The Log-Optimal Portfolio And Risk Management

The theory of portfolio is one of the most important topics in mod-ern finance, the purpose of which is finding a optimal portfolio to minimize the investment risk under a given profit level, or to maximize the investor's expected utility under a given level of investment risk.In the aspect of invest-ment decisions, investors usually invest continuously in financial markets, namely, one re-invest wholly or partly with interest at the end of the cy-cle. Therefore, the problem of multi-period portfolio has caused a number of economists and investors'attention. The log—optimal portfolio model is one of the feasible method to solve multi-period portfolio problem, which is not only easy to handle in mathematics, but the logarithmic function is also a commonly used utility function, indicating that investors are risk-averse.In this thesis, we focus on the log-optimal portfolio, based on theo-retical research-empirical analysis paradigm.Firstly, we study the products of domestic and foreign scholars on the log-optimal portfolio systematically and refine the classic theory and method; secondly, we establish the periodic and multi-period log-and semi-log-optimal portfolio models, and study the solutions and the properties of these models, also discussed the different co-variance matrix under different risk constraints log-optimal portfolio model; finally, using our actual data, we compare the log-and semi-log-optimal port-folio models.The concrete structure is as follows:The first chapter is the Introduction.The main topics of this chapter is the background and significance of this paper.We also present the main contents, methods and structure of this thesis, and point out the major innovations in this chapter. Chapter II is a literature review.In this paper, we study the financial risk measure, optimal investment portfolio and log—ptimal portfolio of re-search literature.Among them, is the aspect of financial risk measure, we introduce the domestic and international classical research theory and meth-ods in all ages, and study the evolution and status of the method of financial risk measurement as well.In the aspect of optimal portfolio, we introduce the log—ptimal portfolio, including the one under different periods, differ-ent risk constraints, and the log—ptimal portfolio and semi-log-optimal portfolio under continuous time, we also study the various algorithms of log—ptimal portfolio at home and abroad.In Chapter III, we investigate the log—ptimal portfolio in detail.First of all, we generalize the notions of utility function, short selling and mar-gin etc; secondly, we introduce the definition the properties of log—Optimal Portfolio; finally, based on the different tolerance levels of risk that differ-ent investors may have in actual investment decision, we present the risk control function and consistent risk measures.Besides, the notions, proper-ties and the difference of Variance, Value-at-Risk (VaR), Expected Shortfall (ES) are given respectively, moreover, we construct the single-periodic and multi-period log-optimal portfolio model based on the variance, the Value at Risk (VaR), Expected Shortfall (ES) constraints, and discuss the nature of the solutions to these models with different risk constraints, we also give the theorems of the existence and uniqueness of the solution and prove them.In Chapter IV, we present the algorithm and empirical research of log-optimal portfolio.First of all, we summarize the classical theory and method of log-optimal portfolio:particle swarm optimization (PSO) was proposed by Kennedy and Eberhart in 1995, and established by the simulation of the simplified social model; the irreducible gradient method is presented by Wolfe in 1962, for non-linear objective function and linear elements issues; genetic algorithms is a global optimization adaptive probabilistic search algorithm by the reference of the biological natural selection and genetic mechanism.Secondly, the data sources and selection are introduced, we select Chi- nese CSI 300 industry index as a research data, including CSI 300 Energy index, CSI 300 Materials index, CSI 300 Industrials, CSI 300 Consumer Dis-cretionary index, CSI 300 Consumer Staples index, CSI 300 Health Care in-dex, CSI 300 Financial index, CSI 300 Information Technology index, CSI 300 Telecommunication Services index and CSI 300 Utilities index.They shall be the sample stock of the empirical research.We describe, analysis and Granger causality test the data, then study them by dividing them into two sets of data where Granger causality exists or not.Finally, we give the empirical research of log-optimal portfolio. The empirical research of single-periodic and multi-period log—ptimal portfolio model under variance, Value at Risk and Expected Shortfall constraints are given under the same confidence level, different levels of risk and different confidence level, the same level of risk, respectively, we also analysis and compare the results we obtained.obtained.In Chapter V, we discusses the sample covariance matrix, the numerical matrix, the two parameter model matrix, the single-index model matrix, the constant correlation matrix as the covariance matrix related to the stock. We also analysis log-optimal portfolio model with the variance, VaR and ES risk function constraints under different covariance matrix, and we discussed the mean and variance standard under different covariance matrix for optimal portfolio. Moreover, we analyse and compare the empirical findings, and give a relatively proper estimated matrix of the covariance matrix.In Chapter VI, we study the semi-log-optimal portfolio model and the empirical research.First of all, we introduce the classical literature and mod-els of semi-log-optimal portfolio, including the nature and empirical analy-sis of semi-log-optimal portfolio model by Vajda (2006), in a stable market assumption.Secondly, the single-periodic and multi-period semi-log-optimal portfolio models are established based on Value at Risk and expected short-fall risk constraints, and we discuss the properties of the solutions to these models under different risk constraints and give the existence and uniqueness theorem and proof of the solutions.Finally, we give the empirical research of the single-periodic and multi-period semi-log-optimal portfolio model for VaR and Expected Shortfall constraints by the application of genetic algorithm, and the results are compared with log-optimal portfolio model.The main conclusions of this thesis are as follows:l.We propose a log-optimal portfolio model under the restriction of ES(Expected Shortfall) risk based on the traditional variance and log-optimal portfolio model under VaR risk.The study of ES risk is at an early stage, and applying ES to portfolio model is still rare.Therefore, our research is in the front of this issues, we also discuss the existence and uniqueness of the solutions of the model in this paper.2.By choosing CSI300 industry index as a research data, we use Granger causality test to investigate the data in part, and give a further analysis on the economic significance of the log-optimal portfolio model under variance, VaR and ES constraints using the numerical simulation.We find that, there exist differences in the ratio of investment under the conditions of not allowing short selling and without risk-free profits:variance model requests a high risk level of single-stock, VaR model is very demanding on the stock returns, while the ES model is a optimal portfolio which has considered both risk and profits.3.We construct a log-optimal portfolio model with multi-period ES risk constraints, and discuss the existence and uniqueness of solutions of the model, we also give the empirical analysis and numerical simulation and point out that, multi-period investment strategy is superior to single-period one at the same data range and under the same constraint functions.4. We use the sample covariance matrix, the numerical matrix, the two parameter model matrix, the single-index model matrix, the constant correlation matrix as the covariance matrix related to the stock to analysis the log-optimal portfolio model with the variance, VaR and ES risk function constraints under different covariance matrix, we also discuss the log-optimal portfolio model of different covariance matrix under mean variance criteria. Moreover, we analyse and compare the empirical findings, find that, for the log-optimal portfolio model with variance constraints, the numerical matrix and two parameter model matrix can not be effectively estimated matrix of the covariance matrix, while other matrices can response investment strategy differently; and for the log-optimal portfolio model with VaR and ES risk constraints, all the matrices except numerical matrix can response investment strategies differently, which has some reference value.5.We establish single-period and multi-period semi-log-optimal portfolio models based on VaR and ES constraints, and discuss the existence and uniqueness of the optimal solution for these models, then give the numerical simulation.From the empirical analysis we find that, the semi-log-optimal portfolio model with risk constraints has similar characteristics as log-optimal one, and the calculation results using them respectively are approximate, however, the semi-log-optimal portfolio model has the advantages that it is easy and convenient to compute.