Empirical Analysis Of Dynamic Portfolio Model Based On CRRA Utility Function

Portfolio selection problem, in brief, is that the investors distribute their wealth into different assets to achieve the purpose that risk diversification, the income or wealth final expected utility maximization. Compared with the classical Markowitz mean-variance static portfolio model, dynamic portfolio selection model can reflect the dynamic behavior of asset prices and the stock market reaction characteristics, describe the investment decision-making process. It provides a way to for economic agents to configure the limited resources achieving the investment objective in an uncertain environment. It also plays a role in promoting the rational expectations of the securities market. Therefore, to study dynamic portfolio selection problem is of great significance.The utility function is quantitative description of preference, so study the portfolio selection problem to maximize the utility function will be more scientific and reality. Constant relative risk aversion (CRRA) utility function is a standard analysis model in classical framework. It could describe the different investors’ utility function by setting different risk aversion coefficient. This article is based on CRRA utility function to establish a dynamic portfolio selection model. Firstly, using Poisson-jump process to describe the asset price dynamics, can effectively describe the peak fat-tail phenomenon of asset return. Secondly, we derive the dynamic process of wealth according to the asset price dynamics process. At last, we solve the dynamic portfolio choice results, to maximize CRRA utility function, getting an implicit solution of nonlinear differential equations. In the empirical analysis, the first we use the maximum likelihood estimation method estimated the value of the parameters of the Poisson jump process, and then the Gauss-Newton iterative method is used to solve the numerical solution of nonlinear equations with weight variable. Finally, many indicators are used to analyze and evaluate of the results of the model.In this article, we analyze the results of portfolio selection problem under different utility function by setting different risk aversion coefficient values, where y=3,5,7. The results showed that:with the increase of the coefficient of risk aversion, the weight of high risk assets reduce. On the contrary, the weight of high returns assets will increase as the coefficient of risk aversion increases. This implies that investors with small risk aversion will distribute the wealth into different assets on an average attitude. They hope that the high risky assets will have a chance to get high returns. However, investors with high risk aversion should adjust their portfolios constantly, making high-risk assets ratio decreased, low-risk assets proportion increased, to avoid the high risk may bring damage in wealth, and to ensure stable return of low-risk assets.In order to test the robustness of the model, we consider the Jensen index, Treynor index and Sharpe index to evaluate the effectiveness of the portfolio selection model. Three indicators’results show that:the portfolio selection result in quarter1,2of2008and2,3,4quarter of2011, is not very satisfactory, only considering Jensen index. But the two phases is in financial crisis, the entire stock market is in a bear market phase, and the result of Treynor Index also show that the portfolio perform better than the market. Considering the above two reasons, the portfolio model in this article is still valid. Moreover, from the return of the portfolio and the benchmark Shanghai Composite Index return Graph can also be seen that whether in the market is in the economic prosperity bull or bear market when economic depression, the portfolio selection model established in this paper can outperform index returns.