Abstract: Since the seminal work of Markowitz (1952), portfolio has drawn more and more attention from academics and practitioners. It is known to all that the risk measure plays an important role in portfolios. So far, there are many risk measures including variance, value-at-risk (VaR) and expected-shortfall (ES). ES, also called mean excess loss, tail VaR, or CVaR, is anyway considered to be a more consistent measure of risk than VaR. Consequently, the mean-ES model has become the focus of portfolio selection. The mean-ES model is traditionally optimized through analytical or scenario-based methods with large numbers of instruments, in which the calculations often come down to linear programming or nonsmooth programming. In order to reduce the computational complexity in the mean-ES portfolio decision, it is transformed to an expectile regression theoretically and a new method is proviede for its solution. The novel approach has at least two advantages. First, the model can be easily optimized and further extended due to the continuity and smoothness of asymmetric quadratic loss function in expectile regression. Second, the ES risk can be precisely measured through the relationship between ES and expectile results produced in expectile regressions. To illustrate the efficacy of our method, empirical studies are condueted on five representative stocks in Shanghai and Shenzhen 300 (HS300) Index and our mean-ES model based on expectile regression with a mean-variance model is compared to that with a mean-VaR model. The data comes from the Genium Finance platform (http://www.genius.com.cn/) and covers the period from Jan 1, 2010 to Jun 26, 2017. The data are split into two parts:in-sample one with size 1212 from Jan 1, 2010 to Jan 5, 2015 and out-of-sample with size 602 from Jan 6, 2015 to Jun 26, 2017. The returns, the risks (standard deviation, VaR, and ES), the Omegas, and the efficient frontiers obtained by solving the portfolio selection problem under different risk measures are studied. The empirical results are promising and show that our method outperforms the others in terms of dispersing tail risk and improving portfolio performance. In practice, it is important to consider a large scale portfolio selection. To this end, it would be necessary to introduce variable selection techniques, such as Lasso, into expectile regressions to form a Lasso expectile regression approach. This approach can be applied to solve the large scale portfolio selection, which does not have in our current method. We leave this for future research.

Key words: portfolio selection, expectile regression, mean-ES model, mean-VaR model, mean-variance model