Abstract: Value at Risk (VaR), which is widely used by fund companies, banks, securities firms and financial supervision institution, is one of the most popular risk measurement tools presently. The estimation methods of VaR and portfolio optimization models with VaR have been one of the hot spots in recent years. Since VaR is not a convex risk measure, it is difficult to obtain the global optimal solution of portfolio selection problems based on VaR. Moreover, the present study on portfolio selection with VaR is mostly carried out under normal or ellipsoidal distribution assumptions, which is not consistent with the reality of financial markets. In this paper, nonparametric kernel estimation method is firstly applied to estimate VaR and a nonparametric kernel estimator for asset portfolio's value at risk (VaR) is gotten with distribution-free specification. Then kernel estimator of VaR is embedded into the mean-VaR portfolio selection models and accomplish the goal that financial risk estimation and portfolio optimization are implemented at the same time. It is easy to show that the objective function of our model is smooth theoretically and easy to solve the optimization problem. Monte Carlo simulations are carried out to compare the accuracy of our method with the accuracy of classical methods. The simulation results show that our model possesses large sample properties, and outperforms empirical distribution method and Cornish-Fisher expansion method which are usually applied in the classical literatures under the asymmetric and thick tail distribution setting. Finally, our models and methods are applied to the Chinese A stock market. The daily data of SSE 50 Index and its constituent stocks are collected. The data window ranges from January 2^nd 2004 to July 8^th 2016, with a total of 3040 daily data. The empirical results show that our model can effectively control risk, as well as obtain excess returns relative to the stock index and support effectiveness of our model and application value of this research. It is acknowledged that, in this study, our nonparametric mean-VaR model has these shortcoming:First, our model requires a large number of samples; Secondly, our model is non-convex optimization problem, which is difficult to find the global optimal solution; Finally, it can be seen from the Monte Carlo simulation, sometimes our model cannot give the optimal asset allocation strategy, especially when the number of assets is large and the sample size is small. These questions are left for further research.

Key words: Investment portfolio, Value at Risk, nonparametric kernel estimation