Generalized Measures Of Correlation For Nonlinearity And Asymmetry

The Pearson correlation coefficient measures the linearity and symmetry between variables,which is easy to understand and handle.Its properties have been studied by many scholars.It measures the degree of a variable in a straight line.The closer its ab?solute value is to 1,the stronger the correlation between the two variables.The closer the value is to 0,the weaker the correlation between the two variables.It has played a huge role in data analysis and has a wide range of applications in finance.But its biggest drawback is that it can only measure the symmetry and correlation between variables,and it can not do anything about the asymmetry and nonlinearity between variables.This is why it performs badly in some cases and sometimes even gets the wrong results.Howev-er,the asymmetry and nonlinearity between the risk assets' returns have been proved by historical data of a large number of markets.Therefore,the mean variance model based on Pearson correlation coefficient performs bad in modern portfolio selection.Modern investment theory places great emphasis on measuring the asymmetry and nonlinear rela-tionship between risky assets' returns.In this paper,we carefully studied the generalized measures of correlation proposed by Zheng,Shi and Zhang based on the variance decomposition formula.It is a very use-ful tool for measuring the nonlinearity and asymmetry between the risky assets' returns.Under special conditions,it can also degenerate into the square of the Pearson correlation coefficient.And a pair of generalized measures of correlation,in general,are not equal,since the relationship between the two variables is not equal,which is consist with the reality.That is the special advantage of generalized measures of correlation.In the first chapter,Markowitz proposed the mean variance model,which developed the modern portfolio theory and was widely used in practice and made great breakthrough-s.In almost all statistical inference problems,it is crucial to choose the right tool to mea-sure the relationship between variables.If the relationship between variables can not be measured correctly and even we got the wrong conclusion.The Pearson correlation co-efficient has a wide range of applications and measured the relationship between stocks'returns in the mean variance model,which is linearity and symmetry.Although this mod-el performs very well at the beginning,it is perform now generally due to the efficient market hypothesis.So a better tool is needed to measure the asymmetry and nonlinear re-lationship between stocks' returns,replacing the Pearson correlation coefficient and build a new model.In the second chapter,with more than 60 years of development,the theoretical re-search and practice of portfolio selection has achieved rich results.Modern finance is a theoretical finance developed in the 1950s.The main contents include Markowitz's portfo-lio selection theory,MM theory of corporate finance,capital asset pricing theory,efficient market hypothesis,option pricing theory and arbitrage pricing theory.We summarize the content of portfolio theory over the past 60 years.In the third chapter,in order to prevent investors from choosing stocks through ex-perience and information,balancing returns and risks is an urgent problem to be solved.Markowitz creatively defines risk as fluctuations of returns,assuming that investors are risk averse and propose the mean variance criterion.That is,a better investment portfolio should have higher returns and lower risk.Finally,the mean variance model is given to balance the risk and returns and find an optimal portfolio.That is the portfolio with the lowest risk is the optimal portfolio under the given returns.This model has been widely used in financial field to help investors find an optimal portfolio.A diversified portfolio can effectively reduce non-systemic risks.In statistical inference,a good model relies on accurately measuring the relationship between variables.In the mean variance model,the covariance matrix plays an important role in measuring the symmetry and linearity between the fluctuations in the risk assets' returns,but according to the of risky assets in developed and emerging financial markets,there are also nonlinearity and asymmetry be-tween different assets' returns,because of this,the mean variance model does not perform well in practice.A tool that measures the nonlinearity and asymmetry between variables is needed to compensate for the shortcomings of the mean variance model.The generalized measures of correlation can be a good alternative to the Pearson correlation coefficient.In addition,in order to avoid taking too many risks,in practice,investors have restrictions on short selling,and then we have also added some restrictions in the model.In the fourth chapter,we first introduce the Pearson correlation coefficient and it-s properties,and then introduce the generalized measures of correlation by the famous variance decomposition formula.In a pair of generalized measures of correlation,it is ob-vious that the two are not equal in most cases.This is due to the fact that the conditional variance is not equal.This is logic and flaw in the Pearson correlation coefficient.When the two variables are linearly related,the generalized measures of correlation degenerates into the Pearson correlation coefficient.If an independent variable has different effects on the two dependent variables,whether it is a linear or nonlinear relationship.The general-ized measures of correlation with strong influence is greater than the weak one.Based on some superiorities,we construct a generalized covariance matrix by generalized measures of correlation.The generalized covariance matrix is asymmetry,and it is used to replace the covariance matrix in the mean variance model.Thus we constructed a new model,which is generalized mean variance model.At the end of this chapter,other properties of generalized measures of correlation are discussed.In the fifth chapter,all the previous discussions are theoretical,because in the real financial market,we do not know the distribution of the risk assets' returns.Using histori-cal data to estimate real values has become an important step in the practical application of theory.Firstly,we use the data of the daily returns over the past period of time to estimate the mean and covariance matrix of future returns.This is relatively simple,but we can not use data with a large time span because the distribution of the risk assets' returns and the relationship between them will change.Then,we use the shrinkage estimation to reduce the error of the covariance matrix.Under a large amount of data,the direct calculation of the Pearson correlation coefficient is complicated.The relationship between the risk assets' returns and the change of the market index can be calculated by a single factor model,and then the covariance matrix of the returns of different risk assets can be quick-ly obtained through the model.The covariance matrix obtained by shrinkage estimation can reduce the error.The final difficulty is the calculation of the generalized covariance matrix.The non-parametric estimation of the generalized measure of correlation is found.Since the distribution function of the returns of the risky asset cannot be obtained,the distribution function can be estimated by the kernel estimation method.And the most important one in the kernel estimation is bandwidth,and cross-validation is used in this paper to achieve an optimal bandwidth.With a nonparametric estimate of the generalized correlation measure,we can calculate its estimate from stock history data.In the sixth chapter,given three assumptions,we study the asymptotics of the gen-eralized measures of correlation.Through a series of derivations,the joint asymptotics is established.According to the central limit theorem,its distribution tends to be normal-ly distributed.Therefore,large sample inference can be used to determine whether the relationship between the two variables is equal or not.In the seventh chapter,in order to verify the superiority of the model,we use 50 s-tocks in the Chinese stock market,a total of two years of historical data,applying in four different models to find the optimal portfolio strategy.These four models are the mean variance model,the mean variance model of the covariance matrix obtained by shrinkage estimation,the equal weight model,and the generalized mean variance model mentioned in the paper.Firstly,through descriptive analysis,it is found that the skewness and kur-tosis of returns indicate that the returns is not in accordance with the normal distribution.And the skewness value is generally not equal to 0,which implies that the returns distribu-tion is left or right.The kurtosis is generally greater than 3.According to the distribution,the characteristics of the peak fat tail are consistent with previous scholars' research.Nex-t,comparing the square root of generalized measures of correlation with the the Pearson correlation coefficient,the difference is usually large.This indicates a nonlinear and asym-metrical relationship between stocks' returns.And the generalized covariance matrix is asymmetrical,which means that the relationship between variables is not equal.Finally,in order to reduce the error caused by the sample,the two-year stock returns is divided into four subsets.Under different given maximum returns,four models are applied to obtain the optimal portfolio in each subset.Apply this portfolio to the sample and compare the returns after 2 days and 5 days.We find that when the stock rises,the generalized mean variance model behaves better than the other models,and the mean variance model of the covariance matrix obtained by the shrinkage estimate is second.When stocks fall,the generalized mean variance model also loses less.This shows that the generalized mean variance model is better than the other three models.In this paper,the analysis of the properties of the generalized correlation measure.It is a powerful tool for measuring asymmetry and nonlinearity.It can accurately measure the relationship between stock returns in financial markets.Through the analysis of Chinese stock historical data in the last chapter,the results of the generalized mean variance model are better than that of the other three models.Therefore,the excellent performance of the generalized mean variance model in practice not only allows investors to obtain greater returns,but also reduces the loss of the stock market decline.It is a model worthy of promotion.One of the assumptions of the mean variance model,investors have the same asset holding period.However this is unrealistic.As a result considering the multi-stage as-set holding period of the portfolio is a very meaningful issue,which can help investors constantly adjust their strategies to deal with different emergencies.In addition,the mean variance model only considers the volatility of the portfolio's returns,but many investors are risk averse.They are more likely to avoid losses,so we can add a measure of loss,such as adding a downside standard deviation to make the composite investment model more suitable for different investors.Finally,there are frictions in stock trading,such as fees,and considering these will make the model more accurate.These issues can all be studied in the future.