>Magister Scientiae - MSc / In this thesis, we discuss methods for optimising the expected rate of return of a portfolio with minimal risk. As part of the work we look at the Modern Portfolio Theory which tries to maximise the portfolio's expected rate of return for a cer- tain amount of risk. We also use Quadratic Programming to optimise portfolios. Generally it is recognised that portfolios with a high expected return, carry higher risk. The Modern Portfolio Theory assists when choosing portfolios with the lowest possible risk. There is a nite number of assets in a portfolio and we therefore want to allocate them in such a way that we're able to optimise the expected rate of return with minimal risk. We also use the Markowian approach to allocate these assets. The Capital Asset Pricing Model is also used, which will help us to reduce our e cient portfolio to a single portfolio. Furthermore we use the Black-Litterman model to try and optimise our portfolio with a view to understanding the current market conditions, as well as considering how the market will perform in the future. An additional tool we'll use is Value at Risk. This enables us to manage the market risk. To this end, we follow the three basic approaches from Jorion [Value at Risk. USA: McGraw-Hills, 2001]. The Value at Risk tool has become essential in calcu- lating a portfolio's risk over the last decade. It works by monitoring algorithms in order to nd the worst possible scenarios within the portfolio. We perform several numerical experiments in MATLAB and Microsoft Excel and these are presented in the thesis with the relevant descriptions.

This thesis is focused on the analysis of performance of chosen russian mutual funds on the basis of achieved yield and risk. After short introduction to the russian market of mutual funds, the paper deals with a theoretical background underlying the performance indicators. Risk perception and following construction of indicators are discussed in detail from the perspective of modern and post-modern portfolio theory. The indicators are interpreted and appropriateness of their application is assessed. The analytic part is devoted to the application of discussed methods on 10 open-ended equity mutual funds. Based on the result, the funds are compared with each other and with selected market index.

The thesis provides a comparison of different portfolio models and tests their performance on the financial markets. Our analysis particularly focuses on comparison of the classical Markowitz modern portfolio theory and the downside risk models of the post-modern portfolio theory. In addition, we consider some alternative portfolio models ending with total eleven models that we test. If the performance of different portfolio models should be evaluated and compared correctly, we must use a measure that is unbiased to any portfolio theory. We suggest solving this issue via a new approach based on the utility theory and utility functions. We introduce the unbiased method for evaluation of the portfolio model performance using the expected utility efficient frontier. We use the asymmetric behavioural utility function to capture the behaviour of the real market investors. The Markowitz model is the leading market practice. We investigate whether there are any circumstances in which some other models might provide better performance than the Markowitz model. Our research is for three reasons unique. First, it provides a comprehensive comparison of broad classes of different portfolio models. Second, we focus on the developed markets in United States and Germany but also on the local emerging markets in Czech Republic and Poland. These local markets have never been tested in such extent before. Third, the empirical testing is based on the broad data set from 2003 to 2012 which enable us to test how different portfolio model perform in different macroeconomic conditions.

Risk and tracking error budgeting was originally adopted by large institutional investors, including pension funds, plan sponsors, foundations, and endowments. More recently, risk and tracking error budgeting have gained popularity among financial advisors, multi-managers, fund of funds managers, high net worth individuals as well as retail investors. These techniques contribute to the portfolio optimisation process by limiting the extent to which a portfolio can deviate from its benchmark with regards to risk and tracking error. This is an ambitious paper that attempts to determine the optimal strategy to practically implement risk and tracking error budgeting as a portfolio optimisation technique in South Africa. This study attempts to bridge the gap between active, passive, and smart-beta investment management styles by introducing a low-cost portfolio construction technique, for core-satellite portfolio management, which contributes to the risk and tracking error budgeting process. Core-satellite portfolios are designed to expose the portfolio to a low-cost primary “core” consisting of passive and enhanced index funds, thus systematic risk “beta”, limiting the tracking error of the portfolio. The secondary “satellite” component is allocated to active and smart-beta managers to exploit expected excess return “alpha”. The primary aim of this research is to construct a rule-based product range of core-satellite portfolios called “replica portfolios”. The product range builds on the foundation of the Association for Savings & Investments South Africa (ASISA) framework. The study identifies three “target portfolios” from ASISA's framework, namely (1) High Risk: SA General Equity, (2) Medium Risk: SA Multi-Asset High Equity and (3) Low Risk: SA Multi-Asset Low Equity. Through this framework, active managers from each category are shortlisted using a Sharpe and Information Ratio filter. A secondary filtering technique, namely Returns Based Style Analysis (RBSA) is used to determine the style, R-squared and alpha-generating ability of active managers versus the passive asset classes and style indices they seek to replicate. Applying Euler's theorem for homogenous functions, we decompose the risk of the coresatellite portfolio into the risk contributed by each of its components. The primary mandate of the core-satellite portfolios in the product range is to allocate risk and tracking error efficiently across several investment management styles and asset classes in order to maximise returns while remaining within the specified risk parameters. iii The results highlighted that active managers, after fees, predominantly failed to outperform their benchmarks and passive building blocks, as identified through RBSA over the sample period (October 2009 – September 2019). However, only a small number of active managers generated superior risk-adjusted returns and were included in the core-satellite range of products. This study recommends to investors that they exploit the “hot-hands effect” by investing in specialised, benchmark agnostic active managers who consistently produce superior risk-adjusted returns. By blending active, passive and smart-beta strategies, investors are exposed to less total risk, less risk per holding and a lower tracking error. The three coresatellite portfolios developed in this study generated absolute and risk-adjusted returns that are more significant than their active and passive counterparts. Fee arbitrage was derived through the range of core-satellite products, resulting in tangible alpha over the sample period. The study encourages investors to use smart-beta strategies alongside active and passive funds since it improves Sharpe and Information ratios while enhancing the original portfolio's characteristics.

Bakgrund och problemformulering: Investmentbolag är ett bolag vars affärsidé är att äga andra bolag. De har en betydande roll i samhället genom att bidra med kapital och att hjälpa driva etablerade företag framåt. I tidigare forskningssammanhang har andra liknande typer av bolag varit i fokus vilket orsakar ett gap i forskningen kring investmentbolag och riskhantering. Det här blir särskilt intressant att studera då investmentbolag har blivit en allt mer vanlig investering samtidigt som de beskrivs ha liknande riskspridning som en aktiefond. För att få en bredare förståelse om hur det skiljer sig mellan olika bolag och länder jämförs svenska investmentbolag med brittiska samt amerikanska investmentbolag.   Syfte: Syftet med denna uppsats är att analysera om investmentbolagens portföljer är effektiva.   Metod: Utgångspunkten i studien är den moderna portföljvalsteorin som utgår ifrån Markowitz tankar och menar att en effektiv portfölj inte enbart består av en lång rad med aktier. Det är i stället korrelationen mellan tillgångarna som är viktiga att ha i åtanke och den optimala portföljen är den mest effektiva samt har den högsta sharpekvoten. För att besvara studiens syfte jämförs den optimala portföljen men den faktiska portföljen och de bolag som har det minsta avståndet anses vara effektivast i sin riskhantering och vice versa.    Resultat: Resultaten visar att det finns förbättringar att göra för de allra flesta investmentbolag och det är ytterst få som håller en portfölj som är lika effektiv som den optimala portföljen. Det finns även skillnader mellan hur de olika investmentbolagen hanterar risker i portföljen och vilka typer av bolag som de investerar i, vilket främst är de brittiska investmentbolagen som främst skiljer sig från mängden. / Background and problematization: An investment company is a firm which business idea is to own other companies. It has a significant role in the society by contributing with capital and help already established companies forward. Within a research context other kind of companies with similar business ideas has been in focus which causes a gap in the research area regarding investment companies and risk management. This is especially interesting to study because investment companies have become a more common investment, meanwhile investment companies are described to have a similar risk diversification as an equity fund. To get a broader understanding on how it differs between different companies and countries a comparison is made between Swedish, British and American investment companies.   Purpose: The purpose of this paper is to analyze if investment companies manage an efficient portfolio.    Method: The main theory of this study is the modern portfolio theory which is based on Markowitz´s ideas of an efficient portfolio that does not only contain a long list of assets but instead consider the correlation between assets. According to this theory the optimal portfolio is the most efficient and has the highest sharperatio. To be able to achieve the purpose of this study the optimal portfolio will be compared to the investment companies’ actual portfolio. The companies with the smallest difference between these portfolios will be considered the most efficient regarding risk management and vice-versa.   Conclusion: The results show that there are room for improvements for most investment companies and there are extremely few that holds a portfolio that is as effective as the optimal portfolio. There are also differences between the companies regarding how they manage their portfolio risk and which types of companies they invest in. The British investment companies are those who stands out in this study.

Traditionally, practitioners use modern portfolio theory to invest optimally. Its appeal lies in its mathematical simplicity and elegance. However, despite its beauty, the theory it is plagued with many problems, which are in combination called the Markowitz curse. Lopéz de Prado introduced Hierarchical Risk Parity (HRP), which deals with the problems of Markwitz’s theory by introducing hierarchical structures into the portfolio allocation step.This thesis is a continuation of the HRP. In contrast to De Prado’s work, we build hierarchical clusters that do not have a predetermined structure and also use portfolio allocation methods that incorporates the mean estimates. We use an algorithm called community detection which is derived from graph theory. The algorithm generates clusters purely from the data without user specification. A problem to overcome is the correct identification of the market mode, whichis non-trivial for futures contracts. This is a serious problem since the specific clustering method we use hinges on correctly identifying this mode. Therefore, in this thesis we introduce a method of finding the market mode for futures data. Finally, we compare the portfolios constructed from the hierarchical clusters to traditional methods. We find that the hierarchical portfolios results in slightly worse performance than the traditional methods when we incorporate the mean and better performance for risk based portfolios. In general, we find that the hierarchical portfolios result in less extreme outcomes. / Traditionellt används modern portföljteori för attinvestera optimalt. Anledningen till detta ligger i dess matematiska enkelhet och elegans. Men trots sina många fördelar är teorin plågad med flertal problem, som i kombination kallas för Markowitz-förbannelsen. Lopéz de Prado introducerade Hierarchical Risk Parity (HRP), som påstås tacköa problemen med Markwitz teori genom att införa hierarkiska strukturer i portföljallokeringssteget. Detta examensarbete är en fortsättning på HRP. I motsats till De Prados arbete bygger vi hierarkiska kluster som inte har en förutbestämd struktur och använder även portföljallokeringsmetoder som inkluderar medelskattningarna. Vi använder en algoritm som kallas communitydetection som härrör från grafteori. Algoritmen genererar kluster enbart från data utan användarspecifikation. Ett problem att överkomma är den korrekta identifieringen av marknadsläget, vilket inte är trivialt för terminskontrakt. Detta är ett allvarligt problem eftersom den specifika klustringsmetoden vi använder hänger samman med att korrekt identifiera detta läge. Därför introducerar vi i denna avhandling en metod för att hitta marknadsläget för terminsdata. Slutligen jämför vi portföljerna konstruerade från de hierarkiska klustren med traditionella metoder. Vi finner att de hierarkiska portföljerna ger något sämre prestandaän de traditionella metoderna när vi tar med medelvärdet och bättre prestanda för riskbaserade portföljer. Generellt finner vi att de hierarkiska portföljerna resulterar i mindre extrema utfall.

The aim of this study is to examine classical Dow-strategies, Dogs of the Dow and Foolish Four relative to each other and OMXS30GI in order to test if promises of substantial returns would be kept on the Swedish stock market during the period 2002-2019. Our empirical findings show no statistically significant excess-return generated by the Foolish Four-strategy over neither the Dogs of the Dow-strategy nor OMXS30GI. Furthermore, we found that the Dogs of the Dow-strategy produced a statistically significant excess-return over benchmark OMXS30GI, however excess-return does not remain after excluding years of great market turmoil.

This study examines whether modern portfolio theory can be used to improve the Magic Formula investment strategy. With the assets picked by the investment strategy we modify the portfolios by weighting the portfolios in accordance with modern portfolio theory. Through the process of creating efficient frontiers and weighting the portfolios differently we create two alternative portfolios each year. One portfolio that aimsfor maximum Sharpe ratio and one that aims for minimum variance. These weighted portfolios produce higher risk-adjusted returns consistently during the examined period of 2010-2020. We conclude that the Magic Formula can be improved by using modern portfolio theory.

This thesis analyzes the mean variance optimization problem with respect to cardinalityconstraints. The aim of this thesis is to figure out how much of an impact transactionchanges has on the profit and risk of a portfolio. We solve the problem by implementingmixed integer programming (MIP) and solving the problem by using the Gurobi solver.In doing this, we create a mathematical model that enforces the amount of transactionchanges from the initial portfolio. Our results is later showed in an Efficient Frontier,to see how the profit and risk are changing depending on the transaction changes.Overall, this thesis demonstrates that the application of MIP is an effective approachto solve the mean variance optimization problem and can lead to improved investmentoutcomes.

Risk management in portfolio construction is a widely discussed topic and the tradeoff between risk and return is always considered before an investment is made. Modern portfolio theory is a mathematical framework which describes how a rational investor can use diversification to optimize a portfolio, which suggests using variance to measure financial risk. However, since variance is a symmetrical metric, the framework fails to correctly account for the loss aversion preferences most investors exhibit. Therefore, the use of downside risk measures were proposed, which only measures the variance of the portfolio below a certain threshold, usually set to zero or the risk-free rate. This thesis empirically investigates the differences in performance between the two risk measures when used to solve a real world portfolio optimization problem. Backtests using the different measures on all major Nordic equity markets are performed to highlight the dynamics between the frameworks, and when one should be preferred over the other. It is concluded that the optimization frameworks indeed provides a useful tool for investors to construct great performing portfolios. However, even though the downside risk framework is more mathematically rigorous, implementing this risk measure instead of variance seems to be of less importance for the actual results. / Riskhantering för aktieportföljer är mycket centralt och en avvägning mellan risk och avkastning görs alltid innan en investering. Modern Portföljteori är ett matematiskt ramverk som beskriver hur en rationell investerare kan använda diversifiering för att optimera en portfölj. Centralt för detta är att använda portföljens varians för att mäta risk. Dock, eftersom varians är ett symmetriskt mått lyckas inte detta ramverk korrekt ta hänsyn till den förlustaversion som de flesta investerare upplever. Därför har det föreslagits att istället använda olika mått på nedsiderisk (downside risk), som endast tar hänsyn till portföljens varians under en viss avkastningsgräns, oftast satt till noll eller den riskfria räntan. Denna studie undersöker skillnaderna i prestation mellan dessa två riskmått när de används för att lösa ett verkligt portföljoptimeringsproblem. Backtests med riskmåtten har genomförts på de olika nordiska aktiemarknaderna för att visa på likheter och skillnader mellan de olika riskmåtten, samt när det enda är att föredra framför det andra. Slutsatsen är att ramverken ger investerare ett användbart verktyg för att smidigt optimera portföljer. Däremot verkar den faktiska skillnaden mellan de två riskmåtten vara av mindre betydelse för portföljernas prestation. Detta trots att downside risk är mer matematiskt rigoröst.