دانشگاه اصفهان مدیریت دارایی و تامین مالی 2383-1189 14 4 2026 12 22 stock selection by using hierarchical risk parity method and minimum spanning tree based on correlation coefficients matrix in the 50 most active companies of Tehran Stock Exchange انتخاب سهام با استفاده از رویکرد برابری ریسک سلسله‌مراتبی و درخت پوشای کمینه مبتنی بر ماتریس ضرایب همبستگی در 50 شرکت فعال‌تر بورس اوراق بهادار تهران 23 46 30093 10.22108/amf.2025.143014.1930 FA سعید فلاح پور دانشیار، گروه مهندسی مالی، دانشکدۀ حسابداری و علوم مالی، دانشکدگان مدیریت، دانشگاه تهران، تهران، ایران علی نمکی دانشیار، گروه مهندسی مالی، دانشکدۀ حسابداری و علوم مالی، دانشکدگان مدیریت، دانشگاه تهران، تهران، ایران 0000-0003-4167-6472 رضا فضلی کارشناس ارشد، دانشکدۀ حسابداری و علوم مالی، دانشکدگان مدیریت، دانشگاه تهران، تهران، ایران Journal Article 2024 10 13 Portfolio optimization represents a central challenge in financial investment. This study evaluates the performance of a proposed stock selection model for the Tehran Stock Exchange, comparing its outcomes to both the market index and the Markowitz model using statistical tests. Portfolios were constructed by applying the Hierarchical Risk Parity (HRP) method combined with marginal asset selection, utilizing two distinct correlation matrices. These portfolios were rebalanced semiannually based on updated correlation coefficients and the composition of the Top 50 Active Companies Index. The results demonstrate that the HRP approach with asset selection generally outperforms both the index and the Markowitz model in terms of portfolio performance, a finding consistent across various market conditions, including recessions and booms. Statistical tests confirm the significant superiority of this method over the Markowitz approach. Furthermore, this research achieved effective diversification within the Tehran Stock Exchange using fewer assets than other methods. The primary innovation of this work lies in the application of two distinct correlation matrices and the implementation of two novel methodologies for asset weighting and selection, offering practical insights for both individual and institutional investors.<br /><strong>Keywords:</strong> Portfolio Optimization, Hierarchical Risk Parity, Minimum Spanning Tree, Asset Selection, Marginal Sssets<br /><strong>JEL Classification:</strong> C61, G11, G32<br /> <br /><strong>Introduction</strong><br />The process of selecting financial assets and constructing an investment portfolio has long been characterized by uncertainty. Portfolio management—which entails the analysis, selection, and ongoing maintenance of a collection of investments—is inherently complex and often lacks a formalized structure. Capital markets allow investors to allocate surplus capital across various assets to generate profit. A primary objective of investment is to distribute capital among a set of assets in a manner that maximizes expected return while minimizing associated risk (Namaki & Kafi Mousavi, 2023). Investors tend to prefer assets or asset combinations that offer the highest expected return for a given level of risk. Consequently, understanding the future price behavior of assets, the factors that influence them, and their associated risks is essential (Noorahmadi & Sadeghi, 2023). The stock market, as a central investment venue, has attracted significant attention, yet the selection of an optimal stock portfolio remains a complex challenge influenced by numerous variables (Tavakoli et al., 2013).<br />Accurately assessing portfolio risk requires a clear understanding of the dynamic relationships among assets. While correlation matrices are commonly used for this purpose, they often contain noise due to market instability. Techniques such as Minimum Spanning Trees (MST) and Planar Maximally Filtered Graphs (PMFG) can help filter this noise and clarify underlying asset linkages (Pozzi et al., 2013). Financial markets also exhibit inherent hierarchical structures, with interconnected industries and sectors. Hierarchical Risk Parity (HRP)—the core methodological framework of this study—capitalizes on such structures to achieve improved diversification and risk control. By integrating HRP with MST and employing two distinct correlation matrices, this research seeks to construct efficient portfolios within the Tehran Stock Exchange, balancing theoretical rigor with practical applicability.<br /><strong> </strong><br /><strong>Materials and Methods</strong><br />The data for this study were sourced from the historical trading records of the Tehran Stock Exchange (TSE), with a specific focus on its 50 most active companies. The analysis covers the period from 2016 to early 2023. All data processing and model implementation were performed using Python. Initially, the logarithmic daily returns for the selected stocks were computed for each six-month interval. These returns were then used to derive covariance and correlation matrices. Two distinct types of correlation matrices were employed: a Full Cross-Correlation (FC) matrix and a Global Motion Subtracted Correlation (GMSC) matrix. The GMSC was obtained by filtering out the market-wide component based on the eigenvalue distribution defined by the Marchenko–Pastur theorem.<br />The Hierarchical Risk Parity (HRP) method was subsequently applied for portfolio construction. The correlation matrices were first converted into metric distance matrices, after which Minimum Spanning Trees (MST) were generated using Kruskal’s algorithm. The HRP algorithm itself follows a three-step procedure: (1) hierarchical clustering to group assets with similar return profiles, (2) quasi-diagonalization of the covariance matrix to order correlated assets contiguously, and (3) recursive bisection for top-down capital allocation, wherein riskier clusters are assigned proportionally lower weights. To improve diversification and streamline the portfolio, a marginal asset selection process was implemented using modified network centrality and periphery metrics—specifically, degree, betweenness, closeness, eccentricity, and eigenvector centrality. Assets identified as peripheral based on a composite of these centrality measures were selected for inclusion. Portfolios of varying sizes (comprising 3, 5, 10, 20, and 30 assets) were formed and rebalanced on a semiannual basis. The performance of the portfolios constructed using the HRP methodology was evaluated against two benchmarks: the TSE Top 50 Index and a classical Markowitz mean–variance optimization model. The Markowitz model was constrained to non-negative weights in accordance with the short-selling restrictions present in the Iranian capital market.<br /><strong> </strong><br /><strong>Findings</strong><br />The analysis was conducted using adjusted price data for the 50 most active companies on the Tehran Stock Exchange from July 2016 to January 2023. From logarithmic and normalized returns, two distinct correlation matrices were constructed: a Full Cross-Correlation (FC) matrix and a Global Motion-Subtracted Correlation (GMSC) matrix. These matrices were used to build Minimum Spanning Trees (MSTs), identify marginal assets, and implement the Hierarchical Risk Parity (HRP) algorithm, which involved hierarchical clustering, quasi-diagonalization, and recursive bisection. Eight distinct portfolio strategies were evaluated based on the number of assets (5, 10, 25, 50) and the type of correlation matrix used. The results indicate that HRP-based portfolios consistently outperformed both the Top 50 Index and the Markowitz model across several key metrics, including cumulative returns, annualized returns, Sharpe ratios, and drawdown reduction. For instance, the GMSC-HRP (10 assets) and FC-HRP (5 assets) strategies achieved cumulative returns of 2966% and 2625%, respectively, compared to the index's return of 2034%. The Wilcoxon signed-rank test confirmed the statistically significant superiority of six out of the eight HRP-based portfolios at the 10% significance level.<br />A regime-based analysis further contextualizes these results. HRP portfolios delivered performance comparable to the index during stable periods, significantly outperformed it during the bull market (2017–2020), and demonstrated a marked ability to mitigate losses during the subsequent bear market (2021–2022). Notably, FC-HRP models were found to be more effective for smaller portfolio sizes, making them suitable for individual investors, whereas GMSC-HRP models performed better with larger portfolios, aligning with the needs of institutional investors. In summary, the HRP approach, combined with MST-based marginal asset selection, provided robust diversification, superior risk-adjusted returns, and demonstrated resilience across varying market conditions.<br /><strong> </strong><br /><strong>Discussion and Conclusion</strong><br />The findings demonstrate that the Hierarchical Risk Parity (HRP) approach, combined with asset selection, generally outperformed the Top 50 Index, resulting in lower portfolio standard deviation and superior risk-adjusted returns, as evidenced by higher Sharpe ratios. Both correlation inputs—the Full Cross-Correlation (FC) and the Global Motion Subtracted Correlation (GMSC) matrices—were found to be effective within the HRP framework. A principal contribution of this study is the novel simultaneous application of the GMSC matrix and the integration of HRP asset weighting with MST-based marginal asset selection, specifically for the Tehran Stock Exchange (TSE). The results suggest a distinct advantage based on portfolio size: the FC matrix yielded superior performance for smaller portfolios (e.g., 5 assets), whereas the GMSC matrix proved more effective for larger portfolios. Consequently, FC-based HRP models are recommended for individual investors with limited capital, while GMSC-based strategies appear better suited for institutional investors. HRP-based portfolios delivered particularly strong results during the bull market (2017–2020) and demonstrated notable resilience during the subsequent downturn beginning in August 2020, effectively mitigating losses.<br />Furthermore, the HRP weighting method, even without asset selection, improved the performance of the full 50-asset portfolio compared to simple index weighting. The use of marginal asset selection via Minimum Spanning Tree (MST) facilitates efficient diversification with a smaller asset count, reducing associated trading costs while preserving portfolio efficiency. Future research could enhance the model by integrating HRP with machine learning techniques or exploring advanced graph-based market structures to improve predictive accuracy and interpretability. Extending this methodology to other asset classes in the Iranian market, such as bonds, gold ETFs, and derivatives, would further test its capacity for risk reduction and return enhancement. مسئلۀ بهینه‌سازی سبد دارایی یکی از چالش‌های اساسی در حوزۀ سرمایه‌گذاری مالی است. هدف این پژوهش، ارزیابی عملکرد مدل ارائه‌شده برای انتخاب سهام در بورس اوراق بهادار تهران و مقایسۀ آن با شاخص و رویکرد مارکوویتز براساس آزمون‌های آماری است. در پژوهش حاضر، با استفاده از روش وزن‌دهی برابری ریسک سلسله‌مراتبی و انتخاب دارایی‌های حاشیه‌ای، سبدهای سهام ساخته شده‌ است. این سبدها در دو حالت مختلف براساس ماتریس‌های همبستگی متفاوت تشکیل شده و سپس هر شش ماه یک‌بار براساس ضرایب همبستگی و ترکیب شرکت‌های عضو شاخص ۵۰ شرکت فعال‌تر، بازنگری و وزن‌دهی مجدد شده‌ است. در ادامه، عملکرد این سبدها با شاخص ۵۰ شرکت فعال‌تر و رویکرد مارکوویتز مقایسه گردیده است. نتایج نشان می‌دهد که رویکرد ریسک سلسله‌مراتبی با انتخاب دارایی توانسته است در بیشتر موارد عملکرد سبد سهام را براساس شاخص‌های ارزیابی در مقایسه با شاخص و رویکرد مارکوویتز بهبود دهد. نتایج آزمون‌های آماری نیز برتری معنادار این روش را در مقایسه با رویکرد مارکوویتز تأیید می‌کند. افزون بر این، پژوهش حاضر توانسته است با انتخاب بهینۀ دارایی و ساخت سبدهایی با تعداد دارایی کمتر در مقایسه باروش‌های مشابه، به تنوع‌بخشی کارآمد در بورس اوراق بهادار تهران دست یابد. نوآوری اصلی این پژوهش در استفادۀ هم‌زمان از دو ماتریس همبستگی متفاوت به‌عنوان ورودی و به‌کارگیری دو روش نوین برای وزن‌دهی و انتخاب دارایی در بازار سهام تهران است؛ به‌گونه‌ای‌که نتایج آن می‌تواند برای سرمایه‌گذاران شخصی و نهادی در استفاده از ماتریس‌های همبستگی مختلف کاربرد داشته باشد. https://amf.ui.ac.ir/article_30093_9b9c03f14981d8dcab4fce6f04bec412.pdf