Forecasting High Frequency Covariance Matrix And Its Application In Portfolio Based On Spectral Decomposition Model

Estimate and forecasting financial assets covariance matrix play an important role in finance-and many other fields such as asset allocation, risk management. Large number of empirical studies have shown that volatility of financial assets have strong memories.It is very important to estimate and forecast assets covariance matrix accurately when we make investment decisions.In recent years, covariance matrix estimation and prediction model of financial metrology has become an important research direction.Andersen and Bollerslev came up with realized covariance matrix in2003. The method of realized covariance matrix makes full use of high-frequency intraday data to. estimate covariance matrix directly.It is with model, no parameter estimation.It overcomes the dimension disaster of the traditional multivariate GARCH when the number of assets are very large. Meanwhile, High frequency data contains more market information than the low frequency data.Therefore, realized covariance matrix has becomes the most widely used covariance matrix estimation method in high frequency data.However, prediction of covariance matrix for high-dimensional assets still has been a difficult point in the field of high frequency research.Recently,portfolios of financial assets contain hundreds of assets, To predict such a high dimension assets covariance matrix prediction is very difficult. Firstly, when the number of assets increases, the number of parameters we need to estimate increases rapidly. Second, we should ensure that the covariance matrix is qualitative. Finally, a large number of empirical studies have shown that the covariance matrix of assets has strong memory. Therefore, the prediction model must capture it’s long memory.This paper studies how to forecast high frequency covariance matrix of the Chinese stock market.The stock market is a barometer of a country economy, the stock market plays an important role in the national economy, through the stock market, the enterprise can raise funds to develop. The Chinese stock market history is not as long as the developed countries.And China’s stock market is in a perfect and mature process. China’s stock market’s volatility is very large in recently years.In recent years, many scholars have researched China’s stock market.However, the research of high frequency data covariance matrix started late in China. Most of the study is on one-dimensional high frequency asset volatility.Compared with the studies abroad, research of predict high dimensional assets of covariance matrix is still blank in China.There is few effective prediction model of high frequency covariance matrix. The dimension disaster and the conditions of qualitative are the difficulties.However, multi-dimensional financial high-frequency covariance matrix of plays a very important role in the portfolio and risk management.For portfolio construction, most of domestic scholars still use traditional low frequency data to estimate covariance matrix. Because of dimension disaster, although the traditional multivariate GARCH has been put forward for a long time, it still can’t be widely used in real application. Realized covariance matrix based on high-frequency data can easily solve dimension disaster.At present, domestic scholars rarely research estimation and prediction of high-frequency covariance matrix. Compare with low frequency data,high frequency data has more market information. Therefore in the financial investment portfolio, risk management, and other fields, high frequency data has more advantages than the low frequency data.It’s a hot research to expand the range of application of the volatility base on high frequency data.Hautsch(2010) model the dynamics of large-dimensional covariance on the basis of a multi-scale spectral decomposition of a realized covariance. Covariance forecasts are constructed based on predicted variances and eigenvalues. Volatilities, correlation eigenvalues and eigenvectors are allowed to evolve on different frequencies. Spectral decomposition of a realized covariance not only can play a role for dimension reduction, but also can make sure the positive of the matrix, at the same time it can capture the long memory of volatility.This paper apply the idea of Hautsch--multi-scale spectral decomposition to the model of forecasting the realized covariance matrix of the stock market in China.The paper choose50stock from Shanghai180stocks, study the50stock covariance matrix estimation and prediction. We can believe that in stock market of China, dynamic spectrum decomposition model can accurately forecast high-dimensional covariance matrix. It can overcome the "dimension disaster", keep qualitative, also can depict the memory of covariance matrix. It performs good in the application of portfolio. In addition, the paper adopted two different estimation methods to estimate covariance matrix. one is the realized covariance matrix, the other one is the improved method of the high frequency noise and price jump called Threshold Pre-averaging Realized Covariance Matrix.Use high frequency data to estimate covariance matrix will be mainly affected by two aspects, one is the market structure noise, another one is the intraday price jumps Market structure noise increased with the number of high frequency data. At one hand, more high frequency data contains more market information, at the other hand,more high frequency data means more market structure noise. As a result, there are two ways to reduce market structure, one is by reducing the sample frequency data, another way is to smooth noise, smoothing noise is more effective without loss of data. In the real financial environment, some significant assets price usually change a lot, we refer it as the price jump. Prices jump and market structure noise can destroy continuous assumption,and realized covariance matrix is no longer a integral consistent estimator.Therefore, this paper chose a method which can not only smooth noise but also can eliminate prices jump. Theory is confirmed that when the market contains Market structure noise and Prices jump, Threshold Pre-averaging Realized Covariance Matrix is consistent estimator of covariance matrix.This paper applies two kinds of high-frequency covariance matrix to construct dynamic portfolio, and compares portfolio performance of the two kinds of different method.Compare the constructed portfolio’s earnings to earnings of the Shanghai180index during the same period. It concludes that portfolio based on high frequency covariance matrix performs better than earnings of the Shanghai180index during the same period. And, the improved method performs better than realized covariance matrix. Through theoretical analysis and empirical research, this paper get the following conclusion, one is that the dynamic spectral decomposition can solve the dimension disaster, keep qualitative, capture the market long memory. It is an effective model to predict large-dimensional covariance.the other one is that the paper constructs a dynamic portfolio with minimum variance.using dynamic spectrum decomposition model to predict higher dimensional covariance matrix,the portfolio’s earnings is higher than earnings of the Shanghai180index during the same period.Based on market structure and noise and jumping improved method perform better than realized covariance matrix.This paper innovation points are as follows:the dynamic spectral decomposition method divide two different parts to predict large-dimensional covariance. Covariance forecasts are constructed based on predicted variances and eigenvalues. Volatilities, correlation eigenvalues and eigenvectors are allowed to evolve on different frequencies. Spectral decomposition of a realized covariance not only can play a role for dimension reduction, but also can make sure the positive of the matrix, at the same time it can capture the long memory of volatility.In this paper, direction for further research is that we can extend the application field of high-dimensional dynamic spectrum decomposition forecasting model, such as it can be applied to the bond market, currency market, options market, etc and in this paper, T only use two different kinds of high-dimensional covariance matrix estimation method, in the further research, we can use more different covariance matrix estimation methods to compares the influence of the market structure noise and price jump.