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This project describes the underlying principles of Modern Portfolio Theory (MPT), the Capital Asset Pricing Model (CAPM), and multi-factor models in detail. It also explores the process of constructing optimal portfolios using Modern Portfolio Theory, as well as estimates the expected return and covariance matrix of assets using the CAPM and multi-factor models. Finally, the project applies these models in real markets to analyze our portfolios and compare their performances.

<p>Introduction: Within the private economy individuals are today accepting an increased individual responsibility for retirement funds and other economic challenges. This is due to the decreased confidence in government programs and that the increased life expectancy raises the risk to outlive the own life savings. The shift from state run security systems to more private responsibility could be spotted in Sweden as well, where one important part of the private economy, the saving system for retirement, has been changed. The pension plan met critics when it was proposed and implemented, especially for the part where some of the responsibility relies on the individual. It was discussed that this huge responsibility might be larger then what many individuals would be able to handle. These factors have increased the importance of successes in the individuals own saving plans. To enhance the chances of a certain level of success, individuals turn to private bankers to plan their wealth and savings. The position of these private bankers and their performance has amplified more then ever before.</p><p>Purpose: The purpose of the thesis is to describe: if Swedish private bankers look on enough features of an investor to be able to prescribe the appropriate portfolio for the investor?</p><p>Methodology: A qualitative research has been used since the purpose and the information gathered demanded it. Cases where the authors created four fictitious investors was sent out by electronic mail and the private bankers where asked to construct suitable portfolios to each investor. The cases were sent out to ten different private bakers, however only two replied within the deadline. The authors have strived to keep high reliability and validity in the paper; however the small response rate lowers the reliability.</p><p>Conclusion: The qualitative research found that Swedish private bankers look on enough features on a client to be able to prescribe an appropriate portfolio for the investor. However the Private bankers’ main focus seems to be time horizon and risk profile of the investor.</p>

In this thesis, we study three topics in optimal portfolio selection that are relevant to the theory of forward investment performance processes. In Chapter 1, we develop a connection between the classical mean-variance optimisation and time-monotone forward performance processes for infinitesimal trading times. Namely, we consider consecutive mean-variance problems and we show that, for an appropriate choice of the corresponding mean-variance trade-off coefficients, the wealth process that is generated converges (as the trading interval goes to zero) to the optimal wealth process generated by a time-monotone forward performance process. The choice of the trade-off coefficients is made in accordance to the evolution of the risk tolerance process of the forward performance process. This result allows us to provide a fresh view on the issue of time-consistency of mean-variance analysis, for we propose a method to update mean-variance risk preferences forward in time. As a by-product, our convergence theorem generalises a result by Gy&ouml;ngy (1998) on the convergence of the Euler scheme for SDEs. We also provide novel results on the Lipschitz regularity of the local risk tolerance function of forward investment performance processes. The material in this chapter is joint work with Marek Musiela and Thaleia Zariphopoulou. Chapter 2 combines forward investment theory and partial information. Specifically, we construct forward investment performance processes in models where the drift is a random variable distributed according to a known distribution. The forward performance processes we consider are of the type U(t,x) = u(t,x, R_t), where R. denotes the process of cumulative excess returns, and u(t,x,z):[0,&infin;) &times; &Ropf; imes &Ropf;^N &xrarr; &Ropf; is such that u(t,.,z) is a utility function satisfying Inada's conditions. We derive the Hamilton-Jacobi-Bellman (HJB) equation for u(.). The HJB equation is linearised into the ill-posed heat equation; then, using the multidimensional version of Widder's theorem, we fully characterise the solutions to this equation in terms of a collection of positive measures; the result is an integral representation of the convex conjugate function of u(t,.,z). We construct several examples, and we show how these can be combined, in the dual domain, to generate mixtures of forward investment performance processes. We also show that the volatility of these processes is intrinsic, in that it is not generated by changes of num&eacute;raire/measure. In Chapter 3, we provide an extension of the Black-Litterman model to the continuous time setting. Our extension is different from, and complements that of, Frey, Gabih, and Wunderlich (2012) and Davis and Lleo (2013). Specifically, we develop a novel robust estimator of instantaneous expected returns which is continuously shrunk towards the predictions of an asset pricing theory, such as the CAPM. We derive this estimator fairly explicitly and study some of its properties. As in the Black-Litterman model, such an estimator can be used to make optimal asset allocation problems in continuous time more robust with respect to estimation errors. We provide explicit solutions to the problem of maximising expected power utility of terminal wealth, when our estimator is used to estimate the drift. As an example, we illustrate our results explicitly in the case of a multifactor model, where Arbitrage Pricing Theory predicts that alphas should be approximately zero.

Syftet med studien är att undersöka bitcoins kapacitet som hedge gentemot den svenska aktiemarknaden. För att identifiera om korrelation existerar mellan avkastningen i bitcoin och SIX30RX (OMXS30 med utdelning) och således besvara forskningsfrågan studeras associationen. Tidsperioden som studeras är 2012-01-02 till 2016-10-21. Associationen undersöks med hjälp av regressionsmodeller. Resultatet visar att bitcoin inte är korrelerat med avkastningen för SIX30RX under den studerade tidsperioden. Bitcoin kan således klassificeras som en hedge gentemot den svenska aktiemarknaden. / This paper examines bitcoin’s capacity as a hedge towards the Swedish stock market. To identify if correlation exists between the returns of bitcoin and SIX30RX (OMXS 30 including dividends) and thus respond to the research question the association is investigated. The time period considered is 2012-01-02 to 2016-10-21. Association is analysed using regression models. The results demonstrate that bitcoin is uncorrelated with the return for SIX30RX during this time period. Therefore, bitcoin can be classified as a hedge against the Swedish stock market.

In this Portfolio Optimization Project, we used Markowitz¡¯s modern portfolio theory for portfolio optimization. We selected fifteen stocks traded on the New York Stock Exchange and gathered these stocks¡¯ historical data from Yahoo Finance [1]. Then we used Markowitz¡¯s theory to analyze this data in order to obtain the optimal weights of our initial portfolio. To maintain our investment in a current tangency portfolio, we recalculated the optimal weights and rebalanced the positions every week. In the CAPM project, we used the security characteristic line to calculate the stocks¡¯ daily returns. We also computed the risk of each asset, portfolio beta, and portfolio epsilons. In the Factor Modeling project, we computed estimates of each asset¡¯s expected returns and return variances of fifteen stocks for each of our factor models. Also we computed estimates of the covariances among our asset returns. In order to find which model performs best, we compared each portfolio¡¯s actual return with its corresponding estimated portfolio return.

In this Portfolio Optimization Project, we used Markowitz¡¯s modern portfolio theory for portfolio optimization. We selected fifteen stocks traded on the New York Stock Exchange and gathered these stocks¡¯ historical data from Yahoo Finance [1]. Then we used Markowitz¡¯s theory to analyze this data in order to obtain the optimal weights of our initial portfolio. To maintain our investment in a current tangency portfolio, we recalculated the optimal weights and rebalanced the positions every week. In the CAPM project, we used the security characteristic line to calculate the stocks¡¯ daily returns. We also computed the risk of each asset, portfolio beta, and portfolio epsilons. In the Factor Modeling project, we computed estimates of each asset¡¯s expected returns and return variances of fifteen stocks for each of our factor models. Also we computed estimates of the covariances among our asset returns. In order to find which model performs best, we compared each portfolio¡¯s actual return with its corresponding estimated portfolio return.

This project describes the underlying principles of Modern Portfolio Theory, the Capital Asset Pricing Model (CAPM), and multi-factor models in detail, explores the process of constructing optimal portfolios using the Modern Portfolio Theory, estimates the expected return and covariance matrix of assets using CAPM and multi-factor models, and finally, applies these models in real markets to analyze our portfolios and compare their performances.

A passive investment management model was developed via a critical literature review of portfolio methodologies. This model was developed based on the fundamental models originated by both Markowitz and Sharpe. The passive model was automated via the development of a computer programme that can be used to generate the required outputs as suggested by Markowitz and Sharpe. For this computer programme MATLAB is chosen and the model’s logic is designed and validated. The demonstration of the designed programme using securities traded is performed on Johannesburg Securities Exchange. The selected portfolio has been sub-categorised into six components with a total of twenty- seven shares. The shares were grouped into different components due to the investors’ preferences and investment time horizon. The results demonstrate that a test portfolio outperforms a risk- free money market instrument (the government R194 bond), but not the All Share Index for the period under consideration. This design concludes the reason for this is due in part to the use of the error term from Sharpe’s single index model. An investor following the framework proposed by this design may use this to determine the risk- return relationship for selected portfolios, and hopefully, a real return.