本篇論文主要是將Black-Litterman模型應用在組合型基金上。從一個組合型基金的基金經理人角度出發，在有限的風險下，如何進行資產配置使其達到報酬極大化的目標？第二章介紹mean-variance模型，以及其模型之缺點。第三章介紹Black-Litterman模型，其不僅可以改善mean-variace模型的缺點，此外允許投資人加入主觀看法，結合數量方法以及投資人的主觀看法是此模型的特色之一。第四章，針對兩個模型的進行比較。最後，我們發現：BLack-Litterman模型不僅符合經濟直覺，進行資產配置時也展現模型的穩定性。 / This paper applies a popular asset allocation model: the Black-Litterman model on a fund of funds. First, an overview is given of the foundations of modern portfolio theory with the mean-variance model. Next, we discuss some improvements that could be made over the mean-variance model. The Black-Litterman model addresses some of these flaws and tries to improve them. Finally, simulation has been performed to compare the performance of the Black-Litterman model to mean-variance optimization. The models have been compared in intuitiveness and stability. The conclusion can be drawn that BL-model improves the mean-variance model, in our simulation, both in intuitiveness and stability.

MASTER THESIS ABSTRACT TITLE: Trading Strategy in Incomplete Market AUTHOR: Tomáš Bunčák DEPARTMENT: Department of Probability and Mathematical Statistics, Charles University in Prague SUPERVISOR: Andrea Karlová We focus on the problem of finding optimal trading strategies (in a meaning corresponding to hedging of a contingent claim) in the realm of incomplete markets mainly. Although various ways of hedging and pricing of contingent claims are outlined, main subject of our study is the so-called mean-variance hedging (MVH). Sundry techniques used to treat this problem can be categorized into two approaches, namely a projection approach (PA) and a stochastic control approach (SCA). We review the methodologies used within PA in diversely general market models. In our research concerning SCA, we examine the possibility of using the methods of optimal stochastic control in MVH, and we study the problem of our interest in several settings of market models; involving cases of pure diffusion models and a jump- diffusion case. In order to reach an exemplary comparison, we provide solutions of the MVH problem in the setting of the Heston model via techniques of both of the approaches. Some parts of the thesis are accompanied with numerical illustrations.

Este estudo considera o problema de controle ótimo multi-período de média-variância para sistemas em tempo discreto com saltos markovianos e ruídos multiplicativos. Inicialmente considera-se um critério de desempenho formado por uma combinação linear da variância nal e valor esperado da saída do sistema. É apresentada uma solução analítica na obtenção da estratégia ótima para este problema. Em seguida são considerados os casos onde os critérios de desempenho são minimizar a variância nal sujeito a uma restrição no valor esperado ou maximizar o valor esperado nal sujeito a uma restrição na variância nal da saída do sistema. As estratégias ótimas de controle são obtidas de um conjunto de equações de diferenças acopladas de Riccati. Os resultados obtidos neste estudo generalizam resultados anteriores da literatura para o problema de controle ótimo com saldos markovianos e ruídos multiplicativos, apresentando condições explícitas e sucientes para a otimalidade da estratégia de controle. São apresentados modelos e simulações numéricas em otimização de carteiras de investimento e estratégias de gestão de ALM (asset liabilities management). / This thesis focuses on the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noise under three kinds of performance criterions related to the nal value of the expectation and variance of the output. In the first problem it is desired to minimize the nal variance of the output subject to a restriction on its nal expectation, in the second one it is desired to maximize the nal expectation of the output subject to a restriction on its nal variance, and in the third one it is considered a performance criterion composed by a linear combination of the nal variance and expectation of the output of the system. The optimal control strategies are obtained from a set of interconnected Riccati dierence equations and explicit sufficient conditions are presented for the existence of an optimal control strategy for these problems, generalizing previous results in the literature. Numerical simulations of investment portfolios and asset liabilities management models for pension funds with regime switching are presented.

Centralized power system planning covers time windows that range from ten to thirty years. Consequently, it is the longest and most uncertain part of power system economics. One of the challenges that power system planning faces is the inability to accurately predict random events; these random events introduce risk in the planning process. Another challenge stems from the fact that, despite having a centralized planning scheme, generation plans are set first and then transmission expansion plans are carried out. This thesis addresses these problems. A joint model for generation and transmission expansion for the vertically integrated industry is proposed. Randomness is considered in demand, equivalent availability factors of the generators, and transmission capacity factors of the transmission lines. The system expansion model is formulated as a two-stage stochastic program with fixed recourse and probabilistic constraints. The transmission network is included via a DC approximation. The mean variance Markowitz theory is used as a risk minimization technique in order to minimize the variance of the annualized estimated generating cost. This system expansion model is capable of considering the locations of new generation and transmission and also of choosing the right mixture of generating technologies. The global tendency is to move from regulated power systems to deregulated power systems. Power pool electricity markets, assuming that the independent system operator is concerned with the social cost minimization, face great uncertainties from supply and demand bids submitted by market participants. In power pool electricity markets, randomness in the cost and benefit functions through random demand and supply functions has never been considered before. This thesis considers as random all the coefficients of the quadratic cost and benefit functions and uses the mean variance Markowitz theory to minimize the social cost variance. The impacts that this risk minimization technique has on nodal prices and on the elasticities of the supply and demand curves are studied. All the mathematical models in this thesis are exemplified by the six-node network proposed by Garver in 1970, by the 21-node network proposed by the IEEE Reliability Test System Task Force in 1979, and by the IEEE 57- and 118-node systems.

Centralized power system planning covers time windows that range from ten to thirty years. Consequently, it is the longest and most uncertain part of power system economics. One of the challenges that power system planning faces is the inability to accurately predict random events; these random events introduce risk in the planning process. Another challenge stems from the fact that, despite having a centralized planning scheme, generation plans are set first and then transmission expansion plans are carried out. This thesis addresses these problems. A joint model for generation and transmission expansion for the vertically integrated industry is proposed. Randomness is considered in demand, equivalent availability factors of the generators, and transmission capacity factors of the transmission lines. The system expansion model is formulated as a two-stage stochastic program with fixed recourse and probabilistic constraints. The transmission network is included via a DC approximation. The mean variance Markowitz theory is used as a risk minimization technique in order to minimize the variance of the annualized estimated generating cost. This system expansion model is capable of considering the locations of new generation and transmission and also of choosing the right mixture of generating technologies. The global tendency is to move from regulated power systems to deregulated power systems. Power pool electricity markets, assuming that the independent system operator is concerned with the social cost minimization, face great uncertainties from supply and demand bids submitted by market participants. In power pool electricity markets, randomness in the cost and benefit functions through random demand and supply functions has never been considered before. This thesis considers as random all the coefficients of the quadratic cost and benefit functions and uses the mean variance Markowitz theory to minimize the social cost variance. The impacts that this risk minimization technique has on nodal prices and on the elasticities of the supply and demand curves are studied. All the mathematical models in this thesis are exemplified by the six-node network proposed by Garver in 1970, by the 21-node network proposed by the IEEE Reliability Test System Task Force in 1979, and by the IEEE 57- and 118-node systems.

The most popular and fundamental portfolio optimization problem is Markowitz&#039 / s one period mean-variance portfolio selection problem. However, it is criticized because of its one period static nature. Further, the estimation of the stock price expected return is a particularly hard problem. For this purpose, there are a lot of studies solving the mean-variance portfolio optimization problem in continuous time. To solve the estimation problem of the stock price expected return, in 1992, Black and Litterman proposed the Bayesian asset allocation method in discrete time. Later on, Lindberg has introduced a new way of parameterizing the price dynamics in the standard Black-Scholes and solved the continuous time mean-variance portfolio optimization problem. In this thesis, firstly we take up the Lindberg&#039 / s approach, we generalize the results to a jump-diffusion market setting and we correct the proof of the main result. Further, we demonstrate the implications of the Lindberg parameterization for the stock price drift vector in different market settings, we analyze the dependence of the optimal portfolio from jump and diffusion risk, and we indicate how to use the method. Secondly, we present the Lagrangian function approach of Korn and Trautmann and we derive some new results for this approach, in particular explicit representations for the optimal portfolio process. In addition, we present the L2-projection approach of Schweizer for the continuous time mean-variance portfolio optimization problem and derive the optimal portfolio and the optimal wealth processes for this approach. While, deriving these results as the underlying model, the market parameterization of Lindberg is chosen. Lastly, we compare these three different optimization frameworks in detail and their attractive and not so attractive features are highlighted by numerical examples.

This report covers a study testing the possibility of adding portfolio optimization by mean-variance analysis as a tool to extend the concept of momentum strategies in contrast to naive allocation formed by Jegadeesh & Titman (1993). Further these active investment strategies are compared with a passive benchmark as well as a randomly selected portfolio over the entire study-period. The study showed that the naive allocation model outperformed the mean-variance model both economically as well as statistically. No indication where obtained for a lagged return effect when letting a mean-variance model choose weights for a quarterly holding period and the resulting investment recommendation is to follow a naive investment strategy within a momentum framework.

Within the scope of this thesis, the Black-Litterman Asset Allocation Model (as presented in He & Litterman, 1999) is compared to the classical mean-variance framework by simulating past performance of portfolios constructed by both models using identical input data. A quantitative investment strategy which favours stocks with high dividend yield rates is used to generate private views about the expected excess returns for a fraction of the stocks included in the sample. By comparing the ex-post risk-return characteristics of the portfolios and performing ample sensitivity analysis with respect to the numerical values assigned to the input variables, we evaluate the two models’ suitability for different categories of portfolio managers. As a neutral benchmark towards which both portfolios can be measured, a third market-capitalization-weighted portfolio is constructed from the same investment universe. The empirical data used for the purpose of our simulations consists of total return indices for 23 of the 30 stocks included in the OMXS30 index as of the 21st of February 2014 and stretches between January of 2003 and December of 2013.   The results of our simulations show that the Black-Litterman portfolio has delivered risk-adjusted return which is superior not only to that of its market-capitalization-weighted counterpart but also to that of the classical mean-variance portfolio. This result holds true for four out of five simulated strengths of the investment strategy under the assumption of zero transaction costs, a rebalancing frequency of 20 trading days, an estimated risk aversion parameter of 2.5 and a five per cent uncertainty associated with the CAPM prior. Sensitivity analysis performed by examining how the results are affected by variations in these input variables has also shown notable differences in the sensitivity of the results obtained from the two models. While the performance of the Black-Litterman portfolio does undergo material changes as the inputs are varied, these changes are nowhere near as profound as those exhibited by the classical mean-variance portfolio.   In the light of our empirical results, we also conclude that there are mainly two aspects which the portfolio manager ought to consider before committing to one model rather than the other. Firstly, the nature behind the views generated by the investment strategy needs to be taken into account. For the implementation of views which are of an α-driven character, the dynamics of the Black-Litterman model may not be as appropriate as for views which are believed to also influence the expected return on other securities. Secondly, the soundness of using market-capitalization weights as a benchmark towards which the final solution will gravitate needs to be assessed. Managers who strive to achieve performance which is fundamentally uncorrelated to that of the market index may want to either reconsider the benchmark weights or opt for an alternative model.

In view of the current situation of fluctuating high crude oil prices, it is now more important than ever for petroleum refineries to operate at an optimal level in the present dynamic global economy. Acknowledging the shortcomings of deterministic models, this work proposes a hybrid of stochastic programming formulations for an optimal midterm refinery planning that addresses three factors of uncertainties, namely price of crude oil and saleable products, product demand, and production yields. An explicit stochastic programming technique is utilized by employing compensating slack variables to account for violations of constraints in order to increase model tractability. Four approaches are considered to ensure both solution and model robustness: (1) the Markowitz???s mean???variance (MV) model to handle randomness in the objective coefficients of prices by minimizing variance of the expected value of the random coefficients; (2) the two-stage stochastic programming with fixed recourse approach via scenario analysis to model randomness in the right-hand side and left-hand side coefficients by minimizing the expected recourse penalty costs due to constraints??? violations; (3) incorporation of the MV model within the framework developed in Approach 2 to minimize both the expectation and variance of the recourse costs; and (4) reformulation of the model in Approach 3 by adopting mean-absolute deviation (MAD) as the risk metric imposed by the recourse costs for a novel application to the petroleum refining industry. A representative numerical example is illustrated with the resulting outcome of higher net profits and increased robustness in solutions proposed by the stochastic models.

>Magister Scientiae - MSc / The Basel Committee published its proposals for a revised capital adequacy framework(the Basel II Capital Accord) in June 2006. One of the main objectives of this framework is to improve the incentives for state-of-the-art risk management in banking, especially in the area of credit risk in view of Basel II. The new regulation seeks to provide incentives for greater awareness of differences in risk through more risk-sensitive minimum capital requirements based on numerical formulas. This attempt to control bank behaviour has a heavy reliance on regulatory ratios like the risk-based capital adequacy ratio (CAR). In essence, such ratios compare the capital that a bank holds to the level of credit, market and operational risk that it bears. Due to this fact the objectives in this dissertation are as follows. Firstly, in an attempt to address these problems and under assumptions about retained earnings, loan-loss reserves, the market and shareholder-bank owner relationships, we construct continuous-time models of the risk-based CAR which is computed from credit and market risk-weighted assets (RWAs) and bank regulatory capital (BRC) in a stochastic setting. Secondly, we demonstrate how the CAR can be optimized in terms of equity allocation. Here, we employ dynamic programming for stochastic optimization, to obtain and verify the results. Thirdly, an important feature of this study is that we apply the mean-variance approach to obtain an optimal strategy that diversifies a portfolio consisting of three assets. In particular, chapter 5 is an original piece of work by the author of this dissertation where we demonstrate how to employ a mean-variance optimization approach to equity allocation under certain conditions.