NI Xuanming, ZHENG Tiantian, SUN Huixia, ZHAO Huimin
According to the arbitrage asset pricing (APT) theory, an ideal risk factor needs to be able to price both the expected return (first-order moment) and variance-covariance matrix (second-order moment) of the cross-sectional assets. Therefore, it is very important to link the order moment information in the process of building risk factors. Based on the asset pricing and portfolio optimization theory, this paper considers the Sharpe ratio to be an ideal indicator for linking the first and second moments, and proposes a sparse principal component analysis algorithm with mean-variance efficiency (MVE-SPCA) to build risk factors. Specifically, the algorithm introduces Sharpe ratio information on the basis of principal component analysis (PCA), that is, the original principal component is sparsely adjusted by introducing the L1 regularization term into the objective function of PCA, and the coefficient of the penalty term is determined by maximizing the Sharp ratio. Through this process, factors can not only capture the co-movement of the stock market, but also price the differences in the expected returns of cross-sectional assets. Further analysis shows that the weight structure of the factors constructed in this paper is similar to that of the classical economic factors, which shows that the statistical factors proposed in this paper are essentially consistent with the financial logic of the classical economic factors. Using the monthly frequency data of the A-share market from April 2002 to May 2022, this paper constructs 5×5 and 10×10 cross-sorted portfolio data sets and single-sorted portfolio data sets based on the common 18 anomalies in the A-share market to test the pricing ability of the proposed algorithm. The results show that the pricing ability of the factor model constructed by the algorithm in this paper is better than the traditional three-factor, four-factor and five-factor models, and also better than the statistical factor models such as PCA and risk premium PCA. The factor model built by the algorithm in this paper has smaller GRS statistics of the test assets. Meanwhile, the factors extracted by the proposed algorithm can achieve a larger optimal Sharpe ratio. This paper further finds that although the factors extracted by MVE-SPCA are statistical factors, they also have obvious economic meanings. The weight structure of the first three principal components extracted from 5×5 cross-sorted portfolios can respectively restore the market factor, size factor and value factor in the traditional factor model. The above conclusion also holds in the 10×10 cross-sorted portfolio data sets and the more general single-sorted portfolio data sets, which shows that the conclusion of this paper is robust.