This thesis aims to investigate different types of optimization methods that can be used when optimizing fund of fund portfolios. Moreover, the thesis investigates which funds that should be included and what their respective portfolio weights should be, in order to outperform the Swedish SIX Portfolio Return Index. The funds considered for the particular fund of funds in this thesis are all managed by a particular company. The optimization frameworks applied include traditional mean variance optimization, min conditional value at risk optimization, as well as optimization methods studying alpha in combination with the risk measures tracking error and maximum drawdown, respectively. All four optimization methods were applied on a ten years data period as well as on a five years data period. It was found that while the funds have different strengths and weaknesses, four of the funds were considered most appropriate for the fund of funds. Geography and sector constraints were also taken into account and it was found that, in this particular case, the healthcare sector constraint affected the allocated portfolio weights the most. / Syftet med detta masterexamensarbete är att undersöka olika typer av optimeringsmetoder som kan användas vid optimering av en fond i fond. Vidare är syftet med optimeringen att utvärdera vilka fonder som bör inkluderas och vilka deras respektive portföljvikter bör vara för att prestera bättre än det svenska SIX Portfolio Return Indexet. Optimeringsmetoderna inkluderar traditionell modern portföljteori, minimering av conditional Value at Risk och optimeringsmetoder som studerar alpha i kombination med riskmåtten tracking error respektive maximum drawdown. Alla fyra optimeringsmetoder applicerades på en tio år lång respektive fem år lång dataperiod. Det visade sig att även om fonderna har olika styrkor och svagheter kunde fyra av fonderna anses vara mest lämpliga att inkluderas i fond i fonden. Geografiska och sektoriella begränsningar beaktades och det konstaterades att sektorbegränsningen för hälsovårdssektorn hade störst påverkan på resultatet.

In 2023, the topic of ongoing inflation is being discussed almost daily as it has become inevitable. The global economy is facing significant uncertainty and downward pressure as several leading developed nations adopted expansionary fiscal policies and quantitative easing monetary policies during the pandemic. Those action has lead to an unprecedented level of inflation today. The purpose of this report is to investigate different portfolio strategies and evaluate how various asset classes perform under varying inflationary conditions. Using regression analysis, the study assesses the performance of different assets during high and low inflation regimes. Additionally, two different portfolio strategies are implemented and compared against the 60/40 portfolio strategy, which is considered a benchmark approach among investors. The first strategy involves a modified version of the Markowitz optimization method, which determines the optimal weights of the portfolio during high and low inflationary environments. The second strategy entails identifying a signal and then dynamically adjusting the portfolio's weights based on the signal's value. The findings indicate that during high inflation periods, oil, gold, energy, basic materials, and technology sectors exhibit strong performance. Furthermore, the results reveal that the first strategy is more effective than the second strategy and the 60/40 benchmark. An interesting topic for further investigation is exploring the impact of short selling on portfolio allocation and strategy, which was not addressed in this report. / Under år 2023 är ämnet om pågående inflation nästan oundvikligt. Den globala ekonomin har stått inför betydande osäkerhet och nedåtgående tryck då flera ledande utvecklade nationer antagit expansiva finanspolitiska åtgärder och kvantitativa lättnadsmonetära åtgärder under pandemin. Dessa åtgärder har lett till en enastående nivå av inflation idag. Syftet med denna rapport är att undersöka olika portföljstrategier och hur olika tillgångsslag presterar under olika inflationsregimer. Med hjälp av regressionsanalys undersöks hur olika tillgångar presterar under hög respektive låg inflation. Därefter genomförs två olika portföljstrategier som sedan jämförs mot en 60/40 portföljstrategi, som anses vara en standardstrategi bland investerare. Den första strategin som genomförs är en modifierad version av Markowitz optimeringsmetod. Metoden används för att identifiera de optimala vikterna av portföljen under hög respektive låg inflationsmiljö. Den andra strategin som undersöks innebär att identifiera en signal och sedan dynamiskt justera portföljens vikter baserat på signalens värde. Resultaten visar att olja, guld, energi-, basmaterial- samt teknologisektorn presterar bra under hög inflation. Resultaten påvisar även att den första strategin är den mest effektiva i jämförelse med den andra strategin och 60/40 portföljstrategin. En aspekt som inte inkluderades i denna rapport är att undersöka hur blankning påverkar portföljallokeringen och strategin. Detta kan vara ett intressant ämne för vidare forskning.\\\\

We aim at generalizing the celebrated portfolio optimization problem "à la Merton", where the asset evolution is steered by a self-exciting jump-diffusion process. We first define the rigorous mathematical framework needed to introduce the stochastic optimal control problem we are interesting in. Then, we provide a proof for a specific version of the Dynamic Programming Principle (DPP) with respect to the general class of self-exciting processes under study. After, we state the Hamilton-Jacobi-Bellman (HJB) equation, whose solution gives the value function for the corresponding optimal control problem. The resulting HJB equation takes the form of a Partial-Integro Differential Equation (PIDE), for which we prove both existence and uniqueness for the solution in the viscosity sense. We further derive a suitable numerical scheme to solve the HJB equation corresponding to the portfolio optimizationproblem. To this end, we also provide a detailed study of solution dependence on the parameters of the problem. The analysis is performed by calibrating the model on ENI asset levels during the COVID-19 worldwide breakout. In particular, the calibration routine is based on a sophisticated Sequential Monte Carlo algorithm.

In this thesis, the traditional methods of only using ESG scores to screen stocks for sustainable portfolios is broadened. The selection of securities for portfolios will instead depend on aggregation, weighting and normalization of a wider set of sustainability variables, in turn creating more all-encompassing sustainability scores. Using these scores, the aim is to implement them in index tracking portfolios. These portfolios combines a hybrid approach between active and passive investment, with the aim of creating sustainable enhanced index funds that can beat the index without adding significant risk. Additionally, this allows for comparison of how different combinations and levels of sustainability affects returns, risk and index tracking. The results that are obtained shows that in the scenario presented in the thesis, it is possible to create a sustainability score which both increases the average sustainability of portfolios, and yields risk adjusted returns. We also studied how a net increase in sustainability scores over a control portfolio results in higher active returns, and eventually a small drop off in information ratio as we apply too strong of a sustainability constraint to our portfolios. The combination of sustainability scores which showed the highest risk adjusted returns was created using equal parts z-scored ESG ratings, ESG risk ratings and ESG momentum. / Detta examensarbete breddar de traditionella metoderna för att skapa hållbara portföljer. Genom att basera urvalet av aktier på aggregering, viktande och normalisering av ett större set av hållbarhetsvariabler, jämfört med traditionell screening baserad på endast ESG betyg, skapas mer omfattande hållbarhetsbetyg. Syftet med studien är att implementera dessa hållbarhetsbetyg vid skapandet av index-portföljer och analysera resultaten. Dessa portföljer kombinerar då både aktiva och passiva investeringsprinciper, med målet att skapa hållbara indexnära fonder som kan prestera bättre än indexet, utan signifikant höjd risk. Dessa hållbarhetsbetyg tillåter även jämförelse av hur olika kombinationer och nivåer av hållbarhet påverkar avkastning, risk och närhet till index. Resultaten visar tydligt att det, inom uppsatsens avgränsningar, är möjligt att skapa hållbarhetsbetyg som ökar både hållbarheten av portföljer i snitt, och skapar riskjusterad avkastning. Det visar även hur en relativ höjning av hållbarhetsbetygen resulterar i högra aktiv avkastning jämfört med en kontroll-portfölj. Vid en viss nivå av höjning sker dock en avtappning av den riskjusterade avkastningen. Den kombinationen av hållbarhetsvariabler som visar högst riskjusterad avkastning när de aggregeras till ett hållbarhetsbetyg är en kombination, i lika delar, av ESG betyg, ESG risk och ESG momentum.

Modern portfolio theory has its attractive characteristics of promoting diversification in a portfolio and can be seen as an easy alternative for setting optimal weights for portfolio managers. Furthermore, as portfolio managers try to beat a defined benchmark for their portfolio the Black-Litterman model allows them to include their own prospects on the future return of markets and securities. This thesis examines how the practical use of the Black-Litterman model can affect portfolios' performance. The analysis was done by calculating historical portfolio weights with investor views, without investor views, and with perfect investor views in the Black-Litterman model. Thereafter, calculating historical return and volatility for six multi-asset portfolios between 2017-09-25 and 2021-01-31. This was then compared with benchmark portfolios, which reflect the practical use. These portfolios included historically used investor views and constraints in the mean-variance optimization. The results showed that investor views had a negative effect on total return (lower return) and a positive effect on volatility (lower risk), however, an increased Sharpe ratio. The constraints in the mean-variance optimization used in the benchmark portfolios resulted in a lower total return. In conclusion, the Black-Litterman model showed robustness and did not generate unintuitive or unreasonable portfolios, and it has great potential with increasing accuracy in the investor views. / Modern portföljteori har attraktiva egenskaper vad gäller att främja diversifiering i en portfölj och kan ses som ett enkelt alternativ för att välja optimala vikter för portföljförvaltare. Eftersom portföljförvaltare försöker slå ett definierat benchmark för sin portfölj tillåter dessutom Black-Litterman modellen dem att inkludera sina egna åsikter angående förväntade avkastningar på marknader och värdepapper. Detta examensarbete undersöker hur den praktiska användningen av Black-Litterman modellen kan påverka portföljernas prestation. Analysen gjordes genom att beräkna historiska portföljvikter med Black-Litterman modellen med och utan invetserarens egna åsikter angående förväntade avkastningar, och med perfekta förväntade avkastningar. Därefter beräknades historiska avkastningar och volatiliteter för sex investeringsportföljer mellan 2017-09-25 och 2020-01-31. Detta jämfördes med benchmarkportföljer, vilka återspeglade den praktiska användningen. Dessa portföljer inkluderade historiskt använda förväntade avkastningar och restriktioner i mean-variance optimeringen. Resultaten visade att investerares åsikter angående förväntade avkastningar hade en negativ effekt på avkastningen (lägre avkastning), positiv effekt på volatiliteten (lägre risk), vilket resulterade i en högre Sharpe kvot. Restriktionerna i mean-variance optimeringen som användes i benchmarkportföjerna resulterade i en lägre totalavkastning. Sammanfattningsvis visade Black-Litterman modellen robusthet och genererade inte ointuitiva eller olämpliga portföljer, och modellen har stor potential med ökad träffsäkerhet i investerarens åsikter angående förväntade avkastningar.

This project investigates the applicability of the originalversion of Markowitz’s mean-variance model for portfoliooptimization to real-world modern actively managed portfolios.The method measures the mean-variance model’s capability toaccurately capture the riskiness of given portfolios, by invertingthe mathematical formulation of the model. The inversion of themodel is carried out both for fabricated data and real-world dataand shows that in the cases of real-world data the model lackscertain accuracy for estimating risk averseness. The method hascertain errors which both originate from the proposed estimationmethods of input variables and invalid assumptions of investors. / Projektet undersöker lämpligheten att använda den ursprungliga versionen av Markowitzs ”Mean-Variance model” för portföljoptimering för moderna aktivt förvaltade portföljer. Metoden mäter modellens förmåga att tillförlitligt beräkna risken för givna portföljer genom att invert-era den matematiska formuleringen av modellen. Inversionen av modellen utförs både för simulerad data och verklig data och visar att i fallet med verkliga data saknar modellen viss noggrannhet för att uppskatta riskpreferens. Metoden har vissa fel som både uppstår från de föreslagna uppskattningsmetoderna för inputvariabler och ogiltiga antaganden för investerare. / Kandidatexjobb i elektroteknik 2021, KTH, Stockholm

[pt] Os grandes agentes do mercado de energia dedicam muitos esforços na avaliação e decisão da alocação ótima de capital para a implementação de projetos, em decorrência do grande número de projetos candidatos em seus portfólios de investimentos. Essas decisões visam escolher o subconjunto de projetos a ser implementado, pois os recursos orçamentários são geralmente menores que o necessário para a implementação de todos eles. Muitos são os riscos apresentados, e quanto mais riscos e incertezas, maiores se tornam as dificuldades de avaliação e decisões de investimento de maneira otimizada. As metodologias clássicas para avaliação de portfólios de projetos de investimento são baseadas em maximizar os retornos (VPL, TIR, etc) e minimizar o risco (desvio-padrão do VPL, variância, etc). Muitas vezes, estes métodos tradicionais de avaliação podem não conseguir tratar adequadamente as flexibilidades gerenciais (Opções Reais) características dos projetos, assim como os riscos e incertezas, devido às possíveis dificuldades de solução e modelagem matemática (multi-variáveis) dos problemas. O desenvolvimento e aplicação de modelos alternativos, tais como os baseados na Teoria de Opções Reais, inclusive com a utilização de métodos de Inteligência Computacional, podem se mostrar mais adequados para estes problemas. Nesta tese é desenvolvida uma metodologia híbrida, apresentando um modelo de Opções Reais Fuzzy para a avaliação de projetos de Revamp por um agente do mercado de Geração Termelétrica de Energia, a partir de um Portfólio de Opções Reais em ambiente de incertezas. Para a seleção do subconjunto de projetos por faixa orçamentária, é aplicado um Algoritmo Genético para otimização multi-critério, através da utilização de um índice de ponderação retorno x risco (lâmbda). / [en] Large players in energy market dedicate many efforts in valuation and optimal capital allocation decision for their project implementation, due the large candidate projects number in their investment portfolios. These decisions aim to choose the projects subset to be implemented, because the monetary resources are generally smaller than necessary for all projects implementation. There are many risks, and with risks and uncertainties, greater become the difficulties in analysis and optimally investment decisions. The classical methods to investment portfolios are based on to maximize returns (NPV, IRR, among others) and to minimize risks (NPV standard deviation, variance, among others). Often, these traditional methods may not be able to handle properly the projects managerial flexibilities (Real Options), as well the risks and uncertainties, due to possible solution difficulties and mathematical modeling problems (multi variables). Alternative models development and implementation, such as those based on Real Options Theory, including the use of Computational Intelligence methods, may be more suitable for these problems. In this thesis, a hybrid methodology is developed, presenting a Fuzzy Real Options model for Revamp projects valuation by a Thermoelectric Power Generation market player, from a Real Options Portfolio in uncertainties environment. For selecting the projects subset by budget range, a multi-criteria Genetic Algorithm optimization is applied, using a weighting return x risk index (lambda).

The increasing penetration of proactive agents in distribution systems (DS) has opened new possibilities to make the grid more resilient and to increase participation of responsive loads (RL) and non-conventional generation resources. On the resiliency side, plug-in hybrid electric vehicles (PHEV), energy storage systems (ESS), microgrids (MG), and distributed energy resources (DER), can be leveraged to restore critical load in the system when the utility system is not available for extended periods of time. Critical load restoration is a key factor to achieve a resilient distribution system. On the other hand, existing DERs and responsive loads can be coordinated in a market environment to contribute to efficiency of electricity consumption and fair electricity tariffs, incentivizing proactive agents' participation in the distribution system. Resiliency and market applications for distribution systems are highly complex decision-making problems that can be addressed using modern optimization techniques. Complexities of these problems arise from non-linear relations, integer decision variables, scalability, and asynchronous information. On the resiliency side, existing models include optimization approaches that consider system's available information and neglect asynchrony of data arrival. As a consequence, these models can lead to underutilization of critical resources during system restoration. They can also become computationally intractable for large-scale systems. In the market design problem, existing approaches are based on centralized or computational distributed approaches that are not only limited by hardware requirements but also restrictive for active participation of the market agents. In this context, the work of this dissertation results in major contributions regarding new optimization algorithms for market design and resiliency improvement in distribution systems. In the DS market side, two novel contribution are presented: 1) A computational distributed coordination framework based on bilateral transactions where social welfare is maximized, and 2) A fully decentralized transactive framework where power suppliers, in a simultaneous auction environment, strategically bid using a Markowitz portfolio optimization approach. On the resiliency side, this research proposed a system restoration approach, taking into account uncertain devices and associated asynchronous information, by means of a two-module optimization models based on binary programming and three phase unbalanced optimal power flow. Furthermore, a Reinforcement Learning (RL) method along with a Monte Carlo tree search algorithm has been proposed to solve the scalability problem for resiliency enhancement. / Doctor of Philosophy / Distribution systems (DS) are evolving from traditional centralized and fossil fuel generation resources to networks with large scale deployment of responsive loads and distributed energy resources. Optimization-based decision-making methods to improve resiliency and coordinate DS participants are required. Prohibitive costs due to extended power outages require efficient mechanisms to avoid interruption of service to critical load during catastrophic power outages. Coordination mechanisms for various generation resources and proactive loads are in great need. Existing optimization-based approaches either neglect the asynchronous nature of the information arrival or are computationally intractable for large scale system. The work of this dissertation results in major contributions regarding new optimization methods for market design, coordination of DS participants, and improvement of DS resiliency. Four contributions toward the application of optimization approaches for DS are made: 1) A distributed optimization algorithm based on decomposition and best approximation techniques to maximize social welfare in a market environment, 2) A simultaneous auction mechanism and portfolio optimization method in a fully decentralized market framework, 3) Binary programming and nonlinear unbalanced power flow, considering asynchronous information, to enhance resiliency in a DS, and 4) A reinforcement learning method together with an efficient search algorithm to support large scale resiliency improvement models incorporating asynchronous information.

[pt] Otimização de portfólio tradicionalmente assume ter conhecimento da distribuição de probabilidade dos retornos ou pelo menos algum dos seus momentos. No entanto, é sabido que a distribuição de probabilidade dos retornos muda com frequência ao longo do tempo, tornando difícil a utilização prática de modelos puramente estatísticos, que confiam indubitavelmente em uma distribuição estimada. Em contrapartida, otimização robusta considera um completo desconhecimento da distribuição dos retornos, e por isto, buscam uma solução ótima para todas as realizações possíveis dentro de um conjunto de incerteza dos retornos. Mais recentemente na literatura, técnicas de distributionally robust optimization permitem lidar com a ambiguidade com relação à distribuição dos retornos. No entanto essas técnicas dependem da construção do conjunto de ambiguidade, ou seja, distribuições de probabilidade a serem consideradas. Neste trabalho, propomos a construção de conjuntos de ambiguidade poliédricos baseado somente em uma amostra de retornos. Nestes conjuntos, as relações entre variáveis são determinadas pelos dados de maneira não paramétrica, sendo assim livre de possíveis erros de especificação de um modelo estocástico. Propomos um algoritmo para construção do conjunto e, dado o conjunto, uma reformulação computacionalmente tratável do problema de otimização de portfólio. Experimentos numéricos mostram que uma melhor performance do modelo em comparação com benchmarks selecionados. / [en] Portfolio optimization traditionally assumes knowledge of the probability distribution of returns or at least some of its moments. However is well known that the probability distribution of returns changes over time, making difficult the use of purely statistic models which undoubtedly rely on an estimated distribution. On the other hand robust optimization consider a total lack of knowledge about the distribution of returns and therefore it seeks an optimal solution for all the possible realizations wuthin a set of uncertainties of the returns. More recently the literature shows that distributionally robust optimization techniques allow us to deal with ambiguity regarding the distribution of returns. However these methods depend on the construction of the set of ambiguity, that is, all distribution of probability to be considered. This work proposes the construction of polyhedral ambiguity sets based only on a sample of returns. In those sets, the relations between variables are determined by the data in a non-parametric way, being thus free of possible specification errors of a stochastic model. We propose an algorithm for constructing the ambiguity set, and then a computationally treatable reformulation of the portfolio optimization problem. Numerical experiments show that a better performance of the model compared to selected benchmarks.

本文提出一個新的混合整數線性規劃模型建立投資組合。這個模型所採用的風險函數為最大損失的絕對值，而不是一般常用的損失變異數。在給定的報酬水準下，模型尋找在觀測期間中最小的最大損失的投資組合，即為大中取小的原則。模型也同時考慮實務上常遇見之情況，如：交易成本、最小交易單位、固定交易費用比率、資產總類數等限制。因此，模型內需使用整數變數及二元變數，導致模型的計算求解過程變得比不含整數變數及二元變數的模型困難許多。我們以固定整數變數的啟發式演算法增進求解的效率，並以台灣股票市場的資料做為實證計算的對象。 / A new mixed integer linear program (MILP) for selecting portfolio based on historical return is proposed. This model uses the downside risk rather than the variance as a risk measure. The portfolio is chosen that minimizes the maximum downside risk over all past observation periods to reach a given return level. That is a mini-max principle. The model incorporates the practical characteristics such as transaction costs, minimum transaction units, fixed proportional transaction rates, and cardinality constraint. For this reason a set of integer variables and binary variables are introduced. The introduction, however, increases the computational complexity in model solution. Due to the difficulty of the MILP problem, a heuristic algorithm has been developed for the solution. The computational results are presented by applying the model to the Taiwan stock market.