**6. Conclusions**

In finance literature, the issues of portfolio selection, and risk measurement have always attracted attention of researchers globally. Accordingly, the present paper set out to review the development of related literature in the above areas and to identify the directions for future research. The study focused on three themes: (a) a review of literature on stylized facts that is, fat tails, volatility clustering and dependence structure of returns data, thereafter (b) a review of literature on portfolio selection and finally on (c) portfolio risk measurement. The objective was not only to trace the historical development but also identify possible research issues for future research.

The two important models for portfolio selection are the mean-variance model and global minimum variance model. The portfolio risk is measured by the covariance matrix in these models. From the literature review of these two models, we stressed that the covariance matrix estimation is important because the optimal portfolio weights rely on the covariance matrix. Accordingly, one of our focuses is on the estimation of covariance matrix. However, the estimation error in the covariance matrix estimation of asset returns is so large that the portfolio weights are likely inefficient. Therefore, the shrinkage methods are adopted to cope the estimation error in the estimation of covariance matrix. The shrinkage methods include Stein-type shrinkage methods and linear shrinkage methods. In linear shrinkage methods, we find that the factor model can be used to estimate the covariance matrix and the estimation is used as the target matrix. Consequently, the factor model is also included in the present paper. Also, to reflect the rapid changes of financial markets, we consider that the time-varying structure of covariance matrix is effective, and we take it as one useful improvement of the estimation of covariance matrix.

In addition to the covariance matrix to measure risk of portfolio, VaR and CVaR is another approach from the quantile perspective. Furthermore, many researchers think the risk is not symmetric and the risk should be the downside risk which measures the risk of falling below a target value. If the investor cares more about the loss, the downside risk measure could be a good solution.

The fat tail feature of financial data has received considerable attention in the relevant literature and many studies are based on the multivariate *t* distribution. In the presence of fat tails, the risk measure becomes more difficult to examine and the dependence of financial data is more important because co-movements exacerbate negative portfolio returns. Consequently, the Copula method has become a popular tool to describe the dependence structure of financial data appropriately.

However, there are many interesting questions that remain unsolved. For instance, in the stein-type shrinkage estimation of covariance matrix for portfolio selection problem, could we give an explicit shrinkage parameters selection method with maximizing investors' utility? How to measure the asymmetric relationship of asset returns and apply it to portfolio selection problems?

One of the co-authors of this paper, Sun et al. (2018) have derived the Stein-type shrinkage strategy for optimal portfolio selection using the Cholesky decomposition of the covariance matrix under the mean-variance framework. The Stein-type shrinkage strategy is applied to simulation experiments and an empirical study to test its feasibility. Their proposed method works well in the simulation study and in the empirical analysis; however, there still exist interesting questions. For future work, the assumption of n > p may be replaced by p > n for high dimensional cases, where n is the sample size and p is the number of variables. A reasonable statistical loss function with a different objective function may be studied to take advantage of the proposed approach. In addition, the assumption of the normal distribution can be extended to elliptically symmetric or skewed distributions and take robustness into consideration as well.

Please note that in the minimum variance model, the covariance matrix plays an important role because it measures the risk and relationship of asset returns simultaneously under the normality assumption. However, as discussed earlier, the distribution of asset returns is non-normal and has an obvious fat tail nature. In addition, the risk is one-sided. Hence it should be beneficial to study further and use a better tool to replace the covariance matrix, by involving the semi variance and distance correlation as discussed by e.g., Huang et al. (2016) and Sun et al. (2019).

Similarly, studies are required to examine the extent to which investment managers in the real-world incorporate the findings from the academic literature in practice.

**Funding:** The first two authors received support from the National Natural Science Foundation of China (11471264).

**Acknowledgments:** The authors would like to thank the editor and referees for the opportunity and their constructive comments which led to an improved version of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.
