## 6 Conclusions  
In this paper, we have presented the \(k\)-covariance estimation which is using the undersampling approach and ensemble learning to improve the stability of the covariance matrix in quantitative trading, particularly for the global minimum variance portfolio. Our approach reduces the impact of outliers in the covariance estimation by manipulating the process at the data level. All elements in our combinations are homogeneous covariance matrices, such as combining multiple sample covariance matrices and they are estimated from different samples. The experimental results show that the \(k\)-covariance approach \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Estimator** & \(\sigma\) **(\%)** & **Sharpe Ratio** & **Portfolio Turnover (\%)** \\ \hline sample & 8.30 & 1.39 & 7.28 \\ STM & 7.24 & 1.94 & 3.49 \\ SSIM & 7.37 & 2.00 & 4.41 \\ SCCM & 7.71 & 1.94 & 3.43 \\ \(k\)-sample & 7.80 & 1.77 & 5.62 \\ \(k\)-STIM & 7.13 & 2.04 & 3.34 \\ \(k\)-SSIM & 7.17 & 2.07 & 3.86 \\ \(k\)-SCCM & 7.52 & 1.91 & 3.23 \\ sTS & 6.79 & 2.24 & 2.88 \\ sTC & 7.46 & 2.01 & 2.89 \\ sSC & 7.47 & 2.01 & 2.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Out-of-sample performance of eleven minimum variance portfolios with different covariance matrix estimations. All available stocks on the HOSE exchange are considered. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Estimator** & \(\sigma\) **(\%)** & **Sharpe Ratio** & **Portfolio Turnover (\%)** \\ \hline sample & 9.81 & 0.82 & 4.30 \\ STIM & 9.34 & 0.99 & 3.17 \\ SSIM & 9.54 & 0.92 & 3.78 \\ SCCM & 9.42 & 1.09 & 3.18 \\ \(k\)-sample & 9.47 & 0.82 & 3.67 \\ \(k\)-STIM & 9.16 & 0.93 & 2.91 \\ \(k\)-SSIM & 9.31 & 0.86 & 3.32 \\ \(k\)-SCCM & 9.29 & 1.01 & 2.90 \\ sTS & 8.90 & 1.13 & 2.47 \\ sTC & 9.28 & 1.10 & 2.59 \\ sSC & 9.28 & 1.10 & 2.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Out-of-sample performance of eleven minimum variance portfolios with different covariance matrix estimations. Only top one hundred assets by market capitalization are considered, \((N=100)\). improves significantly the sample covariance matrix estimation to the performance level of the linear shrinkage estimations. Although the linear shrinkage estimations are robust, applying our approach shows improvements in some cases, such as on the shrinkage to single-index model. In practice, our results suggest that fund managers should focus on two shrinkage estimations, the shrinkage to the identity matrix and the Shrinkage to constant correlation matrix, which are more stable and the performances of the portfolios of estimators are worth analyzing by the research community. The scientific novelty of the research is using machine learning techniques to build and combine multiple weak covariance matrix estimations to achieve a sustainable covariance matrix estimation. We conclude that combining multiple sample covariance matrices by our approach and similarly applying on the weak shrinkage estimation achieve the same level of highest portfolio performances of the shrinkage estimations. In which, the improvement of the shrinkage estimations mostly come from the combination with other matrices while in our approach, we combine only one kind of covariance matrix estimation multiple times which is weak and noisy. Furthermore, the portfolios of covariance estimations that combine the sample covariance matrix and more than two other matrices have exceptional results, we propose the investigation of these portfolios in future work. There are two limitations in our study, the first is the outdated data which is from 2013 to the end of 2019 and the second is our analysis performed on the single market. Because of a COVID-19 crisis at the early of 2020, the financial markets around the world performed differently at that period. Therefore, we focus on the normal scenario of the markets and the crisis scenario is worth studying in a separate paper. Another research direction is to test this approach on other markets to verify our conclusions generally. Conceptualization, T.T. and N.N.