This project has three parts. The first part is to use the efficient frontier and find the tangency portfolio to form our optimal portfolio. We built our portfolio using the Interactive Brokers software and rebalanced every week for 4 holding periods to see the relationship between our projected returns and actual market returns. In the second part we considered the Capital Asset Pricing Model (CAPM) and ran linear regressions on the stocks we chose in the first part of the project. This process is based on our idea of finding the systematic risk in each stock to improve our stock choosing ability. In the last part we introduce the concept of factor models and add more factors into our original CAPM model. Via a back-testing method, we test the reasonability of our factors and give advice to further improve our portfolio optimization project.

Long term asset creates more revenue, however it is riskier in a liquidity sense. Our question is: How does a liquidity constrained bank make decisions between profitability and liquidity? We present a computable DSGE model of banks optimal portfolio choices under liquidity constraints. Our theory predicts that liquidation plays an important role in a bank's portfolio model. Even though liquidation is an off-equilibrium phenomenon, banks can have rich loan portfolios due to the possibility of liquidation. Liquidity condition is a key factor in banks portfolio. In a moderate liquidity situation, a bank can lend more profitable longer term loans, however, if a shock in liquidity is expected, then the bank lends more loans in short term. According to the liquidity conditions, the bank can have medium term loans which are different from other previous literature. In addition, we extend our model to the bank's securities business where the bank's debts are largely short term deposit. Our theory predicts that the bank securities business produces a chasm between a real liquidity of economy and market liquidity. Banks can have more liquidity by selling their securitized loans, and as our model already pointed out, a good liquidity condition makes the bank have more profitable but less liquid long term loans. As a consequence, long term loans are accumulated with this securitization, simply because a long term loan gives higher revenue. Any market turbulence can invoke a problem in economy wide liquidity.

This project has three parts. The first part is to use the efficient frontier and find the tangency portfolio to form our optimal portfolio. We built our portfolio using the Interactive Brokers software and rebalanced every week for 4 holding periods to see the relationship between our projected returns and actual market returns. In the second part we considered the Capital Asset Pricing Model (CAPM) and ran linear regressions on the stocks we chose in the first part of the project. This process is based on our idea of finding the systematic risk in each stock to improve our stock choosing ability. In the last part we introduce the concept of factor models and add more factors into our original CAPM model. Via a back-testing method, we test the reasonability of our factors and give advice to further improve our portfolio optimization project.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.

In this diploma thesis, selected techniques for construction of optimal portfo- lios are presented. Risk measures and other criteria (Markowitz approach, Value at risk, Conditional value at risk, Mean absolute deviation, Spectral risk measure and Kelly criterion) are defined in the first part. We derived analytical solution for some cases of optimization problems, in some other cases there exists numeri- cal solution only however. Advantages and disadvantages, theoretical properties and practical aspects of software implementation in Wolfram Mathematica are also mentioned. Simulation methods suitable for portfolio optimization are brie- fly presented with their motivation in the second part. Multivariate distributions: normal, t-distribution and skewed t-distribution are presented in the third part with connection to optimization of portfolio with assumption of multivariate dis- tribution of financial losses. Optimization methods are illustrated on real data in the fourth part of this thesis. Analytical methods are compared with numerical ones. 1

This empirical study has shown that optimal portfolios need approximately 10 securities to diversify away the unsystematic risk. This challenges previous studies of randomly chosen portfolios which states that at least 30 securities are needed. The result of this study sheds light upon the difference in risk diversification between random portfolios and optimal portfolios and is a valuable contribution for investors. The study suggests that a major part of the unsystematic risk in a portfolio can be diversified away with fewer securities by using portfolio optimization. Individual investors especially, who usually have portfolios consisting of few securities, benefit from these results. There are today multiple user-friendly software applications that can perform the computations of portfolio optimization without the user having to know the mathematics behind the program. Microsoft Excel’s solver function is an example of a well-used software for portfolio optimization. In this study however, MATLAB was used to perform all the optimizations. The study was executed on data of 140 stocks on NASDAQ Stockholm during 2000-2014. Multiple optimizations were done with varying input in order to yield a result that only depended on the investigated variable, that is, how many different stocks that are needed in order to diversify away the unsystematic risk in a portfolio. / <p>Osäker på examinatorns namn, tog namnet på den person som skickade mejl om betyg.</p>

Dans la premiere partie de cette these, nous donnons un point de vue financier sur l'etude des taux d'interet long terme. En finance, les modeles classiques de taux ne s'appliquent plus pour des maturites longues (15 ans et plus). Nous montrons que les techniques de maximisation d'utilite esperee permettent de retrouver la regle de Ramsey (qui relie la courbe des taux a l'utilite marginale de la consommation optimale). En marche incomplet, il est possible de montrer un analogue de la regle de Ramsey et nous examinons la maniere dont la courbe des taux est modifiee. Ensuite nous considerons le cas ou il y a une incertitude sur un parametre du modele, puis nous etendons ces resultats au cas ou les fonctions d'utilites sont stochastiques. D'autre part nous proposons dans cette these une nouvelle maniere d'apprehender la consommation, comme des provisions que l'investisseur met de cote pour les utiliser en cas d'un evenement de defaut. Alors le probleme de maximisationn de l'utilite esperee de la richesse et de la consommation peut etre vu comme un probleme de maximisation de l'utilite esperee de la richesse terminale avec un horizon aleatoire. La deuxieme partie de cette these concerne l'analyse stochastique des processus ponctuels determinantaux. Les processus determinantaux et permanentaux sont des processus ponctuels dont les fonctions de correlations sont donnees par un determinant ou un permanent. Les points de ces processus ont respectivement un comportement de repulsion ou d'attraction: ils sont tres loin de la situation d'absence de correlation rencontree pour les processus de Poisson. Nous etablissons un resultat de quasi-invariance: nous montrons que si nous perturbons les point le long d'un champ de vecteurs, le processus qui en resulte est toujours un determinantal, dont la loi est absolument continue par rapport a la distribution d'origine. En se basant sur cette formule et en suivant l'approche de Bismut du calcul de Malliavin, nous donnons ensuite une formule d'integration par parties.

In this thesis we study models, which search for an optimal portfolio from a set of stocks. On the contrary to the classical approach focusing only on expected return and variance, we examine models where an additional crite- rion of skewness is included. Furthermore we formulate a model for measuring performance of a portfolio defined as the distance from the Pareto efficient frontier. In numerical experiments we apply the models on historical prices and stock data from the electronic stock market NASDAQ. We analyze the stock data from companies listed in the index NASDAQ-100. We conclude by comparing of optimal portfolios created using different models among each other, with trivial single-stock portfolios and the with NASDAQ-100 index itself.

This master thesis is focused on portfolio diversification. In the Czech Republic, the majority of the population still deposits their free funds to current accounts, but the yield is not sufficient to cover the devaluation caused by inflation. In addition, investments in securities enable these funds to be better valued (naturally with a higher risk). The aggregate of all investments is called the investment portfolio. Harry Markowitz is the founder of modern portfolio theory. The aim of the thesis is to compile an optimal portfolio from chosen financial assets. The theoretical part of the thesis describes the terms such as the financial market, its nature and function and the basic elements of the investment strategy - profitability, risk and liquidity. On top of that, this part describes problems of portfolio theory with a focus on the Markowitz model of optimization. In total 15 stocks-issuing companies are selected from various industries. These companies are traded both on the Czech and American stock markets. The practical part is focused on creating optimal portfolio of selected financial assets. For different attitudes of the investor to risk and its selected strategy the optimal portfolio according to Markowitz is compiled. The weights of individual securities are determined as well as the yield and risk of the portfolios created and an effective boundary is demarcated.