Study On Mean-variance Portfolio Model Based On Complex Network Theory

In recent years, complex network theory is a hotspot of research, and its applied research has a wide range of areas. Financial and securities market is an important research area of complex network. At present the theory of complex network applications in the securities market are concentrated in the study of Chinese securities market statistical properties, such as average path length, clustering coefficient, small-world effect of network, scale-free property and community structure. In this paper, we improve the mean-variance portfolio model combined with the complex network theory.Investment portfolio theory is an effective way to disperse risk and increase investment revenue, namely we can minimize the risk at a certain level of income or maximize investment revenue at a certain level of risk. The mean-variance model is the most traditional method to solve the portfolio problem. But in practice, the traditional mean-variance portfolio model faces with heavy and complex calculations when we determine the portfolio. When covariance matrix between stocks in this model is singular matrix, determining the weights of the portfolio will become more difficult, which results that the traditional mean-variance model faced significant limitations in practical application. Based on the block structure for the covariance matrix of asset returns and complex network, a new portfolio optimal model was put forward when the minimum of variance was taken as objective function and the expectant portfolio profit was taken as constraint. The new model can reduce computation and resolve the portfolio optimal problem when the covariance matrix was singular. The covariance matrix in this model is a covariance matrix in block structure, which is calculate based on complex network, and it must be non-singular matrices if it meets certain conditions. Since the covariance of all stocks in the same block is the same value, in practice the model significantly reduces the amount of calculation. Defining the stock as node and correlation coefficient of stock price as the side, complex network of stocks is built a specified threshold. Hierarchical clustering method is used to divide the community of the complex network. The stocks in the same community are assigned into the same block, so covariance between blocks in the same community is the same value. Clustering method using chromatographic methods communities divided into several equity-linked network communities, the same communities in the stock ownership of the same block in the same community in which the covariance between stocks is the same. Under the condition that the covariance matrix is nonsingular, we use the Lagrange multiplier method to solve the improved mean-variance model. The numerical results show that the given model is reasonable and the result is effective. And, the portfolios are compared under the different number of communities of network. The result shows that when the community number is four, we get the optimal portfolio.The improved model based on complex network theory can not only eliminate the limit of the singular matrix, but also greatly reduces the amount of computation. The research has provided a new perspective on securities market portfolio theory and its practical application.