基于RMT去噪法股票投资组合风险优化研究

发布时间：2018-04-02 10:44

本文选题：Markowitz股票投资组合　切入点：组合风险优化　出处：《哈尔滨工业大学》2013年博士论文

【摘要】：Markowitz证券投资组合理论是当代金融理论的重要支柱之一，它也是对实际证券投资具有最大指导价值的投资理论。作为Markowitz模型的关键输入参数，证券收益率历史相关矩阵和历史协方差矩阵的噪声会通过组合风险的增大和组合风险预测准确率的下降而导致组合风险的恶化。由于噪声对证券组合风险的影响随证券数的增加而增加，因此，在目前组合中证券数呈增大趋势的背景下，Markowitz模型在实践中应用的可行性正逐渐下降甚至接近于完全失效，噪声的影响已成为当今证券投资风险空前增大的重要原因之一。如何减小噪声对组合风险的影响是当前迫切需要关注的研究问题。越来越多的学者将对证券收益历史相关矩阵和历史协方差矩阵的去噪作为解决方法，并进行详细研究。和针对金融相关矩阵和协方差矩阵的其他去噪方法相比，基于随机矩阵理论的去噪法（RMT去噪法）具有决定模型最佳维度、操作技术难度低和适用范围广泛等优点。然而，基于RMT相关或多元波动率的去噪及其在股票投资组合风险优化中的应用的研究目前尚处于较为初级的阶段，有很多研究工作有待于进一步开展。以随机矩阵理论和现代投资组合理论为理论基础，采用蒙特卡洛模拟法、toy模型法、模拟退火算法、主成分分析法、一般数理分析法、理论研究和实证研究相结合等多种研究方法深入研究应用RMT去噪法实现股票投资组合风险优化的问题不仅有助于完善RMT去噪理论，而且对扫除Markowitz模型的应用障碍，从而促进Markowitz证券投资组合理论的进一步发展有重要的理论意义。在证券投资已经成为全球社会经济生活的一个重要构成部分的今天，该研究明显也具有很强的现实意义。本文从阐释Markowitz股票投资组合风险的含义入手，说明了以投资组合风险的大小和投资组合风险预测的准确率衡量投资组合风险的优劣，分析了股票收益历史相关矩阵和协方差矩阵的噪声对Markowitz投资组合风险优化水平的影响，，并选取RMT去噪法作为解决噪声影响的方法，进而构建了应用RMT去噪法实现Markowitz股票投资组合风险优化的整体思路框架，其包括四个角度：第一，通过对面向股票收益相关矩阵的RMT去噪法原理和算法的改进实现股票投资组合风险优化；第二，通过建立对已有去噪研究未曾涉及的股票收益率多元波动率模型去噪的一般方法实现股票投资组合风险优化；第三，通过弥补面向收益样本协方差矩阵的现有RMT去噪法在小组合条件下的缺陷实现股票投资组合风险优化；第四，根据对总体协方差矩阵特征值进行估计的新去噪思路建立面向收益样本协方差矩阵的RMT去噪法来实现股票投资组合风险的优化。从这四个角度出发，论文的主要研究内容和成果包括如下方面。首先，在介绍面向股票收益相关矩阵的现有RMT去噪法的基础上，在理论层面分析指出了KR方法是去噪原理最为合理和最有利于组合风险优化的RMT去噪法。对股票收益相关矩阵的特征向量最小扰动稳定性进行了数理推导，进而提出了KRMIN去噪法。KRMIN法吸纳了KR法的以相关矩阵特征向量稳定性的提高为核心的思想，弥补了KR法的原理和算法存在的缺陷，是一种更有利于改进股票投资组合风险的RMT去噪法。实证结果表明基于金融相关矩阵特征向量的Krzanowski稳定性的KR法和KRMIN法的组合风险优化效果好于其他RMT去噪法，且投资组合风险存在一种随收益相关矩阵特征向量的最小扰动稳定性提高而减小的趋势。其次，基于RMT提出了对多元波动率模型去噪的一般方法，并通过定性分析和定量推导说明该方法对股票投资组合风险优化可能带来的好处。为了进一步验证该方法改进组合风险优化水平的有效性，建立了将基于RMT的相关矩阵估计和波动率结合在一起的两种多元波动率模型即SC-GARCH模型和IO-GARCH模型，并以这两种模型为去噪对象进行了股票投资组合风险优化的实证研究。结果表明RMT去噪法能对多元波动率模型的最佳维度进行正确的确定，从而实现股票组合风险的最优化。再次，为了解决小组合风险优化条件下面向协方差矩阵的现有RMT去噪法因噪声特征值边界界定误差而产生的效力下降的问题，采用蒙特卡洛模拟法确定噪声特征值的边界，从而设计了蒙特卡洛RMT去噪法。通过实证分析方法，当股票收益序列长度和衰减因子不变时，在不同股票数量下对LCPB法、PG+法和KR法等已有RMT去噪法和蒙特卡洛RMT法的组合风险优化效果进行了对比研究。实证结果表明，在小组合条件下，蒙特卡洛RMT法能够弥补已有RMT方法的噪声特征值边界界定误差增大的缺陷，从而对已有RMT方法的组合风险优化作用的下降起到改进作用。最后，以股票收益总体协方差矩阵和样本协方差矩阵特征谱矩的关系为理论基础，采用模拟退火算法估计总体协方差矩阵的特征值，进而提出了对样本协方差矩阵去噪的矩法。不同于已有的RMT去噪法，矩法并没有采取对噪声特征值替换的做法，而是通过估计总体协方差矩阵的特征值来实现协方差矩阵的降噪，从而引入了一种RMT去噪的新思路。通过构建toy模型，对矩法去噪效果进行了模拟研究。结果表明矩法对股票收益总体协方差矩阵特征值的估计误差一般都能被控制在小于10%的范围内，且矩法的去噪效果受到样本序列长度和样本协方差矩阵数量的影响。采用模拟方法，通过设定符合现实经济情况的总体协方差矩阵模型，在理论层面分析了矩法对股票投资组合风险优化水平的作用。结果表明矩法对股票投资组合风险的优化作用随噪声对组合风险影响的增大而提高，并且受到对总体协方差矩阵的部门数或特征值数的猜测的制约而存在饱和现象。使用bootstrap方法，在理想化和现实化条件下对矩法的组合风险优化效果进行了实证分析。结果表明，矩法的组合风险优化效果好于常用的RMT去噪法。
[Abstract]:The portfolio theory is one of the important pillars of modern financial theory . It is also the key input parameter to the real security investment . As the key input parameter of the markwitz model , the noise of the historical correlation matrix and the historical covariance matrix of the stock return leads to the deterioration of the portfolio risk . The effect of noise on the portfolio risk is gradually falling or even close to the complete failure .

Compared with other denoising methods for financial correlation matrix and covariance matrix , the method of de - noising based on random matrix theory has the advantages of determining the best dimension of model , low difficulty of operation and wide application range . However , based on the theory of stochastic matrix theory and modern portfolio theory , it is important to perfect RMT denoising theory .

In this paper , the author starts with the definition of the risk of the portfolio risk , and illustrates the influence of the historical correlation matrix of stock return and the noise of covariance matrix on the risk optimization level .
Secondly , the stock portfolio risk optimization is realized by establishing the general method of denoising the stock yield multiple volatility model which has not been involved in the existing de - noising research ;
Thirdly , the stock portfolio risk optimization is realized by making up the defect of the existing RMT de - noising method facing the covariance matrix of the return sample under the condition of small combination ;
Fourthly , according to the new denoising method for estimating the characteristic value of the total covariance matrix , the RMT denoising method for the covariance matrix of the return sample is established to realize the optimization of the stock portfolio risk . From the four angles , the main research contents and achievements of the thesis include the following aspects .

First , on the basis of introducing the existing RMT de - noising method for stock return correlation matrix , this paper points out that KR method is the most reasonable and most beneficial to combine risk optimization RMT denoising method .

Secondly , based on RMT , the general method for de - noising of multivariate volatility model is proposed , and the advantages of this method to the optimization of stock portfolio risk are presented by qualitative analysis and quantitative deduction . In order to further validate the effectiveness of the method to improve the optimal level of portfolio risk optimization , two kinds of multivariate volatility models , namely SC - ARCH model and IO - ARCH model , which are combined together based on the correlation matrix estimation and fluctuation rate of RMT are established . The results show that the RMT denoising method can accurately determine the optimal dimension of the multivariate volatility model , and thus the optimization of the portfolio risk is realized .

Thirdly , in order to solve the problem that the existing RMT denoising method of the covariance matrix is reduced due to the boundary of the noise characteristic value , the Monte Carlo simulation method is adopted to determine the boundary of the noise characteristic value , so as to design the Monte Carlo RMT denoising method . By the empirical analysis method , the existing RMT denoising method , such as the LCPB method , the PG + method and the KR method , has been compared with the existing RMT denoising method and the Monte Carlo RMT method under different stock numbers .

In the end , based on the relationship between the total covariance matrix of stock return and the characteristic spectral moment of sample covariance matrix , this paper presents a new idea of denoising the stock portfolio by using simulated annealing algorithm .

【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：F830.91;F224

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