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# System prepended metadata

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The stock market is a complex system with interactions between individuals, groups, and institutions at different levels. In financial crises, the risk can quickly propagate among these interconnected institutions. Therefore, the analysis of the correlations between shares is important for the understanding of interactive mechanism of the stock market and the portfolio risk estimation [1, 2]. Variety of works have been done to reveal the information contained in the internal correlations among stocks, and the methods generally used in the research of stock cross correlations include the random matrix theory [3, 4], the principal component analysis [5, 6], and the hierarchical structure [7, 8].

The statistical properties of the cross-correlations in some stock markets, such as South African market and Korean stock market had been studied [9, 10].  The eigenvalues, eigenvectors and the properties of the cross-correlation matrix were studied [11–14]. In addition to this, the non-linearity and non-stationarity in real-world data have also been incorporated into the studies, hence new methods based on detrendization have been proposed [15–17].

These current studies of correlations in stock market mainly focused on the static correlations, and this poses a research gap in studying the dynamic correlations.

The classical Markowitz mean-variance portfolio selection approach [18] has been widely applied in investment decision-making since its introduction in 1952. It helps investors in developing an efficient portfolio with the lowest risk at a given target return. But the computational instability of the classical Markowitz portfolio selection framework has annoying consequences found by practitioners and scholars: large changes in optimal portfolio weights could arise due to slight changes in assets’ expected returns, volatilities, or correlations [19, 20]. 

To avoid this drawback, various portfolio optimization procedures has been proposed. Gotoh and Takeda [21] and Olivares-Nadal and DeMiguel [22] find that norm-constrained portfolio selection can be reformulated as robust optimization problem with certain type of uncertainty set over the return vector, implying similar effects of the two approaches in reducing the model sensitivity. Yang et al. [23] prove that the Euclidean distance between the equal-weighted portfolio and a mean-variance portfolio obtained under squared l_2-norm penalty with penalty parameter λ is proportional to 1/√λ; Chen et al. [24] adopted the SGLasso regularization in [25] and introduced a general formulation for sparse-group portfolio selection. In their paper, they showed theoretically that the l_2-norm in their group lasso penalty has equalizing effect within a sector and the optimal portfolio’s stability is enhanced by allocating equal weights to assets in the same sector.

In our project, we will adopt the SGLasso regularized optimisation to construct portfolio wth a time-varying correlation.
