We propose a clustering-based dimensionality reduction approach to minimum variance portfolio optimization. Rather than constructing the global minimum variance (GMV) portfolio over the full stock universe, which is subject to severe estimation error due to high-dimensionality, we apply Ward’s hierarchical clustering to partition stocks into groups of similarly behaving assets, select one representative per cluster, and optimize on the resulting low-dimensional sub-universe. We show theoretically that clustering preserves the factor structure and yields a better-conditioned covariance matrix than random selection. Empirically, on the Chinese A-share market, the proposed strategies substantially outperform the full-universe benchmark, with gains robust to transaction costs.

Taking the national carbon market and seven local carbon markets in China, we use DEA model to measure market efficiency, and then classify them by hierarchical cluster method. Efficiency of the national carbon market and local carbon markets of Beijing, Shenzhen, Hubei and Shanghai are leading, while Guangdong is in the middle; Chongqing and Tianjin are left behind. Room for improvement and scale returns are further analyzed, and suggestions for each carbon market are proposed finally.