In this thesis, we focus our study on the multi-period portfolio selection problems with different investment conditions. We first analyze the mean-variance multi-period portfolio selection problem with stochastic investment horizon. It is often the case that some unexpected endogenous and exogenous events may force an investor to terminate her investment and leave the market. We give the assumption that the uncertain investment horizon follows a given stochastic process. By making use of the embedding technique of Li and Ng (2000), the original nonseparable problem can be solved by solving an auxiliary problem. With the given assumption, the auxiliary problem can be translated into one with deterministic exit time and solved by dynamic programming. Furthermore, we consider the mean-variance formulation of multi-period portfolio optimization for asset-liability management with an exogenous uncertain investment horizon. Secondly, we consider the multi-period portfolio selection problem in an incomplete market with no short-selling or transaction cost constraint. We assume that the sample space is finite, and the number of possible security price vector transitions is equal to the number of securities. By introducing a family of auxiliary markets, we connect the primal problem to a set of optimization problems without no short-selling or without transaction costs constraint. In the no short-selling case, the auxiliary problem can be solved by using the martingale method of Pliska (1986), and the optimal terminal wealth of the original constrained problem can be derived. In the transaction cost case, we find that the dual problem, which is to minimize the optimal value for the set of optimization problems, is equivalent to the primal problem, when the primal problem has a solution, and we thus characterize the optimal solution accordingly. / Yi, Lan. / Adviser: Duan Li. / Source: Dissertation Abstracts International, Volume: 72-11, Section: A, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 133-139). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

In this thesis, we mainly concentrate on the mean-variance portfolio selection problems with cardinality constraint and/or quantity constraints. These combinatorial problems are NP-hard in general. The first model is the Sharpe ratio portfolio selection problem (2.4) which is a single-period assets selection optimization problem maximizing the Sharpe ratio of a portfolio containing exactly k stocks which are selected from n stocks in the market, and shorting is allowed in this model. We provide an approximation solution for the Sharpe ratio portfolio optimization problem with a worst-case performance guarantee. In the second model, we consider the portfolio selection problem which takes into account both the cardinality constraint and the quantity constraint, i.e., limiting the number of assets and the minimal and maximal shares of each individual asset in the portfolio, respectively, which is reformulated as mixed 0-1 conic programming. In the third model, we consider the random portfolio selection scheme, i.e., we randomly select some stocks into our portfolio either with constant probability or by controlling the probability. In the last model, we assume that investors only would like to either invest in an asset with a substantial amount (represented by some threshold value) or discard it. With the help of the SDP relaxation, a screening algorithm, and a randomized rounding procedure, we find approximative solutions whose worst-case guaranteed performance bound is O( m3). Branch-and-bound method is also considered to find the exact optimal solution for this model. / Keywords. portfolio selection, cardinality, quantity, threshold, SDP relaxation, random rounding procedure, mixed 0-1 conic programming. / Xie Jiang. / "July 2006." / Adviser: Shuzhong Zhang. / Source: Dissertation Abstracts International, Volume: 68-03, Section: B, page: 1910. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (p. 162-172). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Li, Rui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 69-71). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Portfolio Selection Models --- p.1 / Chapter 1.1.1 --- Single Period Models --- p.2 / Chapter 1.1.2 --- Multi-Period Models --- p.4 / Chapter 1.1.3 --- Continuous-Time Model --- p.5 / Chapter 1.2 --- Description and Motivation of New Model --- p.6 / Chapter 1.3 --- Major Contributions --- p.7 / Chapter 1.4 --- Thesis Organization --- p.8 / Chapter 2 --- Formulation and General Methodology --- p.9 / Chapter 2.1 --- Formulation --- p.9 / Chapter 2.1.1 --- Dynamics --- p.10 / Chapter 2.1.2 --- General Form --- p.13 / Chapter 2.1.3 --- Assumptions --- p.13 / Chapter 2.2 --- Methodology --- p.15 / Chapter 2.2.1 --- Weighting Problem --- p.15 / Chapter 2.2.2 --- Search For Optimal Weighting Coefficient --- p.19 / Chapter 3 --- Model I: A Trade-off Between Risk and Return Is Given --- p.22 / Chapter 3.1 --- Problem Formulation --- p.22 / Chapter 3.2 --- Solution to the Parameterized Weighting Problem (PWP(γ)) --- p.23 / Chapter 3.2.1 --- "Construction of the Auxiliary Problem A(γ, λ)" --- p.24 / Chapter 3.2.2 --- Discussion on Parameter A --- p.29 / Chapter 3.3 --- Algorithm --- p.39 / Chapter 4 --- Model II: Expected Return Level Is Specified --- p.42 / Chapter 4.1 --- Problem Formulation --- p.42 / Chapter 4.2 --- Optimal Max-Min Solution --- p.44 / Chapter 4.3 --- Discussion on Parameter λ --- p.50 / Chapter 4.4 --- Algorithm --- p.55 / Chapter 5 --- Numerical Examples --- p.58 / Chapter 6 --- Conclusions --- p.67 / Bibliography --- p.71

Chiu Chun Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 112-114). / Abstracts in English and Chinese. / Abstract --- p.i / 論文摘要 --- p.ii / Acknowledgment --- p.iii / List of Figures n --- p.i / List of Tables n --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.4 / Chapter 3 --- Review of the Minimax Model --- p.7 / Chapter 4 --- Portfolio Optimization with Shorting --- p.13 / Chapter 4.1 --- Formulation of Minimax Model with Shorting --- p.13 / Chapter 4.2 --- A Simple Optimal Investment Strategy --- p.14 / Chapter 4.2.1 --- All Assets Are Risk --- p.14 / Chapter 4.2.2 --- Some Assets Are Riskless --- p.31 / Chapter 4.3 --- Tracing Out the Efficient Frontier --- p.34 / Chapter 4.3.1 --- No Riskless Assets Are Involved --- p.34 / Chapter 4.3.2 --- Riskless Assets Are Involved --- p.43 / Chapter 4.4 --- Chapter Summary --- p.44 / Chapter 5 --- Portfolio Optimization with Investment Limit --- p.50 / Chapter 5.1 --- Formulation of Minimax Model with Investment Limit --- p.51 / Chapter 5.2 --- Optimal Solution to POI(λ) --- p.52 / Chapter 5.2.1 --- All Assets Are Risky --- p.52 / Chapter 5.2.2 --- Some Assets Are Riskless --- p.67 / Chapter 5.3 --- Chapter Summary --- p.71 / Chapter 6 --- Numerical Analysis --- p.72 / Chapter 6.1 --- Data Analysis --- p.72 / Chapter 6.2 --- Experiment Description and Discussion --- p.75 / Chapter 6.2.1 --- Short-Selling is Allowed --- p.75 / Chapter 6.2.2 --- Comparison Between the Cases With Short-Selling and Without Short-Selling --- p.77 / Chapter 6.3 --- Chapter Summary --- p.79 / Chapter 7 --- Conclusion --- p.39 / Chapter A --- List of Companies Included in Numerical Analysis --- p.82 / Chapter B --- Graphical Result of Section 6.21 --- p.84 / Chapter C --- Graphical Result of Section 6.22 --- p.93 / Bibliography --- p.112

Yung Yan Keung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 144-147). / Chapter Chapter 1: --- Introduction --- p.1 / Chapter 1.1 --- Review on Portfolio Theory --- p.3 / Chapter - 1.1.1 --- Expected Return and Risk of a Security --- p.3 / Chapter -1.1.2 --- Expected Return and Risk of a Portfolio --- p.4 / Chapter -1.1.3 --- The Feasible Set --- p.5 / Chapter - 1.1.4 --- Assumptions on the Investor --- p.6 / Chapter -1.1.5 --- Efficient Portfolios --- p.6 / Chapter -1.1.5.1 --- Bounds on the Return and Risk of a portfolio --- p.6 / Chapter -1.1.5.2 --- Concavity of the Efficient Set --- p.8 / Chapter -1.1.6 --- The Market Model --- p.9 / Chapter -1.1.7 --- Risk-free Asset --- p.11 / Chapter - 1.1.8 --- Portfolio involving Risk-free Asset --- p.12 / Chapter -1.1.9 --- The Sharpe Ratio --- p.14 / Chapter 1.2 --- Review on Some Trading Models --- p.19 / Chapter -1.2.1 --- Buy and Hold Model --- p.19 / Chapter -1.2.2 --- Trading Model with Prediction Criteria --- p.20 / Chapter -1.2.2.1 --- Two School of Theories --- p.20 / Chapter - 1.2.2.2 --- Prediction of the stock price movement --- p.20 / Chapter -1.2.2.3 --- The Use of Neural Network in Prediction --- p.21 / Chapter -1.2.2.4 --- Single Step and Multi-step Prediction --- p.23 / Chapter - 1.2.2.5 --- Trading Model based on Prediction Criteria --- p.25 / Chapter - 1.2.2.6 --- For More Accurate Prediction --- p.25 / Chapter -1.2.3 --- Weigend's Model --- p.26 / Chapter - 1.2.3.1 --- Introduction --- p.26 / Chapter -1.2.3.2 --- The Model Setup --- p.26 / Chapter -1.2.3.3 --- The Objective Functions --- p.27 / Chapter - 1.2.3.4 --- The Gradient Ascending Algorithm --- p.27 / Chapter -1.2.3.5 --- The Gradient of the Sharpe Ratio --- p.27 / Chapter - 1.2.3.6 --- The Training Procedure --- p.28 / Chapter - 1.2.3.7 --- Some Properties of the Sharpe Ratio Training --- p.28 / Chapter -1.2.4 --- Bengio's Model --- p.29 / Chapter -1.2.4.1. --- Overview --- p.29 / Chapter -1.2.4.2. --- The Trading System --- p.29 / Chapter - 1.2.4.3 --- The Objective Function: the Portfolio Return --- p.31 / Chapter - 1.2.4.4. --- The Training Process --- p.32 / Chapter - 1.2.4.5 --- Computer Simulation --- p.34 / Chapter - 1.2.4.6 --- Discussion --- p.36 / Chapter Chapter 2: --- The Naive Sharpe Ratio Model --- p.38 / Chapter - 2.1 --- Introduction --- p.39 / Chapter - 2.2 --- Definition of the Naive Sharpe Ratio --- p.39 / Chapter - 2.3 --- Gradient of Naive Sharpe Ratio with respect to the portfolio weighting: --- p.40 / Chapter - 2.4 --- The Training Process --- p.40 / Chapter - 2.5 --- Analysis of the Gradient --- p.41 / Chapter -2.6 --- Compare with Bengio's and Weigend's Model --- p.42 / Chapter -2.7. --- Computer Simulations --- p.43 / Chapter -2.7.1 --- Experiment 1: How the Sharpe Ratio is Maximized --- p.43 / Chapter -2.7.1.1 --- Experiment 11 --- p.44 / Chapter -2.7.1.2 --- Experiment 12 --- p.45 / Chapter -2.7.1.3 --- Experiment 13 --- p.46 / Chapter -2.7.2 --- Experiment 2: Reducing the Unique Risk --- p.49 / Chapter -2.7.3 --- Experiment 3: Apply to the Stock Market --- p.52 / Chapter -2.8 --- Redefining the Naive Sharpe ratio with down-side risk --- p.56 / Chapter -2.8.1 --- Definitions --- p.56 / Chapter -2.8.2 --- Gradient of the Downside Nai've Sharpe Ratio --- p.57 / Chapter -2.8.3 --- Analysis of the gradient of the new Sharpe ratio --- p.57 / Chapter -2.8.4 --- Experiment: Compared with Symmetric Risk --- p.59 / Chapter -2.8.4.1 --- Experimental Setup --- p.59 / Chapter -2.8.4.2 --- Experimental Result --- p.60 / Chapter -2.8.4.3 --- Discussion --- p.62 / Chapter - 2.9 --- Further Discussion --- p.63 / Chapter Chapter 3: --- The Total Sharpe Ratio Model --- p.64 / Chapter - 3.1 --- Introduction --- p.65 / Chapter -3.2 --- Defining risk of portfolio in terms of component securities' risk --- p.65 / Chapter -3.2.1. --- Return for Each Security and the Whole Portfolio at Each Time Step --- p.65 / Chapter -3.3.2. --- Covariance of the Individual Securities' Returns --- p.66 / Chapter -3.2.3. --- Define the Sharpe Ratio and the Objective Function --- p.66 / Chapter -3.2.3.1. --- The Excess Return --- p.66 / Chapter -3.2.3.2. --- The Risk --- p.67 / Chapter -3.2.3.3. --- The Sharpe Ratio at Time t --- p.67 / Chapter -3.2.3.4. --- The Objective Function: the total Sharpe ratio --- p.67 / Chapter -3.2.3.5. --- The Training Process --- p.68 / Chapter -3.3 --- Calculating the Gradient of the Total Sharpe Ratio --- p.69 / Chapter -3.4. --- Analysis of the Total Sharpe Ratio Gradient --- p.70 / Chapter -3.4.1 --- The Gradient Vector of the Sharpe Ratio at a Particular Time Step --- p.70 / Chapter -3.4.2 --- The Gradient Vector of the Risk --- p.70 / Chapter - 3.5 --- Computer Simulation: --- p.72 / Chapter -3.5.1 --- Apply to the Stock Market1 --- p.72 / Chapter -3.5.1.1 --- Objective --- p.72 / Chapter - 3.5.1.2 --- Experimental Setup --- p.72 / Chapter -3.5.1.3 --- The Experimental Result --- p.73 / Chapter -3.5.2 --- Apply to the Stock Market2 --- p.78 / Chapter -3.5.2.1 --- Objective --- p.78 / Chapter -3.5.2.2 --- Experimental Setup --- p.78 / Chapter -3.5.2.3 --- The Experimental Result --- p.79 / Chapter -3.6 --- Defining the Total Sharpe Ratio in terms of Downside Risk --- p.84 / Chapter - 3.6.1. --- Introduction --- p.84 / Chapter -3.6.2. --- Covariance of the individual securities' returns --- p.84 / Chapter -3.6.3. --- Define the Downside Risk Sharpe ratio and the objective function --- p.85 / Chapter -3.6.3.1. --- The Excess Return --- p.85 / Chapter -3.6.3.2. --- The Downside Risk --- p.85 / Chapter -3.6.3.3. --- The Sharpe ratio at time T --- p.85 / Chapter -3.6.3.4. --- The Objective function --- p.85 / Chapter -3.6.4. --- The Training Process --- p.85 / Chapter -3.7 --- Total Sharpe Ratio involving Transaction Cost --- p.86 / Chapter -3.7.1 --- Introduction --- p.86 / Chapter -3.7.2 --- Return for each stock and the whole portfolio at each time step --- p.86 / Chapter -3.7.3 --- Linear Approximation of the Portfolio's return --- p.88 / Chapter -3.7.4 --- Covariance of the individual securities' returns --- p.89 / Chapter -3.7.5 --- Define the Sharpe ratio and the objective function --- p.90 / Chapter -3.7.5.1 --- The Excess Return --- p.90 / Chapter -3.7.5.2 --- The Risk --- p.90 / Chapter -3.7.5.3 --- The Sharpe Ratio at time T --- p.90 / Chapter -3.7.5.4 --- The Objective Function --- p.90 / Chapter -3.7.6 --- Calculation of the gradient of the Total Sharpe ratio --- p.91 / Chapter -3.7.7. --- Analysis of the Total Sharpe Ratio Gradient --- p.94 / Chapter -3.7.7.1 --- The Gradient Vector of the Sharpe Ratio at a Particular Time Step --- p.94 / Chapter -3.7.7.2 --- The Gradient Vector of the Risk --- p.94 / Chapter -3.7.8 --- Experiment 1: Compare with Buy and Hold Method --- p.96 / Chapter -3.7.8.1 --- Experiment 11 --- p.96 / Chapter -3.7.8.2. --- Experiment 12 --- p.102 / Chapter -3.7.9 --- Experiment 2: Compared with Naive Sharpe Ratio --- p.108 / Chapter -3.7.9.1 --- Objective --- p.108 / Chapter -3.7.9.2. --- Experimental Setup --- p.108 / Chapter -3.7.9.3. --- The Experimental Result --- p.109 / Chapter - 3.7.10 --- Experiment 3: Compared with other models --- p.113 / Chapter - 3.7.10.1 --- Experiment 31 --- p.113 / Chapter - 3.7.10.2. --- Experiment 32 --- p.117 / Chapter -3.7.11 --- Experiment 4: Apply to the Stock Market --- p.121 / Chapter -3.7.11.1 --- Objective --- p.121 / Chapter - 3.7.11.2. --- Experimental Setup --- p.121 / Chapter -3.7.11.3. --- The Experimental Result --- p.121 / Chapter Chapter 4: --- Conclusion --- p.126 / Appendix A --- p.130 / Appendix B --- p.139 / Appendix C --- p.141 / Appendix D --- p.142 / Reference --- p.144

by Wan-Lung Ng. / Thesis submitted in: November 1997. / On added t.p.: January 19, 1998. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 114-119). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Overview --- p.1 / Chapter 1.2 --- Organization Outline --- p.5 / Chapter 2 --- Literature Review --- p.7 / Chapter 2.1 --- Modern Portfolio Theory --- p.7 / Chapter 2.1.1 --- Mean-Variance Model --- p.9 / Chapter 2.1.2 --- Setting-up the relationship between the portfolio and its component securities --- p.11 / Chapter 2.1.3 --- Identifying the efficient frontier --- p.12 / Chapter 2.1.4 --- Selecting the best compromised portfolio --- p.13 / Chapter 2.2 --- Stochastic Optimal Control --- p.17 / Chapter 2.2.1 --- Dynamic Programming --- p.18 / Chapter 2.2.2 --- Dynamic Programming Decomposition --- p.21 / Chapter 3 --- Multiple Period Portfolio Analysis --- p.23 / Chapter 3.1 --- Maximization of Multi-period Consumptions --- p.24 / Chapter 3.2 --- Maximization of Utility of Terminal Wealth --- p.29 / Chapter 3.3 --- Maximization of Expected Average Compounded Return --- p.33 / Chapter 3.4 --- Minimization of Time to Reach Target --- p.35 / Chapter 3.5 --- Goal-Seeking Investment Model --- p.37 / Chapter 4 --- Multi-period Mean-Variance Analysis with a Riskless Asset --- p.40 / Chapter 4.1 --- Motivation --- p.40 / Chapter 4.2 --- Dynamic Mean-Variance Analysis Formulation --- p.43 / Chapter 4.3 --- Auxiliary Problem Formulation --- p.45 / Chapter 4.4 --- Efficient Frontier in Multi-period Portfolio Selection --- p.53 / Chapter 4.5 --- Obseravtions --- p.58 / Chapter 4.6 --- Solution Algorithm for Problem E (w) --- p.62 / Chapter 4.7 --- Illstrative Examples --- p.63 / Chapter 4.8 --- Verification with Single-period Efficient Frontier --- p.72 / Chapter 4.9 --- Generalization to Cases with Nonlinear Utility Function of E (xT) and Var (xT) --- p.75 / Chapter 5 --- Dynamic Portfolio Selection without Risk-less Assets --- p.84 / Chapter 5.1 --- Construction of Auxiliuary Problem --- p.88 / Chapter 5.2 --- Analytical Solution for Efficient Frontier --- p.89 / Chapter 5.3 --- Reduction to Investment Situations with One Risk-free Asset --- p.101 / Chapter 5.4 --- "Multi-period Portfolio Selection via Maximizing Utility function U(E {xT),Var (xT))" --- p.103 / Chapter 6 --- Conclusions and Recommendations --- p.108 / Chapter 6.1 --- Summaries and Achievements --- p.108 / Chapter 6.2 --- Future Studies --- p.110 / Chapter 6.2.1 --- Constrained Investment Situations --- p.110 / Chapter 6.2.2 --- Including Higher Moments --- p.111

As we know, the deterministic LQ problems are well-posed if the state weighting matrix and the control weighting matrix are nonnegative and positive definite in the cost function, respectively. Some practical problems, however, often include indefinite weighting matrices in their cost functions such as mean-variance portfolio selection problem. This inspires us to further study the indefinite LQ problems in detail. / In this thesis, we study indefinite stochastic linear-quadratic (LQ) control with jumps and present some financial applications of this new development. / The results of the above LQ control problems are employed to deal with a mean-variance portfolio selection model in an incomplete financial market. An optimal analytical investment strategy is directly derived and the expression of its risk is explicitly presented. In addition, a mean-variance portfolio selection model in a financial market where shorting is not allowed is investigated in detail via the stochastic LQ problem with nonnegative controls. In particular, the explicit expression of the efficient frontier enables an investor to better understand the relation between the expected terminal wealth and the risk in a stock market with no-shorting. / The weighting matrices in the cost function are allowed to be indefinite (in particular, negative) when the diffusion term linearly depends on the control variable in the state equation. In this case, indefinite stochastic LQ control problems with jumps may still be sensible and well-posed. In an infinite time horizon, solvability of coupled generalized algebraic Riccati equations (CGAREs) is sufficient for the well-posedness of the stochastic LQ control problem with jumps. Moreover, an approach algorithm is devised to solve the CGAREs via semi-definite programming over linear matrix inequalities. On the other hand, it is shown that the well-posedness of the stochastic LQ control problem in a finite time horizon with jumps is equivalent to solvability of coupled generalized Riccati equations. / Li Xun. / "November 2000." / Advisers: Cai Xiaoqiang; Zhou Xunyu. / Source: Dissertation Abstracts International, Volume: 61-10, Section: B, page: 5541. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 115-122). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Sen Sui. / Thesis submitted in: August 2003. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 88-92). / Abstracts in English and Chinese. / Abstract --- p.ii / Acknowledgement --- p.iv / Dedication --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation and Objective --- p.1 / Chapter 1.2 --- Contributions --- p.4 / Chapter 1.3 --- Thesis Organization --- p.5 / Chapter 2 --- Background of Risk Management --- p.7 / Chapter 2.1 --- Measuring Return --- p.8 / Chapter 2.2 --- Objectives of Risk Measurement --- p.11 / Chapter 2.3 --- Simple Statistics for Measurement of Risk --- p.15 / Chapter 2.4 --- Methods for Value-at-Risk Measurement --- p.16 / Chapter 2.5 --- Conditional VaR --- p.18 / Chapter 2.6 --- Portfolio VaR Methods --- p.18 / Chapter 2.7 --- Coherent Risk Measure --- p.20 / Chapter 2.8 --- Summary --- p.22 / Chapter 3 --- Selection of Independent Factors for VaR Computation --- p.23 / Chapter 3.1 --- Mixture Convolution Approach Restated --- p.24 / Chapter 3.2 --- Procedure for Selection and Evaluation --- p.26 / Chapter 3.2.1 --- Data Preparation --- p.26 / Chapter 3.2.2 --- ICA Using JADE --- p.27 / Chapter 3.2.3 --- Factor Statistics --- p.28 / Chapter 3.2.4 --- Factor Selection --- p.29 / Chapter 3.2.5 --- Reconstruction and VaR Computation --- p.30 / Chapter 3.3 --- Result and Comparison --- p.30 / Chapter 3.4 --- Problem of Using Kurtosis and Skewness --- p.40 / Chapter 3.5 --- Summary --- p.43 / Chapter 4 --- Mixture of Gaussians and Value-at-Risk Computation --- p.45 / Chapter 4.1 --- Complexity of VaR Computation --- p.45 / Chapter 4.1.1 --- Factor Selection Criteria and Convolution Complexity --- p.46 / Chapter 4.1.2 --- Sensitivity of VaR Estimation to Gaussian Components --- p.47 / Chapter 4.2 --- Gaussian Mixture Model --- p.52 / Chapter 4.2.1 --- Concept and Justification --- p.52 / Chapter 4.2.2 --- Formulation and Method --- p.53 / Chapter 4.2.3 --- Result and Evaluation of Fitness --- p.55 / Chapter 4.2.4 --- Evaluation of Fitness using Z-Transform --- p.56 / Chapter 4.2.5 --- Evaluation of Fitness using VaR --- p.58 / Chapter 4.3 --- VaR Estimation using Convoluted Mixtures --- p.60 / Chapter 4.3.1 --- Portfolio Returns by Convolution --- p.61 / Chapter 4.3.2 --- VaR Estimation of Portfolio Returns --- p.64 / Chapter 4.3.3 --- Result and Analysis --- p.64 / Chapter 4.4 --- Summary --- p.68 / Chapter 5 --- VaR for Portfolio Optimization and Management --- p.69 / Chapter 5.1 --- Review of Concepts and Methods --- p.69 / Chapter 5.2 --- Portfolio Optimization Using VaR --- p.72 / Chapter 5.3 --- Contribution of the VaR by ICA/GMM --- p.76 / Chapter 5.4 --- Summary --- p.79 / Chapter 6 --- Conclusion --- p.80 / Chapter 6.1 --- Future Work --- p.82 / Chapter A --- Independent Component Analysis --- p.83 / Chapter B --- Gaussian Mixture Model --- p.85 / Bibliography --- p.88

Chen Li. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 76-78). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgements --- p.iii / Table of Contents --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature review --- p.4 / Chapter 3 --- The semi-mean target tracking model --- p.9 / Chapter 3.1 --- Introduction --- p.9 / Chapter 3.2 --- The robust optimization problem --- p.11 / Chapter 3.3 --- Portfolio selection methods --- p.14 / Chapter 3.3.1 --- Jensen's inequality approach --- p.15 / Chapter 3.3.2 --- The robust optimization approach --- p.17 / Chapter 3.3.3 --- Empirical method --- p.22 / Chapter 3.4 --- How to evaluate a portfolio? --- p.24 / Chapter 3.4.1 --- Tight bounds --- p.24 / Chapter 3.4.2 --- The semidefinite programming bounds --- p.25 / Chapter 3.4.3 --- Conclusions --- p.28 / Chapter 3.5 --- Numerical results --- p.29 / Chapter 3.5.1 --- The analysis of the data --- p.29 / Chapter 3.5.2 --- Jensen's inequality approach --- p.31 / Chapter 3.5.3 --- The robust optimization approach --- p.34 / Chapter 3.5.4 --- The empirical linear programming method --- p.34 / Chapter 3.6 --- Comparisons and conclusions --- p.39 / Chapter 4 --- The semi-variance target tracking model --- p.45 / Chapter 4.1 --- Introduction --- p.45 / Chapter 4.2 --- The portfolio selection methods --- p.46 / Chapter 4.2.1 --- The robust optimization method --- p.47 / Chapter 4.2.2 --- The empirical method --- p.50 / Chapter 4.3 --- Evaluating a selected portfolio --- p.52 / Chapter 4.3.1 --- Computing SDP bounds --- p.52 / Chapter 4.3.2 --- Conclusions --- p.55 / Chapter 4.4 --- Numerical results --- p.55 / Chapter 4.4.1 --- The robust optimization method --- p.56 / Chapter 4.4.2 --- The empirical second order cone programming method --- p.61 / Chapter 4.4.3 --- Comparisons and conclusions --- p.61 / Chapter 4.5 --- Summary and future work --- p.69 / Appendix A --- p.70 / Bibliography --- p.76

Liu, Weiyang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 111-113). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction and Literature Review --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- Background and Motivating examples --- p.2 / Chapter 1.1.2 --- Basic Concepts --- p.4 / Chapter 1.1.3 --- Outline of the thesis --- p.6 / Chapter 1.2 --- Literature Review --- p.8 / Chapter 2 --- Multi-period Cooperative Re-investment Games: The Basic Model --- p.11 / Chapter 2.1 --- Basic settings and assumptions --- p.11 / Chapter 2.2 --- The problem --- p.13 / Chapter 3 --- Three sub-models and the allocation rule of Sub-Model III --- p.17 / Chapter 3.1 --- Three possible sub-models of the basic model --- p.17 / Chapter 3.1.1 --- Sub-model I --- p.17 / Chapter 3.1.2 --- Sub-model II --- p.18 / Chapter 3.1.3 --- Sub-model III --- p.19 / Chapter 3.2 --- The allocation rule of Sub-model III --- p.19 / Chapter 4 --- A two period example of the revised basic model --- p.25 / Chapter 4.1 --- The two period example with two projects --- p.25 / Chapter 4.2 --- The algorithm for the dual problem --- p.29 / Chapter 5 --- Extensions of the Basic Model --- p.35 / Chapter 5.1 --- The model with stochastic budgets --- p.36 / Chapter 5.2 --- The core of the model with stochastic budgets --- p.39 / Chapter 5.3 --- An example: the two-period case of models with stochastic bud- gets and an algorithm for the dual problem --- p.46 / Chapter 5.4 --- An interesting marginal effect --- p.52 / Chapter 5.5 --- "A Model with stochastic project prices, stochastic returns and stochastic budgets" --- p.54 / Chapter 6 --- Multi-period Re-investment Model with risks --- p.58 / Chapter 6.1 --- The Model with l1 risk measure --- p.58 / Chapter 6.2 --- The Model with risk measure --- p.66 / Chapter 7 --- Numerical Tests --- p.70 / Chapter 7.1 --- The affects from uncertainty changes --- p.71 / Chapter 7.2 --- The affects from budget changes --- p.71 / Chapter 7.3 --- The affects from the budget changes of only one group --- p.71 / Chapter 8 --- Conclusive Remarks --- p.77 / Chapter A --- Original Data and Analysis for Section 7.1 (Partial) --- p.79 / Chapter B --- Data Analysis for Section 7.2 (Partial) --- p.95 / Chapter C --- Data Analysis for Section 7.3 (Partial) --- p.98