Research On Portfolio Problem Based On The Robust Optimization Method

With the continuous acceleration of economic globalization and financial integration,the continuous enhancement of the correlation between different countries and capital markets has gradually intensified the spillover effect of capital risk contagion.At a time when the investment environment is increasingly diversified and complicated and affected by the fiscal and monetary policy environment of various countries,black swan events similar to the 2008 economic crisis and the 2015 stock market crash in Shanghai and Shenzhen are emerging one after another,and global investors are even more faced with the accelerated switching of the market cycle and the accompanying challenges of drifted investment style.Till now,whether it is a single market stock investment or crossindustry asset allocation,or even mixed cross-regional and cross-border investment,investors need to consider the volatility risk of the underlying asset in the process of wealth management,and can effectively obtain excess returns in the complex market competition.The urgent needs of practical investment have prompted investors to consider more deeply risk control,and have also spawned the continuous exploration and in-depth research on the theory of portfolio optimization.By adopting quantitative rather than subjective measures of return and risk,the traditional portfolios represented by the mean-variance model have received extensive attention from academia and industry.However,traditional portfolios do not take into account the uncertainty and volatility of asset returns in real financial markets,which will lead to problems such as sensitivity to input data,lack of diversification,and poor out-of-sample performance,which further hinders its practical application.Aiming at the shortcomings of these traditional portfolios,the robust portfolio makes the relevant parameters change within a reasonable and intuitive range by setting the uncertainty set,and solves the above problems in the worst-case scenario.As a more robust asset allocation method,robust portfolio optimization has become a core issue in quantitative finance research.Although the introduction of robustness improves the ability of the portfolio to risk reduction,it also leads to problems such as harsh model assumptions,difficulty inefficient integration with other models,and inevitable conservatism.Specifically,most robust portfolios set uncertainty sets based on the assumption of normal distribution,which is inconsistent with the skewed and fat-tailed distribution characteristics of asset returns,and it is not compatible with more flexible frameworks,such as fuzzy decision-making model and the introduced investors’ sentiment.Aiming at the characteristics and defects of the traditional portfolios and the robust portfolios,this thesis takes the improvement of the robust portfolio as the main line of our research,takes the factor models commonly used by funds and securities companies in the real financial market as the breakthrough point,and takes the step-by-step process of relaxing the relevant assumptions between the robust optimization method and the factor model.In the process of building a more reasonable and flexible robust asset allocation model,the Bayesian factor portfolio optimization problem,the time-varying dependence problem of the highdimensional asset return factor model,and the fusion problem of the robust portfolio and the fuzzy portfolio,and the problem of how to use non-convex optimization methods to measure investor sentiment factors and develop related robust applications are considered.For all the robust portfolios proposed,this thesis also considers how to overcome their conservatism based on a two-stage optimization method.This thesis includes seven chapters.Chapter 1 is the introduction which elaborates the research background,research significance,research method,research content,and the main academic innovation.Chapter 2 is the literature review,which sorts out the historical and latest literature on traditional portfolio optimization theory,robust portfolio optimization theory,asset dependency structure theory,and fuzzy optimization theory.Chapter 3 is the two-stage robust factor portfolio model.The traditional portfolio optimization model is robustly-factorized by setting linear and elliptic uncertainty sets,and then the concept of ridge regression and shrinkage estimation is combined and the robust factor portfolio model is introduced.Furthermore,the shrinkage strength is set as a quadratic optimization parameter to expand the number of worst-case scenarios considered by the robust optimization model,and the conservative problem of the robust factor model is solved by using the perturbation effect of the shrinkage strength parameter on the optimal weight by the two-stage optimization method.Chapter 4 is the twostage Bayesian robust factor portfolio model.Based on the research in Chapter 3,the robust factor portfolio is combined with the Bayesian model.The matrix-valued normal inverse-Wishart prior distribution is used as investor information and the market information is combined to obtain the matrix-valued conjugate posterior distribution.Then using the sampling method and iterative algorithm to estimate the parameters of the prior distribution and the related parameters of the Bayesian robust factor model are derived.The conservative problem of the Bayesian robust factor model is solved by introducing variable return constraints and variable investor confidence degree and setting the related variable range as the quadratic optimization parameters.Chapter 5 is the two-stage robust inverse optimization portfolio model based on the time-varying factor copula model.Based on Chapter 4,the assumption of factor distribution is relaxed to the time-varying factor model with fat-tailed and skewed distribution.By introducing the robust inverse optimization method based on the convex optimization method for the incompatibility of the Bayesian robust factor framework,the model can still reflect investor information and market information.Furthermore,the robust inverse optimization is solved by setting the investor view(information)matrix as the weight of the classic portfolio optimization model with risk diversification characteristics,and by setting the confidence level of the investor view matrix as a two-stage optimization parameter,thereby solving the Conservative problem of the model.Chapter 6 is the two-stage robust CVa R model based on the fuzzy sentiment factor.First,the assumption of asset return distribution is further relaxed to the fuzzy sentiment factor model.Then using the genetic algorithm based on the inverse optimization method to solve the relevant parameters which measure the overall sentiment of the equilibrium market and the heterogeneous sentiment of investors for a single asset.Further,according to the behavioral finance framework,the investor sentiment adjustment matrix and influence function are set,and the coefficient of the investor sentiment influence function is set as a two-stage optimization parameter to be introduced into the robust CVa R portfolio,to solve its conservative problem.Chapter 7 is the conclusion of this thesis,which summarizes the research results,the relevant investment suggestions,and the feasible research directions in the future.In chapters 3 to 6,in addition to establishing four two-stage robust portfolio models,this thesis also conducts a detailed empirical analysis to test the performance of these models.The main conclusions are as follows:(1)It is necessary to consider the influence of parameter uncertainty in asset allocation,but it is often impossible to obtain an effective risk-return trade-off only by directly using the classical robust portfolios.Based on the traditional portfolios,the robust improvement by setting the uncertainty set can effectively improve the risk control ability and out-of-sample performance.However,the classical robust portfolios take too much into account the risk reduction,so that investment opportunities cannot be effectively captured.In a specific data set and sample period,the simple robust modification will cause the out-of-sample performance of the model to decline.Therefore,this thesis does not recommend investors apply the classic robust portfolio model to allocate their assets.(2)The robust portfolio has different adaptation degrees and performances in the stock markets of developed and emerging countries.Investors should choose appropriate asset allocation schemes respectively.Stock markets in developed countries are less volatile,and robust portfolios can obtain stable returns in various economic cycles;stock markets in emerging countries are more volatile,and robust portfolios can effectively control risks in economic downturns,but the conservativeness prevents it from effectively capturing significant excess returns during an economic upswing or boom cycle.Therefore,the robust portfolio has a limited ability to control risk,and it must be further improved.(3)The distribution of asset returns has a decisive impact on the results of robust portfolio optimization.The relevant modeling process needs to balance the three aspects of conforming to financial reality,preventing overfitting,and fast calculation speed.Specifically,for the asset return distribution,if the assumption is too detailed(such as assigning a specific distribution with more parameters),the generalization ability of the model will be weak,and the optimal out-of-sample performance cannot be obtained;extracting key market features or giving the model more flexibility(such as introducing investor sentiment indicators)can help the model effectively improve its out-of-sample returns.(4)Two-stage optimization methods are of great value for improving the outof-sample performance of robust portfolios.By expanding the number of worst-case scenarios that the robust optimization method can consider,the two-stage robust portfolio model constructed based on considering the quadratic trade-off between investment risk and return can effectively solve its conservatism problem.The empirical research based on stock return data in different countries shows that the two-stage optimization method can effectively improve the out-of-sample returns of robust portfolios while controlling risks.For investors who plan to use a robust portfolio to allocate their assets,this thesis proposes a dynamic modification by selecting appropriate parameters for the specific characteristics of the model used to adaptively improve model performance.(5)The various two-stage robust portfolio models proposed in this thesis can effectively solve many real asset allocation problems.The empirical research results show that the models proposed in this thesis have significant advantages over traditional portfolios and robust portfolios in performance indicators such as out-ofsample mean,standard deviation,Sharpe ratio,and maximum drawdown.During the 15-year sample period of daily returns in China and the United States stock market,the models proposed in this thesis effectively withstand the test at different stages such as the economic crisis and the economic recovery when considering the high-dimensional asset allocation problem,and obtain a good risk-risk trade-off.Based on the modeling process in this thesis,the main theoretical contributions are as follows:(1)For the traditional portfolio,especially the Kelly model with almost no robust modification,this thesis for the first time improves it based on the robust factor optimization method and the Bayesian robust factor optimization method,thus solving its problem of sensitivity to input data and significantly improves its out-of-sample performance in short-term applications;for the robust CVa R portfolio,this thesis for the first time modifies it based on investor sentiment and significantly improves its flexibility.(2)For the robust factor portfolio,this thesis combines the shrinkage estimation and the ridge regression,which not only takes into account the variance and bias but also solves the problem that the robust optimization method cannot be efficiently integrated with other models.Different from the previous shrinkage estimation methods,the two-stage optimization method used in this thesis not only makes the shrinkage strength an objective optimization indicator but also avoids the investors’ interference to solve its conservative problem.(3)For the Bayesian robust factor portfolio,this thesis proposes for the first time to combine the sampling method with the iterative algorithm to estimate the prior distribution parameters,making the prior information more objective and effective.This thesis introduces variable income constraint and variable investor confidence level and sets the relevant variation range as a quadratic optimization parameter,and then solves the conservative problem of the Bayesian robust factor method.By introducing an extended model of the Bayesian robust optimization method: the robust inverse optimization model,this thesis solves the problem that the time-varying factor copula model is not compatible with the robust portfolio model.Through the investor view matrix setting and principal component analysis method,our model reflects both investor views and market dynamics.By setting the investor view matrix as a classical portfolio weight vector with a high degree of risk diversification,this thesis addresses the compatibility problem of the robust inverse optimization model with other portfolios.By setting the investor confidence level as a quadratic optimization parameter,this thesis addresses the conservative problem of robust inverse optimization models.(4)Aiming at the problem that the consistent trapezoidal fuzzy number model cannot describe the investor’s attitude towards the market as a whole and a single asset,this thesis proposes the fuzzy sentiment factor model.Aiming at the problem that the traditional fuzzy number model cannot accurately measure the investor’s sentiment,this thesis proposes a genetic algorithm based on the inverse optimization method and constructs a general algorithm framework that can be applied to various triangular and trapezoidal fuzzy number models.Aiming at the problem that the fuzzy optimization method is difficult to combine with the robust optimization method,this thesis proposes the concept of dynamic investor sentiment influence function and solves the conservative problem of the robust CVa R portfolio.In addition,this thesis introduces an accurate equilibrium fuzzy sentiment factor in the robust optimization method,which not only gives investors using the relevant robust portfolio model higher flexibility but also gives a new idea to solve the compatibility problem between robust optimization model and other optimization methods.There are still some limitations in the research work of this thesis.For example,although the empirical data of stock markets based on multiple economic cycles in emerging market countries and developed market countries are considered,followup research should add more types of assets or industry data to further expand the models;because of time and personal ability constraints,this thesis does not consider the portfolio optimization problem involving continuous-time model,nor does build a robust multi-objective portfolio optimization model based on more realistic constraints.The research results of this thesis not only help to further deepen the understanding of the process of robust asset allocation,but also overcome the conservative problem of the robust portfolio by improving the factor model commonly used in the industry and using the two-stage optimization method,and provide a useful reference for practical asset management.