High Dimensional Portfolio Selection Under Conditional Constraints

Since the mean-variance model for portfolio seclection pineered by Markowitz, the theory has been enriched, improved and developed in literatures. With the devlopment of financial theory and practice, the choice of large-scale financial assets has become a difficult problem for investors. We propose a new portfolio method through adding constraints to the standard portfolio selection model. On the one hand, it can solve the problem of computation and error accumulation effect in high dimensional portfolio selection and improve modeling efficiency. On the other hand, it can choose financial assets and reduces the cost of asset management. CVaR posesses some good mathematical natures, which overcomes the shortcomings of VaR. In this thesis, we solve the problem of high dimension portfolio decision with constraints, and focus on the CVaR risk high dimension portfolio with norm constraint or weight constraint. The mean-variance framework is extended to the mean-CVaR version.In order to overcome the difficulty of financial asset pool management derived from extreme portfolio investment position in the traditional portfolio, we add constraints to the standard CVaR portfolio model and get portfolio selection model under norm constraints. Our method consists of three parts. First, we prove that the process of solving the CVaR portfolio selection model is equivalent to a quantile regression problem. Second, we solve the CVaR portfolio selection model with norm constraints through LASSO quantile regression approach. Third, we compare selection criterions for optimal number of financial assets through Monte Carlo numerical simulations. For illustration, we conduct empirical analysis on Shanghai and Shenzhen 300 index. The empirical results show that our method is efficient for solving high dimension portfolio selection and performs well in dispersing tail risk of portfolio only using a small amount of financial assets.Further, we add constraints to the standard CVaR portfolio and get portfolio selection model under weight constraint. The weight constraints contain many kinds of forms: LASSO, SCAD, adaptive LASSO, elastic net constraint, and so on. All of them choose financial assets through variable selection method. Similarly, we use penalized quantile regression to solve the large CVaR based portfolio model. With the weight constraints, the method can not only overcome the problem of extreme weight in the traditional method, but also solve the problem of high dimension portfolio investment and disperse the risk of the tail.Base on the existing portfolio investment theory, we establish CVaR risk based portfolio model through the penalized quantile regression method. The proposed model can not only solve the problem of high dimensional portfolio selection, but also select a small number of financial assets to manage that effectively reduces the cost of financial assets management and improves the financial risk dispersion effect. The results enriches the theory and method of portfolio decision and has very important significances for the managers of investment institutions.