基于高频夏普比率和稳健相关系数的资产选择.pdf

第 40 卷第 6 期2023 年 11 月中 国 科 学 院 大 学 学 报Journal of University of Chinese Academy of SciencesVol.40NovemberNo.62023 Supported by National Natural Science Foundation of China(12171454)Corresponding author,E-mail:sgzhang 文章编号:2095-6134(2023)06-0834-09Brief ReportAsset selection based on high frequency Sharpe ratio and robust correlation coefficientZHANG Shanhua,ZHANG Sanguo(School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China)(Received 14 January 2022;Revised 14 April 2022)Zhang S H,Zhang S G.Asset selection based on high frequency Sharpe ratio and robust correlation coefficientJ.Journal of University of Chinese Academy of Sciences,2023,40(6):834-842.DOI:10.7523/j.ucas.2022.039.Abstract High frequency Sharpe ratio,a measure of return and risk,is commonly used in current portfolio construction method since it can avoid covariance matrix in high dimensional analysis.The newly proposed D-SEV measures the correlation between stocks return and high frequency Sharpe ratio index to further construct portfolio.However,there are some problems with the measure used in D-SEV,such as its lack of robustness and slow computational speed.In this paper,we propose to use a new correlation coefficient proposed by Sourav Chatterjee instead.The new correlation coefficient guarantee robustness,specifically it can reduce the impact of abnormal data on correlation,such as significant events that have a large impact on the asset prices.It is also extremely fast in its calculations.Extensive simulation demonstrate that new correlation coefficient outperforms D-SEV and other traditional methods in several different models.Actual Shanghai Securities Exchange(SSE)and Shenzhen Securities Exchange(SZSE)stock market data for 2019 and 2020 also show that the assets selected by new correlation coefficient earns 8%more excess annualized return than D-SEV,while it also owns a higher Sharpe ratio.Keywords portfolio;high frequency Sharpe ratio;robust correlation coefficientCLC number:O212.7,F830.91 Document code:A DOI:10.7523/j.ucas.2022.039基于高频夏普比率和稳健相关系数的资产选择 张杉桦,张三国(中国科学院大学数学科学学院,北京 100049)摘 要 高频夏普比率是衡量收益和风险的指标,可以避免估计高维分析中的协方差矩阵,所以在目前的投资组合构建方法中被普遍使用。近年提出的 D-SEV 方法,通过测量股票收益和高频夏普比率指数之间的相关性,来进一步构建投资组合。然而,D-SEV 中用来度量股票与高第 6 期ZHANG Shanhua,ZHANG Sanguo:Asset selection based on high frequency Sharpe ratio and robust correlation coefficient频夏普比率的相关性的方法存在一些问题,如缺乏稳健性和计算速度慢。在本文中,使用由Sourav Chatterjee 提出的新的相关系数来代替。新的相关系数保证了稳健性,特别是它可以降低异常值对相关性的影响,比如对资产价格有很大影响的重大事件。同时它的计算速度也非常快。大量的模拟表明,新的相关系数在几个不同的模型中的表现优于 D-SEV 和其他传统方法。2019 年和 2020 年的上证和深证股市数据也显示,新的相关系数选择的资产组合比 D-SEV 选择的资产组合的年化收益高出 8%,同时也拥有较高的夏普比率。关键词 资产组合;高频夏普比率;稳健相关系数 The portfolio is a classical problem of finance,which was fist addressed and solved by Markowitz1 using mean-variance paradigm.This method is still used today as a standard method.The basis of the mean-variance model is to estimate the mean and covariance of the returns of underlying assets.Once the estimators are inaccurate,the results of the model will be very imprecise.Kan and Zhou2 showed that the estimators of mean-variance paradigm have some shortcomings in the situation of large-scale mean-variance models.Michaud3 proved that the error of expected returns by this model is over estimated.To solve these problems,extensive literature focus on improving the accuracy of estimator.For instance,the shrinkage estimation method is proposed by James and Stein4.Sharpe5 and Chan et al.6 impose a factor structure for the covariance among assets to reduce the number of free parameters of the covariance matrix.Other methods involve reduction of the dimension.Ao et al.7 took advantage of LASSO to achieve dimension reduction.Chen and Yuan8 provided a framework of portfolio selection under subspace and adopted factor model to achieve the global Sharpe Ratio.Although the aforementioned methods improved the original method from different aspects,there are still two problems to be solved.First,these methods employ long historical data,it is not compatible with the current rapidly changing market.The second problem is caused by the dramatic increase in the number of modern assets,the amount of assets facing by modern investors is much larger than traditional market.The essence of this problem is feature screening,it is first proposed by Fan and Lyu9,they investigated the data for which the number of features p is much more than the number of samples n and propose a screening method called sure independence screening(SIS)to quickly reduce the ultrahigh dimension to suitable dimension.Following Fan and Lyu9,the screening methods have been explored in two directions,Fan and Song10 studied the specific model,Zhu et al.11 and Liu et al.12 did the research of model-free.However,the SIS methods have a premise:independent and identically distributed samples,which is invalid in the financial filed.To tackle these problems,Wang et al.13 proposed an asset selection method based on high frequency Sharpe ratio and D-SEV,which do not need the assumption of identically distributed samples.They used D-SEV to calculate the correlation between stocks and high frequency Sharpe ratio,the variables here can be seen as a time series.Their method is reasonable because the D-SEV tells the proportion of the variance explained by the selected assets.Yet,it also has two disadvantages,the lack of robustness and the high computation cost.To deal with these issues,we propose to introduce the robust correlation coefficient instead of D-SEV to evaluate the correlation between assets and high frequency Sharpe ratio index.Simulations and empirical analysis show that our method outperforms D-SEV and other traditional method in several different situations.1 High frequency Sharpe ratio index1.1 Sharpe ratio Sharpe ratio,proposed by Sharpe5,is a well-accepted measure of assets in financial field,defined asSRp=E(Rp)-Rfp,(1)538中国科学院大学学报第 40 卷where Rf is the annualized risk-free interest rate and is commonly set as benchmark,p is the standard deviation of annualized return of portfolio,and E(Rp)is expected annualized rate of return of portfolio.The Sharpe ratio is a measure of excess return earned over the risk-free rate return for every unit of risk taken.High risk return is the ultimate goal for portfolio construction.It can be seen that high Sharpe ratio leads to high excess return earned for a given amount of risk.1.2 High frequency Sharpe ratio With the development of high frequency data,high frequency stock volatility can be calculated as a proxy for the standard deviation.Integrated volatility can be defined asRVi,t=Mj=1(rit,j)2,(2)where M is the number of the time period divided in a day,rit,j=lnPit,j-lnPit,j-1,j=1,2,M,is the intra-day high frequency return of stock i during the time period from j-1 to j of the day t,Pit,j represents the j transaction price data for the day t.Consider stock i=1,2,p,for the day t=1,2,T,high frequency Sharpe ratio of this stock is defined by the daily data of the stock prices and market index yield:SRi,t=Ri,t-Rm,tRVi,t,(3)Here Ri,t is the yield of the stock i during day t,Rm,t is the market index yield during day t as the benchmark,here we choose market index yield instead of risk-free rate return.Because our portfolio contains stocks only,our benchmark can only contain stocks as well,RVi,t is the integrated volatility as described above.1.3Choose assets to construct a return time series of an indexSimilar as original Sharpe ratio,stocks with higher high frequency Sharpe ratio can also be considered as an indication of better performance.Thus,High frequency Sharpe ratio can be used to serve as a bases of portfolio construction.Similar to D-SEV proposed by Wang et al.13,we first calculate high frequency Sharpe ratio for each stock and rank them in descending order for each day,SRd1,t SRd2,t SRdn,t,n stands for the number of the stocks in the market,di(i=1,2,n)stands for the sorted location of the stocks high frequency Sharpe ratio from high to low.With the pre-specified value d,we choose the top d stocks based on ranked high frequency Sharpe ratio and construct a portfolio as:RSR,t=di=1mi,tRdi,t,(4)mi,t=(MVi,t)di=1MVi,t()is the stock i s proportion of the market value in day t,the market value in day t of the stock i is MVi,t.Rdi.t stands for the return of asset di corresponding to SRdi,t in day t.We define the return of this portfolio as high frequency Sharpe ration index.This index,a combination of best performance stocks,has a high risk-return than market benchmark.Although high frequency Sharpe ratio index has an excellent return,it can not be held directly because the index of the day will only be available after the market closed.Thus we can not directly use high frequency Sharpe on the same day to construct portfolio.The only data available is the high frequency Sharpe index over the past time.2 Portfolio established by correlation coefficient As we mentioned in previous section,we can only calculate high frequency Sharpe ratio index over past period.Thus we can not directly apply this index to filter assets and construct portfolio.To overcome this,D-SEV calculates the correlation between each stock and high frequency Sharpe ratio over a pre-specified period and select the stock with higher correlation to construct portfolio.2.1 Dependent sure explained variability(D-SEV)Wang et al.13 proposed the D-SEV as the correlation coefficient to choose assets.The dependent sure explained variability of(X,Y)is 638第 6 期ZHANG Shanhua,ZHANG Sanguo:Asset selection based on high frequency Sharpe ratio and robust correlation coefficientdefined asD-SEV(X,Y)=1-EY-E(Y X)2var(Y).(5)The use of D-SEV to measure the correlation between stock and index may not be efficient for two reasons.First,D-SEV is not a robust correlation coefficient.Our simulation results show that D-SEV cannot handle abnormal data which is commonly encountered in real data from stock market.For example,there are significant events that cause dramatic fluctuations in stock price,and these fluctuations are usually short-term and will not affect the future trend of stock price,so these stocks removed due to dramatic fluctuations should not be excluded from the portfolio.The second reason is the computation time,since D-SEV relies on the kernel estimation of the density function,the estimation process will take a long time with different kinds of kernel functions.This may cause some difficulties since the number of stocks need to be selected is large.To solve these problems,we propose to use a new correlation coefficient proposed by Sourav Chatterjee instead to measure the relationship between the return of the stock and the high frequency Sharpe ratio index to filter assets.2.2 A robust correlation coefficientChatterjee14 propose a new Coefficient of Correlation,which is defined asn(X,Y)=1-nn-1i=1ri+1-ri2ni=1li(n-li).(6)Let(Xi,Yi)be a time series.Rearrange the series as(X(1),Y(1),(X(n),Y(n)such that X(1)X(2)X(n).Let ri be the rank of Y(i),defined as the number of j such that Y(j)Y(i).li is defined as the number of k such that Y(k)Y(i).The robust correlation coefficient is more suitable for the problem of choosing assets correlated to high frequency Sharpe ratio index for two reasons:First,It is a function of ranks,which guarantees robustness.We have discussed the existence of dramatic fluctuations from significant events which can be considered as abnormal data.Using rank to build the robust correlation coefficient can partially solve problem and we will illustrate this in following simulations.Second,the computational time of the new correlation coefficient is much less than D-SEV,since it does not involve complicate density function estimation.2.3 Establish a portfolio of assets held In this part,portfolio selected by robust correlation coefficient between the return of the stock and the high frequency Sharpe ratio index will be established.To begin with,the robust correlation coefficient between the return of every stock in the market and the high frequency Sharpe ratio index will be calculated.Then the stocks will be ranked by the correlation coefficient,the stocks corresponding to the largest r value of n(Xk,Y),k=1,2,p will be chosen to establish a portfolio and be held for a period of time.Markowitzs portfolio theory shows that investors should diversify their holdings of assets to reduce risks,while pursed high returns,so the number of assets holding is very critical.Theoretically,if the number of assets is small,the dispersion degree will be low,resulting in greater volatility,if the number of assets is large,although the dispersion effect is more obvious and the volatility is lower,its income will be greatly affected.In our article,the percentage of the total assets is used instead of threshold value based on the correlation coefficient.The reason is that according to the market experience,we generally believe that top 1%of the assets are high-quality assets.Here we refer to the parameter setting by Wang et al.13 which also use top 1%of the assets to construct portfolio.The stock of the portfolio constructed above can be weighted in two ways,equal weights and weights under the minimum variance portfolio.Both of them can achieve good yields.2.4 AlgorithmFinally,we will introduce the complete algorithm for establishing asset portfolio.1)Calculate the integrated volatility of every stock for every day in Eq.(2).2)Calculate the high frequency Sharpe ratio of 738中国科学院大学学报第 40 卷every stock for every day in Eq.(3).3)Rank high frequency Sharpe ratios in descending order and choose the top d stocks based on ranked high frequency Sharpe ratio for every day.4)Use the d stocks to construct a portfolio weighted by market value in Eq.(4),the portfolio is high frequency Sharpe ratio index for every day.5)Calculate the robust correlation coefficient between the return of every asset in the market and high frequency Sharpe ratio index in Eq.(6)through z periods of data,one period is w days.6)Rank robust correlation coefficients and select the largest r assets based on ranked robust correlation coefficient.7)Establish a portfolio by the r assets of equal weights,the portfolio is the ultimate portfolio we will hold for a period exactly after the z periods in step 5).8)Circulate the step 5)to step 7),we can obtain the portfolio held in every period for a long time.3 Simulation of the correlation coefficient Before applying the robust correlation coefficient method to construct the portfolio,we will examine its performance by simulated data.First generate a p dimensional random variables X=(X1,X2,Xp),X N(0,),the covariance matrix =(ij)pp,where ij=i-j.Then we construct two linear models as follows:Y1=c1X1+c2X2+c3X12+c5exp(X22),(7)Y2=c1X1+c2X2+c3I(X12 0)+c4X22+.(8)We generate n samples to perform simulation experiments.In Eq.(7),we add a exponential terms to compare the robustness.The indicator function is used to simulate the abnormal data in Eq.(8).To avoid the active features