Abstract:

The estimation errors in the mean vector and covariance matrix make it challenging to accurately identify efficient portfolios in real stock markets. To mitigate these issues， prior studies have proposed various shrinkage estimators. However， these estimators overlook the fact that stock return parameters vary across market conditions. How to develop shrinkage estimators for the mean vector and covariance matrix under a mixed-normal distribution is investigated， which can capture stock returns in both bull and bear markets. New shrinkage estimators are constructed based on the law of total expectation and the variance decomposition formula， with the optimal shrinkage intensities and targets determined by minimizing quadratic loss functions. Compared with existing shrinkage estimators， the new shrinkage estimators accounts for estimation errors arising from changes in market conditions and enables nonlinear shrinkage of the eigenvalues of covariance matrix. In the simulation analysis， the new shrinkage estimators reduce estimation errors for the mean vector and covariance matrix by 65% and 6%， respectively. Moreover， the reliability and robustness of efficient portfolios based on the new estimators improve by 24% and 60%， respectively. When applied to the Chinese A-share market， and with the number of assets ranging from 10 to 100， efficient portfolios based on the new shrinkage estimators consistently achieve a monthly net Sharpe ratio above 0.1. As the number of assets increases， the improvement in the net Sharpe ratio delivered by the new shrinkage estimators becomes more pronounced relative to existing estimators. Therefore， it is recommended that investors use shrinkage estimators under mixed normal to construct efficient portfolios， particularly in settings with a large number of assets and limited sample sizes.

Key words: stock market, efficient portfolios, shrinkage estimation, mixture of normal