. Finally, we invest different weights among three transactions according to each expected regime. We find the result that although the return of the strategy is just a little higher than the carry trade, the risk is much lower compared to other transactions. In our out-of-sample testing, we analyze the performance by using the data of the regime two which begins September, 2012. With the respect to the return, most of other risky transactions have negative return, but we get positive return by adjusting the long position and short position according to the result of the mean-variance anal-ysis. However, we can not effectively reduce risk by using the strategy, and in the meantime it can explain the high uncertainty investors face toward the next period.

Die Gewichte eines Portfolios liegen meist als als Kombination des Produkts der Präzisionsmatrix und des Erwartungswertvektors vor. In der Praxis müssen diese Parameter geschätzt werden, allerdings ist die Beschreibung der damit verbundenen Schätzunsicherheit über eine Verteilung dieses Produktes eine Herausforderung. In dieser Arbeit wird demonstriert, dass ein geeignetes bayesianisches Modell nicht nur zu einer leicht zugänglichen Posteriori-Verteilung führt, sondern auch zu leicht interpretierbaren Beschreibungen des Portfoliorisikos, wie beispielsweise einer Ausfallwahrscheinlichkeit des gesamten Portfolios zu jedem Zeitpunkt. Dazu werden die Parameter mit ihren konjugierten Prioris ausgestatet. Mit Hilfe bekannter Ergebnisse aus der Theorie multivariater Verteilungen ist es möglich, eine stochastische Darstellung für relevante Ausdrücke wie den Portfoliogewichten oder des effizienten Randes zu geben. Diese Darstellungen ermöglichen nicht nur die Bestimmung von Bayes-Schätzern der Parameter, sondern sind auch noch rechentechnisch hoch effizient, da Zufallszahlen nur aus bekannten und leicht zugänglichen Verteilungen gezogen werden. Insbesondere aber werden Markov-Chain-Monte-Carlo Methoden nicht benötigt. Angewendet wird diese Methodik an einem mehrperiodigen Portfoliomodell für eine exponentielle Nutzenfunktion, am Tangentialportfolio, zur Schätzung des effizienten Randes, des globalen Minimum-Varianz-Portfolios wie auch am gesamten Mittelwert-Varianz Ansatzes. Für alle behandelten Portfoliomodelle werden für wichtige Größen stochastische Darstellungen oder Bayes-Schätzer gefunden. Die Praktikabilität und Flexibilität wie auch bestimmte Eigenschaften werden in Anwendungen mit realen Datensätzen oder Simulationen illustriert. / Usually, the weights of portfolio assets are expressed as a comination of the product of the precision matrix and the mean vector. These parameters have to be estimated in practical applications. But it is a challenge to describe the associated estimation risk of this product. It is demonstrated in this thesis, that a suitable Bayesian approach does not only lead to an easily accessible posteriori distribution, but also leads to easily interpretable risk measures. This also includes for example the default probability of the portfolio at all relevant points in time. To approach this task, the parameters are endowed with their conjugate priors. Using results from the theory of multivariate distributions, stochastic representations for the portfolio parameter are derived, for example for the portfolio weights or the efficient frontier. These representations not only allow to derive Bayes estimates of these parameters, but are computationally highly efficient since all th necessary random variables are drawn from well known and easily accessible distributions. Most importantly, Markov-Chain-Monte-Carlo methods are not necessary. These methods are applied to a multi-period portfolio for an exponential utility function, to the tangent portfolio, to estimate the efficient frontier and also to a general mean-variance approach. Stochastic representations and Bayes estimates are derived for all relevant parameters. The practicability and flexibility as well as specific properties are demonstrated using either real data or simulations.

Student Number : 9005994G - MSc research report - School of Construction Economics and Management - Faculty of Engineering and the Built Environment / The objectives of this research are to assess the returns, risks and correlation of a mixed-asset portfolio, establish the role of real estate in a mixed-asset portfolio and suggest an appropriate real estate allocation in the South African pension fund industry. The issue of low real estate allocation has been a subject of interest to practitioners and academics, both locally and internationally, despite the diversification benefit that real estate provides in a mixed-asset portfolio. A statistical approach was considered most the appropriate tool for analyzing returns. Solver optimizer in the excel spreadsheet package was used to generate efficient frontiers and the associated values of portfolios. Real estate provided a lower return of 1.25% and a lower standard deviation of 4.90% compared to equities with a return of 1.39% and the highest standard deviation of 6.23%, whilst bonds provided the best risk-return trade off, with a return of 1.42% and the lowest standard deviation of 2.64%. An equally -weighted portfolio consisting of bonds and stocks and a portfolio consisting bonds, stocks and real estate was simulated. The equally -weighted portfolio of bonds and stocks provides a return of 1.41% and a standard deviation of 3.76%. The minimum variance with bias to bonds provides a higher return of 1.42% at a lower level of risk of 2.62%. The equally -weighted portfolio consisting of bonds, stocks and real estate provides a return of 1.35%, with a lower risk of 3.49%. The minimum variance with bias to bonds provides almost the same return of 1.40% at a lower level of risk of 2.54% compared to the bond and share portfolio. The Chi-Square statistical tool was used to test the diversification benefit of real estate. It can be concluded that the standard deviation of the portfolio with property is close enough to the standard deviation without property of 3.76% and cannot statistically say that it is different given the 5% level of significance. The Sharpe ratio was used to test the favourable risk adjusted returns offered by real estate. It concluded that property provides favourable risk adjusted returns and diversification benefits, as illustrated with the increasing portfolio return from 7.44% to 8.66% on Sharpe ratio basis and standard deviation of the portfolio decreasing from 3.76% to 2.54%. The literature review generally supported the view that real estate has a role in a mixed-asset portfolio. Research topics such as securitized versus unsecuritized real estate, real estate allocation and diversification, returns and risk, inflation hedging, modern portfolio theory and the efficient -frontier were analysed and related to the research report. The empirical analysis supports the hypothesis that real estate provides diversification benefits. The property cycle is positive and it is supported by positive property fundamentals like (vacancies are at lowest levels, capitalisation rates are strengthening, the property cycle is turning positive and a stable interest rate environment). The positive property fundamentals will lead to earnings growth. An allocation of between 10% based on the lower end of the minimum variance and 15% based on the lower end of the risk/return ratio is recommended for a mixed-asset portfolio.

Com a crescente desregulamentação dos mercados de energia, os diferentes participantes dos mercados se deparam com a necessidade de gerenciar de maneira eficiente seus investimentos em energia elétrica. Nesse cenário, a otimização das Carteiras de Contratos mostra-se uma técnica interessante no planejamento estratégico dos agentes de mercados de energia. Os mercados estão frequentemente expostos a riscos de diversas fontes, assim, a mitigação dos mesmos é fundamental. A Teoria do Portfólio, proposta por Harry Markowitz, tem sido utilizada em análises envolvendo diversos mercados. Este trabalho analisa um problema de Gestão de Carteiras de Contratos de energia elétrica, com Gestão de Risco. A relação contratual entre a ANDE (Administración Nacional de Electricidad Paraguai) e Itaipu Binacional é utilizada como estudo de caso. A metodologia proposta para tratar o problema extende a teoria de Markowitz em um contexto de tomada de decisão multiobjetivo, no qual se busca minimizar os gastos da ANDE em contratação de energia (via programação não-linear) e também o risco do portfólio, avaliado por meio da variância do mesmo. Por meio do modelo proposto é possível obter a decisão contratual ótima de ANDE, que minimiza o custo de seu portfólio para cada nível de risco. Os resultados obtidos indicam que o modelo é eficiente em termos de redução de custos e risco. / Due to the increasing deregulation of electricity markets, different market participants were faced with the necessity to effectively manage their investment in electricity. In this scenario, portfolio optimization is a relevant technique that can be investigated for strategic planning by agents on energy markets. In general, markets are exposed to risks from multiple sources, the mitigation of such risks, thus, is important. The portfolio theory proposed by Harry Markowitz has been used in analyses involving several markets. This work analyzes the problem of electricity Portfolio Management, with Risk Management. The contractual relationship between ANDE and Itaipu Binacional is used as a study case. The methodology proposed for addressing the problem extends Markowitz´s theory (applying non-linear programming) for a context of multi-objective decision making, searching for the minimization of ANDEs power contract costs, as well the portfolio risk, evaluated by its variance. With the proposed model, it is possible to obtain the optimal contract decision, which minimizes the portfolio cost for each risk level. Results indicate that the model proposed is efficient in cost and risk minimization.

A geração hidrelétrica é dependente da afluência, no entanto, é possível minimizar a variação da energia natural afluente por meio dos reservatórios. Por sua vez, a geração eólica tem como desvantagem a volatilidade devido a sua dependência em relação ao vento. Nesse sentido, uma carteira otimizada de projetos eólicos possibilita a redução da volatidade da energia gerada pelo conjunto, na medida em que aproveita as complementariedades do vento. No Brasil, os Estados da Bahia, Rio Grande do Norte, Ceará, Rio Grande do Sul e Piauí concentram 90% da capacidade instalada das usinas eólicas em operação, em construção ou contratada, com uma previsão da fonte atingir 11,6% de participação na matriz elétrica. A pesquisa tem como objetivo desenvolver uma metodologia de apoio baseada na teoria de portifólio de Markowitz que poderá ser utilizada pelo órgão de planejamento energético brasileiro para a definição da quantidade de energia a ser contratada por fonte e local, por meio de leilões de energia regionais e por fonte, com o objetivo de se obter uma carteira otimizada de empreendimentos, que reduza a volatilidade. O método também pode servir de apoio ao investidor para se obter um portfólio de usinas que minimize o risco de exposição financeira no mercado de curto prazo. Nenhum estudo aplicando a teoria de portifólio de Markowitz em usinas eólicas do Brasil foi encontrado na literatura. Os resultados obtidos demonstram que a carteira formada pelas usinas eólicas existentes não está na fronteira eficiente e poderia ser otimizada com aumento da expectativa de geração ou redução do risco. No mesmo sentido, a otimização da carteira também reduziu o risco de exposição ao mercado de curto prazo. / Even though the hydroelectric generation is highly dependent on the river flows, it is possible to minimize the volatility of the energy generation in a given period using the storage capacity of the reservoirs. In contrast, to minimize the volatility of the wind generation is burdensome due to its dependency on wind. Accordingly, an optimized portfolio of wind projects all together allows the reduction of the volatility of the energy generation for the complementarity of wind from different locations. In Brazil, the states of Bahia, Rio Grande do Norte, Ceara, Rio Grande do Sul and Piauí concentrate 90% of the installed capacity of wind power plants in operation, under construction or contracted with a font forecast to reach 11.6% share the electric matrix. The Thesis aims to develop a support methodology based in portfolio theory of Markowitz that can be used by the Brazilian-planning agency in future, to define the amount of energy to be contracted by source and location, through regional and source energy auctions, to obtain an optimized portfolio projects, with reduced volatility. The methodology can also serve to support the investor to obtain a portfolio of plants that minimize the risk of financial exposure to short-term market. No study applying Markowitz\'s portfolio theory in wind farms of Brazil was found in the literature. The results show that the portfolio of the existing wind farms is not on the efficient frontier and could be optimized with increased expectation of generating or reducing the risk. Similarly, the optimization of the portfolio also reduced the risk of exposure to short-term market.

Made available in DSpace on 2016-04-25T18:39:45Z (GMT). No. of bitstreams: 1 Carlos Eduardo Furlanetti.pdf: 957434 bytes, checksum: 4d15c672f25586845118ea42ab2bd576 (MD5) Previous issue date: 2011-12-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work analyzes the mean quarterly returns produced by portfolios, selected between 2002 and 2010, compounded by stocks traded in the BM&FBovespa, based on the use of two popular multiples, Price/Earnings (P/E), or Price/Book Value (P/B), aiming at verifying whether these returns were consistently higher than the mean valuation of the Bovespa‟s index. Thus, by investigating the possible existence of a market anomaly, this study fits in the field of controversial academic debate: the Efficient Market Hypothesis (Fama, 1970). For each selected date, the shares were sorted by the selected multiple, and were divided, then, into four portfolios (by quartiles). To analyze the results, descriptive statistics were calculated, Jarque-Bera (JB) and Student tests were performed. The results suggest that portfolios formed by stocks 'P / E Very Low' (below the first quartile) were able to produce, over that period, quarterly average returns higher than the Bovespa‟s index, within a confidence interval of approximately 96.2%. In addition, portfolios formed by stocks 'P / B Low‟ or P / E Low‟ (between the 1st and 2nd quartile) produced good performance as well, but at a much lower level of confidence, set between 82,2 and 83,5%, respectively / Este trabalho analisa os retornos médios trimestrais produzidos por carteiras de ações negociadas na BM&FBovespa, montadas entre 2002 e 2010, a partir do uso dos múltiplos Preço/Lucro (P/L) ou Preço/Valor Contábil (P/B). Investiga a existência de possível anomalia de mercado ao verificar se retornos produzidos por carteiras formadas por ações de baixo P/L ou P/B podem ser consistentemente superiores à valorização do Ibovespa. Assim, este estudo transita em campo de controvertido debate acadêmico: a Hipótese de Mercado Eficiente (Fama, 1970). Para cada data selecionada, as ações foram ordenadas de acordo com o múltiplo escolhido e divididas em quatro carteiras (por quartis). Para a análise dos resultados, foram calculadas estatísticas descritivas e realizados testes de normalidade Jarque-Bera (JB) e paramétricos t de Student. Os resultados obtidos sugerem que carteiras de ações formadas por ações de P/L Muito Baixo‟ (abaixo do 1º quartil) foram capazes de produzir, no período analisado, retornos médios trimestrais superiores ao Ibovespa, dentro de um intervalo de confiança de aproximadamente 96,2%. Carteiras formadas por ações de P/B Baixo ou P/L Baixo‟ (entre o 1º e o 2º quartil) tiveram boa performance, porém a um nível de confiança bem menor, fixado entre 82,2 e 83,5%, respectivamente

Com a crescente desregulamentação dos mercados de energia, os diferentes participantes dos mercados se deparam com a necessidade de gerenciar de maneira eficiente seus investimentos em energia elétrica. Nesse cenário, a otimização das Carteiras de Contratos mostra-se uma técnica interessante no planejamento estratégico dos agentes de mercados de energia. Os mercados estão frequentemente expostos a riscos de diversas fontes, assim, a mitigação dos mesmos é fundamental. A Teoria do Portfólio, proposta por Harry Markowitz, tem sido utilizada em análises envolvendo diversos mercados. Este trabalho analisa um problema de Gestão de Carteiras de Contratos de energia elétrica, com Gestão de Risco. A relação contratual entre a ANDE (Administración Nacional de Electricidad Paraguai) e Itaipu Binacional é utilizada como estudo de caso. A metodologia proposta para tratar o problema extende a teoria de Markowitz em um contexto de tomada de decisão multiobjetivo, no qual se busca minimizar os gastos da ANDE em contratação de energia (via programação não-linear) e também o risco do portfólio, avaliado por meio da variância do mesmo. Por meio do modelo proposto é possível obter a decisão contratual ótima de ANDE, que minimiza o custo de seu portfólio para cada nível de risco. Os resultados obtidos indicam que o modelo é eficiente em termos de redução de custos e risco. / Due to the increasing deregulation of electricity markets, different market participants were faced with the necessity to effectively manage their investment in electricity. In this scenario, portfolio optimization is a relevant technique that can be investigated for strategic planning by agents on energy markets. In general, markets are exposed to risks from multiple sources, the mitigation of such risks, thus, is important. The portfolio theory proposed by Harry Markowitz has been used in analyses involving several markets. This work analyzes the problem of electricity Portfolio Management, with Risk Management. The contractual relationship between ANDE and Itaipu Binacional is used as a study case. The methodology proposed for addressing the problem extends Markowitz´s theory (applying non-linear programming) for a context of multi-objective decision making, searching for the minimization of ANDEs power contract costs, as well the portfolio risk, evaluated by its variance. With the proposed model, it is possible to obtain the optimal contract decision, which minimizes the portfolio cost for each risk level. Results indicate that the model proposed is efficient in cost and risk minimization.

A geração hidrelétrica é dependente da afluência, no entanto, é possível minimizar a variação da energia natural afluente por meio dos reservatórios. Por sua vez, a geração eólica tem como desvantagem a volatilidade devido a sua dependência em relação ao vento. Nesse sentido, uma carteira otimizada de projetos eólicos possibilita a redução da volatidade da energia gerada pelo conjunto, na medida em que aproveita as complementariedades do vento. No Brasil, os Estados da Bahia, Rio Grande do Norte, Ceará, Rio Grande do Sul e Piauí concentram 90% da capacidade instalada das usinas eólicas em operação, em construção ou contratada, com uma previsão da fonte atingir 11,6% de participação na matriz elétrica. A pesquisa tem como objetivo desenvolver uma metodologia de apoio baseada na teoria de portifólio de Markowitz que poderá ser utilizada pelo órgão de planejamento energético brasileiro para a definição da quantidade de energia a ser contratada por fonte e local, por meio de leilões de energia regionais e por fonte, com o objetivo de se obter uma carteira otimizada de empreendimentos, que reduza a volatilidade. O método também pode servir de apoio ao investidor para se obter um portfólio de usinas que minimize o risco de exposição financeira no mercado de curto prazo. Nenhum estudo aplicando a teoria de portifólio de Markowitz em usinas eólicas do Brasil foi encontrado na literatura. Os resultados obtidos demonstram que a carteira formada pelas usinas eólicas existentes não está na fronteira eficiente e poderia ser otimizada com aumento da expectativa de geração ou redução do risco. No mesmo sentido, a otimização da carteira também reduziu o risco de exposição ao mercado de curto prazo. / Even though the hydroelectric generation is highly dependent on the river flows, it is possible to minimize the volatility of the energy generation in a given period using the storage capacity of the reservoirs. In contrast, to minimize the volatility of the wind generation is burdensome due to its dependency on wind. Accordingly, an optimized portfolio of wind projects all together allows the reduction of the volatility of the energy generation for the complementarity of wind from different locations. In Brazil, the states of Bahia, Rio Grande do Norte, Ceara, Rio Grande do Sul and Piauí concentrate 90% of the installed capacity of wind power plants in operation, under construction or contracted with a font forecast to reach 11.6% share the electric matrix. The Thesis aims to develop a support methodology based in portfolio theory of Markowitz that can be used by the Brazilian-planning agency in future, to define the amount of energy to be contracted by source and location, through regional and source energy auctions, to obtain an optimized portfolio projects, with reduced volatility. The methodology can also serve to support the investor to obtain a portfolio of plants that minimize the risk of financial exposure to short-term market. No study applying Markowitz\'s portfolio theory in wind farms of Brazil was found in the literature. The results show that the portfolio of the existing wind farms is not on the efficient frontier and could be optimized with increased expectation of generating or reducing the risk. Similarly, the optimization of the portfolio also reduced the risk of exposure to short-term market.