Abstract: In practice, monitoring and managing a portfolio with many assets is not only time consuming but also expensive. It is therefore ideal to pick a reasonable number of stocks to address these two issues. However, this has not been considered in traditional portfolio methods. In addition, the traditional portfolio methods often cause too extreme long and short positions, which implies a high trading cost. To this end, a new method of portfolio selection through adding norm constraints is proposed to the standard CVaR portfolio investment selection model. The basic idea of our method comes from variable selection procedure like LASSO in statistics and contains three important aspects. First, it is illustrated that the process of solving the CVaR portfolio selection model is equivalent to a classical quantile regression problem. As we all know, quantile regression approach is efficient to describe the behavior of a financial asset across quantiles, which is corresponding to a CVaR value. Second, the CVaR portfolio selection model with norm constraints is solved through LASSO quantile regression approach. Third, selection criterions for optimal number of financial assets are compared through Monte Carlo numerical simulations considering two cases:n>p and np and n p case. For illustration, we also do empirical analysis on Shanghai and Shenzhen 300 (HS300) index. The sample period spans from Apr 11, 2011 to Nov 11, 2013. Note that the 300 constituent stocks in HS300 are always changing in the sample period since the sample is adjusted every half year. Those stocks are intersected and ultimately 230 constituent stocks are kept for a portfolio candidate. It shows that with norm constraints, our method avoids two extreme positions effectively. Moreover, our method is efficient for solving high dimension portfolio selection and outperforms some popular method like L1-Variance model in dispersing tail risk of portfolio only using a small amount of financial assets. For example, our method reduces the VaR by 30.28% in sample and 0.69% out of sample, while reduces the CVaR by 44.92% in sample and 10.42% out of sample. To sum up, our new method is a general one that includes the standard CVaR-based portfolio selection model as a special case. It is certainly that our method can be improved by utilizing some alternative constraints like SCAD (Softly Clipped Absolute Deviation) penalty. This penalty will bring an unbiased results, which does not have in our current method. This is left for future research.

Key words: portfolio selection, CVaR based high dimensional portfolio selection, norm constraints, LASSO quantile regression