Sparsity Estimation For Large Portfolios

Portfolio optimization is one of the research directions of modern finance.Starting from the mean-variance theory proposed by Markowitz,portfolio optimization pursues the balance of return and risk.The information used is the historical return rate information of each stock in the stock pool.The optimization goal is either to limit the portfolio risk level with maximize the return,or to minimize the portfolio risk under the specified investment portfolio return level,follow-up research scholars will also impose industry-neutral and market-value-neutral restrictions on the investment portfolio in order to compare the investment portfolio with the benchmark,where investment portfolio and the benchmark portfolio have the same industry exposure and market value exposure.However,there are two problems in the research process of portfolio optimization based on mean-variance theory.First,the estimation of the mean covariance is inaccurate and unstable.Some scholars have carried out research on the estimation of the covariance matrix to make the covariance estimator more stable;Second,the mean-variance model only uses the historical return information of the stock,which is the time-series longitudinal information of the stock itself,but in fact,the stock has information that can be obtained by comparing it with other stocks horizontally,which is also portfolio optimization theory can be enriched and deepened.Aiming at the above two problems,this paper uses the statistical compression method of Sparse Group Lasso,and combines the factor characteristic information and return information of stocks to construct an investment portfolio.The basic model of this paper is a regression model,and the explanatory variable is represented by the return of the investment portfolio.Specifically,it can be known from the multi-factor investment theory that factor information has a significant indicating effect on stock investment.The greater the effective positive factor value corresponding to a stock,the greater the possibility that the stock will have a positive return in the next period.Therefore,this paper uses the factor characteristic information of stocks as the basis for the weight construction of stocks in the portfolio,and the weight of each stock is used as a group structure,where the structure is a linear combination of effective factors.After obtaining the weight of each stock,multiply the weight of the stock with the return of the corresponding stock in the next period to obtain the return of the portfolio in the next period.The explained variable in the regression model is the combination of the Sharpe ratio of the portfolio stock pool and the risk objective,which is obtained by solving the equivalence problem with the mean variance.During the solution process,the model will screen out many stocks with small holding weights,and when there are too many stocks with small weights,it is not conducive to the management of the portfolio and is unstable,so there will be a sparsity assumption in the actual investment portfolio,that is,most stocks do not will appear in the portfolio.This article has the following three advantages: 1.Under the research framework of this method,there is no need to estimate the covariance matrix of the stock pool,which can avoid problems caused by inaccurate estimation;2.By introducing stocks Factor information,not just the historical return information of the stock pool,can make the establishment of investment portfolios have more effective information and increase the stability of portfolio optimization;3.Compress and estimate by using the Sparse Group Lasso method.The individual stocks with small portfolio contributions are compressed to zero,avoiding the phenomenon of excessive micro-weighting in the portfolio and making portfolio management more efficient.In the data simulation and empirical analysis,this paper compares our method with previous optimization method and equal weight investment method.It can be found that the method of this article has greatly improved the investment portfolio rate of return,but the investment portfolio has a mediocre performance in risk control.