Research On Stock Portfolio Risk Optimization Based On The RMT Denoising Methods

Markowitz Portfolio Theory is one of important pillars of modern financetheory， which is the investment theory giving maximum guidance for actualinvestments in securities. As the key input parameters of the Markowitz model,correlation and covariance matrices of historical security return rates contain a lotof noise, which will lead to the deterioration of portfolio risks by the increase ofportfolio risks and the decline of portfolio risk forecast accuracy. Because theimpact of noise on the securities portfolio risk increases with the increase in thenumber of securities, in the context of the present increasing trend of the number ofsecurities in the portfolio,the feasibility of the practical application of theMarkowitz model is gradually declining and even close to complete failure.Nowadays the effect of noise has become one of important reasons for anunprecedented increase of securities investment risks. How to reduce the impact ofnoise on portfolio risks is a research problem which urgently needs to be focusedon. More and more scholars regard denoising correlation matrices and covariancematrices of securities returns as a solution and carry out a detailed study.Compared with other denoising methods of correlation matrices and covariancematrices, random matrix theory (RMT) filters have the advantages of deciding theoptimal model dimension, the low operation difficulty and a wide application range.However, the research on denoising correlation matrices and covariance matricesbased on RMT and applying RMT denoising in portfolio risk optimization is still ata relatively early stage, and there is a lot of work to be further carried out. Thispaper takes random matrix theory and modern portfolio theory as theoretical basis,uses many research techniques,such as Monte Carlo simulation method, toy model,simulated annealing algorithm, principal component analysis, general mathematicalannalysis, qualitative and quantitative analysis,etc., and deeply studies the keyquestions of adopting RMT denoising to achieve equity portfolio risk optimization,which not only helps to improve the denoising theory based on RMT, but also hasimportant theoretical significance for sweeping away the obstacles of applying theMarkowitz model to promote the further development of Markowitz portfoliotheory. Nowadays, as investment in the securities has become an important part ofglobal social and economic life, the study obviously has great practical significancetoo.Start from interpreting the meaning of Markowitz portfolio risks, the paperexplains that the quality of portfolio risks should be measured by the size and forecast accuracy of portfolio risk, analyzes the impact of noise in historicalsecurity return correlation matrices and covariance matrices on the optimizationlevel of Markowitz stock portfolio risks, and selects RMT denoising as a solutionfor the noise effect. Then the paper constructs a theoretical framework ofoptimization of Markowitz stock portfolio risks based on RMT denoising whichconsists of four angles: First, equity portfolio risk optimization by improvingprinciples and algorithms of RMT denoising methods toward stock returncorrelation matrices. Second, equity portfolio risk optimization by establishing ageneral denoising method for multivariate volatility model which has not beeninvolved in recent research. Third, equity portfolio risk optimization by making upfor the disadvantage of the existing RMT denoising methods for sample covariancematrices of small portfolios. Fourth, equity portfolio risk optimization byestablishing a RMT denoising method toward sample covariance matrices based ona new denoising idea of estimating eigenvalues of population covariance matrices.From the above four angles, the paper includes the following research content andresults.Firstly, on the basis of describing the existing RMT denoising methods towardsecurity return correlation matrices, the paper theoretically points out that amongthem the KR method is most reasonable and most beneficial to portfolio riskoptimization. After mathematically deducting the minimum perturbation of acertain eigenvector in a correlation matrix when the corresponding eigenvaluechanges, the paper proposes KRMIN denoising method. The KRMIN methodabsorbs the idea of the KR method focusing on improvement of eigenvectorstability of a correlation matrix and compensates the defects of the KR method inthe principles and algorithms. Therefore The KRMIN method is more conducive toequity portfolio risk optimization in contrast with other methods. The empiricalresults show that: the portfolio risk optimization effect of KR and KRMINconsidering Krzanowski stability of eigenvectors of earnings correlation matrices issuperior to other RMT methods. Portfolio risks have a decreasing tend with theimprovement of stability measured by minimum perturbation of correlation matrixeigenvectors.Secondly, the paper presents a RMT denoising method for multivariate volatilitymodel and demonstrates its benefits to stock portfolio risk optimization throughqualitative analysis and quantitative derivation. In order to demonstrate theeffectiveness of the method for portfolio risk optimization, this paper constructstwo classes of multivariate volatility models combining the volatility process withthe correlation estimate, SC-GARCH model and IO-GARCH model. Taking the twokinds of models as denoising objects, an empirical study of stock portfolio risk optimization is carried out. It is shown in the experiment that the RMT denoisingmethod can precisely determine the optimal dimension of multivariate volatilitymodels and thus produce the best stock portfolio risk.Thirdly, in order to solve the decline of effectiveness of existing RMT denoisingmethods toward covariance matrices of small portfolios due to the error indetermining the noise eigenvalue boundary, the Monte Carlo method is used todetermine the noise eigenvalue boundary and then the RMT denoising methodbased on Monte Carlo simulation is designed. Through empirical approach, whenstock return sequence length and attenuation factor are unchanged, the papercompares portfolio risk optimization effect of existing RMT denoising methodssuch as LCPB, PG+and KR etc and the RMT denoising method based on MonteCarlo simulation in a varying number of shares. Experimental results show thatunder the conditions of small portfolios the RMT method based on Monte Carlosimulation can cover the existing RMT denoising methods’inferiority indetermining the noise eigenvalue boundary and thus hinder the decline of their rolein risk improvement.Finally, the paper takes the exact relation between the eigenvalue spectrummoments of a population covariance matrix and those of its estimator as theoreticalbasis, uses the simulated annealing algorithm to estimate eigenvalues of thepopulation covariance matrix and thereby presents the moment method used fordenoising sample covariance matrices.Unlike existing RMT denoising methods, themoment method does not replace noisy eigenvalues, but estimates eigenvalues ofthe population covariance matrix to denoise sample covariance matrices, therebyintroducing a new way of RMT denoising. Through a toy model, a simulation studyon denoising effect of the moment method is conducted. The results prove that byuse of the moment method, the error in the estimation of eigenvalues of stockreturns population covariance matrices can generally be controlled in the range ofless than10%and the denoising effect of the moment method is influenced by thesample sequence length and the number of sample covariance matrices. By settingthe population covariance matrix model in line with economic reality, the paperadopts the simulation method to theoretically analyze the effect of the momentmethod on the optimization level of stock portfolio risks. The results show that theoptimizing effect of the moment method on stock portfolio risks improves when theimpact of noise increases and the saturated phenomenon of the moment methodoccurs because of constraints of guess on the department number or the eigenvaluenumber of the population covariance matrix. Under idealized and realisticconditions, by using the bootstrap method the paper conducts empirical analysis ofoptimizing effect of the moment method on portfolio risks. It turns out that the optimizing effect of the moment method on portfolio risks is better than thefrequently used RMT denoising method.