{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T18:21:57Z","timestamp":1773253317973,"version":"3.50.1"},"reference-count":51,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T00:00:00Z","timestamp":1748476800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"scientific research budget of the University of Oradea"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In order to evaluate the impact of k-means clustering on portfolio optimization, this study groups enterprises based on profitability, liquidity, and solvency indicators. The study confirms the positive correlation between risk, return, and risk-adjusted performance through an analysis of historical financial records. After the companies were divided into two groups, equal-weighted portfolios were created using these groupings. Although they produced higher returns, cluster 1 portfolios, which included more risky companies, also showed more volatility. Cluster 0 portfolios, on the other hand, offered less risk and more consistent results. Portfolios clustered by ROA, OCFM, and GPM outperformed the market benchmark and produced the highest returns adjusted for risk, according to Sharpe Ratio analysis. Furthermore, the study emphasizes that although solvency and liquidity metrics play a role in portfolio selection, increased liquidity does not always translate into improved risk-adjusted performance. In terms of methodology, Silhouette Analysis outperformed the Elbow technique in determining the optimal number of clusters. All things considered, the results show how data-driven clustering techniques may be used to align portfolio strategies to investors\u2019 risk tolerances.<\/jats:p>","DOI":"10.3390\/sym17060847","type":"journal-article","created":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T06:25:04Z","timestamp":1748499904000},"page":"847","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["K-Means Clustering for Portfolio Optimization: Symmetry in Risk\u2013Return Tradeoff, Liquidity, Profitability, and Solvency"],"prefix":"10.3390","volume":"17","author":[{"given":"Marcel-Ioan","family":"Bolo\u0219","sequence":"first","affiliation":[{"name":"Faculty of Economic Sciences, University of Oradea, 410087 Oradea, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3825-9686","authenticated-orcid":false,"given":"\u0218tefan","family":"Rusu","sequence":"additional","affiliation":[{"name":"Doctoral School of Economic Sciences, University of Oradea, 410087 Oradea, Romania"}]},{"given":"Marius","family":"Leordeanu","sequence":"additional","affiliation":[{"name":"Institute of Mathematics of the Romanian Academy (IMAR), Calea Grivitei 21, 010702 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4813-4556","authenticated-orcid":false,"given":"Claudia Diana","family":"Sab\u0103u-Popa","sequence":"additional","affiliation":[{"name":"Faculty of Economic Sciences, University of Oradea, 410087 Oradea, Romania"}]},{"given":"Diana Claudia","family":"Per\u021bica\u0219","sequence":"additional","affiliation":[{"name":"Faculty of Economic Sciences, University of Oradea, 410087 Oradea, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-4368-6989","authenticated-orcid":false,"given":"Mihai-Ioan","family":"Cri\u0219an","sequence":"additional","affiliation":[{"name":"Faculty of Economics and Business Administration, Babe\u0219-Bolyai University, 400084 Cluj-Napoca, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1086\/257177","article-title":"The Utility of Wealth","volume":"60","author":"Markowitz","year":"1952","journal-title":"J. Political Econ."},{"key":"ref_2","first-page":"243","article-title":"Fund of Funds Selection of Mutual Funds: Superior Knowledge versus Family and Management Goals","volume":"52","author":"Elton","year":"2017","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_3","unstructured":"(2024, February 15). Council of Europe\u2014History of Artificial Intelligence. Available online: https:\/\/web.archive.org\/web\/20240214013651."},{"key":"ref_4","unstructured":"Didur, K. (2025, February 15). Machine Learning in Finance: Why, What and How. Available online: https:\/\/medium.com\/towards-data-science\/machine-learning-in-finance-why-what-how-d524a2357b56."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/0022-0531(76)90046-6","article-title":"The Arbitrage Theory of Capital Asset Pricing","volume":"13","author":"Ross","year":"1976","journal-title":"J. Econ. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/rof\/rfaa004","article-title":"An Augmented q-Factor Model with Expected Growth","volume":"25","author":"Hou","year":"2021","journal-title":"Rev. Financ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jfineco.2014.10.010","article-title":"A Five-Factor Asset Pricing Model","volume":"116","author":"Fama","year":"2015","journal-title":"J. Financ. Econ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"383","DOI":"10.2307\/2325486","article-title":"Efficient Capital Markets: A Review of Theory and Empirical Work","volume":"25","author":"Fama","year":"1970","journal-title":"J. Financ."},{"key":"ref_9","first-page":"1","article-title":"Relationship between Profitability Ratios and Stock Prices: An Empirical Analysis on BIST100","volume":"6","author":"Mirgen","year":"2017","journal-title":"Press. Proceedia"},{"key":"ref_10","first-page":"91","article-title":"The Relationship Between Profitability and Stock Prices: Evidence from the Saudi Banking Sector","volume":"10","author":"Alaagam","year":"2019","journal-title":"Res. J. Financ. Acc."},{"key":"ref_11","first-page":"73","article-title":"The Effect of Profitability Ratio, Solvability Ratio, Market Ratio on Stock Return","volume":"15","author":"Nalurita","year":"2017","journal-title":"Bus. Entrep. Rev."},{"key":"ref_12","first-page":"695","article-title":"The Effect of Profitability on Stock Return","volume":"5","author":"Nadyayani","year":"2021","journal-title":"Am. J. Humanit. Soc. Sci. Res."},{"key":"ref_13","first-page":"261","article-title":"The Effect of Financial Ratios Toward Stock Returns Among Indonesian Manufacturing Companies","volume":"3","author":"Wijaya","year":"2015","journal-title":"iBuss Manag."},{"key":"ref_14","first-page":"101","article-title":"Exploring the Relationship between Financial Ratios and Market Stock Returns","volume":"11","author":"Musallam","year":"2018","journal-title":"Eur. J. Bus. Econ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"44","DOI":"10.4038\/kjm.v4i2.7500","article-title":"Predictability of Stock Returns Using Financial Ratios: Empirical Evidence from Colombo Stock Exchange","volume":"4","author":"Wijesundera","year":"2016","journal-title":"Kelaniya J. Manag."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Guo, X. (2021, January 28). Clustering of NASDAQ Stocks Based on Elbow Method and K-Means. Proceedings of the 4th International Conference on Economic Management and Green Development, Southwest University, Chongqing, China.","DOI":"10.1007\/978-981-16-5359-9_11"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"He, H., Chen, J., Jin, H., and Chen, S.-H. (2007). Trading Strategies Based on K-Means Clustering and Regression Models. Computational Intelligence in Economics and Finance, Springer.","DOI":"10.1007\/978-3-540-72821-4_7"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1980247","DOI":"10.1080\/23311975.2021.1980247","article-title":"Banking Stock Price Movement and Macroeconomic Indicators: K-Means Clustering Approach","volume":"8","author":"Zuhroh","year":"2021","journal-title":"Cogent Bus. Manag."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Li, T., Liu, Z., Shen, Y., Wang, X., Chen, H., and Huang, S. (2024, January 26). MASTER: Market-Guided Stock Transformer for Stock Price Forecasting. Proceedings of the AAAI Conference on Artificial Intelligence, Vancouver, BC, Canada.","DOI":"10.1609\/aaai.v38i1.27767"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zhang, Y., and Fill, H.D. (2024). TS-GRU: A Stock Gated Recurrent Unit Model Driven via Neuro-Inspired Computation. Electronics, 13.","DOI":"10.3390\/electronics13234659"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.ejor.2024.08.032","article-title":"Industry Return Prediction via Interpretable Deep Learning","volume":"321","author":"Zografopoulos","year":"2025","journal-title":"Eur. J. Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gailani, A., Al-Greer, M., Short, M., and Crosbie, T. (2020). Degradation Cost Analysis of Li-Ion Batteries in the Capacity Market with Different Degradation Models. Electronics, 9.","DOI":"10.3390\/electronics9010090"},{"key":"ref_23","first-page":"1","article-title":"Clustering Stock Market Companies via K-Means Algorithm","volume":"4","author":"Momeni","year":"2015","journal-title":"Arab. J. Bus. Manag. Rev."},{"key":"ref_24","unstructured":"Marvin, K. (2024, May 04). Creating Diversified Portfolios Using Cluster Analysis. Independent Work Report 2015. Available online: https:\/\/web.archive.org\/web\/20240504021513."},{"key":"ref_25","unstructured":"Bin, S. (2024, May 04). K-Means Stock. Clustering Analysis Based on Historical Price Movements and Financial Ratios. CMC Senior Theses 2020, 2435. Available online: https:\/\/scholarship.claremont.edu\/cmc_theses\/2435."},{"key":"ref_26","first-page":"1431","article-title":"Sectoral Portfolio Optimization by Judicious Selection of Financial Ratios via PCA","volume":"24","author":"Dhingra","year":"2023","journal-title":"Optim. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Tsai, P.-F., Gao, C.-H., and Yuan, S.-M. (2023). Stock Selection Using Machine Learning Based on Financial Ratios. Mathematics, 11.","DOI":"10.3390\/math11234758"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3007","DOI":"10.1007\/s10462-019-09754-z","article-title":"A Systematic Review of Fundamental and Technical Analysis of Stock Market Predictions","volume":"53","author":"Nti","year":"2019","journal-title":"Artif. Intell. Rev."},{"key":"ref_29","first-page":"95","article-title":"Accounting Information and Stock Returns in Vietnam Securities Market: Machine Learning Approach","volume":"17","author":"Hung","year":"2022","journal-title":"Contab. Neg."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"109358","DOI":"10.1016\/j.knosys.2022.109358","article-title":"Construction of Stock Portfolios Based on K-Means Clustering of Continuous Trend Features","volume":"252","author":"Wu","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Delcea, C., Cotfas, L.-A., Bradea, I.-A., Bolo\u0219, M.-I., and Ferruzzi, G. (2020). Investigating the Exits\u2019 Symmetry Impact on the Evacuation Process of Classrooms and Lecture Halls: An Agent-Based Modeling Approach. Symmetry, 12.","DOI":"10.3390\/sym12040627"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"90041","DOI":"10.1109\/ACCESS.2024.3418510","article-title":"A Multifaceted Approach to Stock Market Trading Using Reinforcement Learning","volume":"12","author":"Ansari","year":"2024","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"837","DOI":"10.55493\/5002.v12i10.4623","article-title":"Investigating the Influence of Financial Indicators on Stock Returns in the Presence of the COVID-19 Pandemic","volume":"12","author":"Almansour","year":"2022","journal-title":"Asian Econ. Financ. Rev."},{"key":"ref_34","first-page":"30","article-title":"Return on Assets, Return on Equity, Earnings per Share, Dividend Yield, and Book-to-Market Ratio\u2019s Effects on Stock Return","volume":"10","author":"Atmariani","year":"2024","journal-title":"Sosiohumaniora"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2021\/6672677","article-title":"The Measurement Method of Investor Sentiment and Its Relationship with Stock Market","volume":"2021","author":"Hu","year":"2021","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Bolo\u0219, M.-I., Bradea, I.-A., and Delcea, C. (2019). Modeling the Performance Indicators of Financial Assets with Neutrosophic Fuzzy Numbers. Symmetry, 11.","DOI":"10.3390\/sym11081021"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Bolo\u0219, M.-I., Bradea, I.-A., and Delcea, C. (2019). A Fuzzy Logic Algorithm for Optimizing the Investment Decisions within Companies. Symmetry, 11.","DOI":"10.3390\/sym11020186"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Aldhyani, T.H.H., and Alzahrani, A. (2022). Framework for Predicting and Modeling Stock Market Prices Based on Deep Learning Algorithms. Electronics, 11.","DOI":"10.3390\/electronics11193149"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Li, W., Hu, C., and Luo, Y. (2023). A Deep Learning Approach with Extensive Sentiment Analysis for Quantitative Investment. Electronics, 12.","DOI":"10.3390\/electronics12183960"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Yeh, W.-C., Hsieh, Y.-H., Hsu, K.-Y., and Huang, C.-L. (2022). ANN and SSO Algorithms for a Newly Developed Flexible Grid Trading Model. Electronics, 11.","DOI":"10.3390\/electronics11193259"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Febrian, S.S., and Mutasowifin, A. (2025). Selection of Agricultural Industry Stocks by Application of K-means Algorithm with Elbow Method. BIO Web Conf., 171.","DOI":"10.1051\/bioconf\/202517104003"},{"key":"ref_42","first-page":"2825","article-title":"Scikit-learn: Machine Learning in Python","volume":"12","author":"Pedregosa","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"ref_43","unstructured":"Banerji, A. (2024, November 11). K-Mean: Getting the Optimal Number of Clusters, sur Analyfics Vidhya. Available online: https:\/\/www.analyticsvidhya.com\/blog\/2021\/05\/k-mean-getting-the-optimal-number-of-clusters\/."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"49","DOI":"10.3905\/jpm.1994.409501","article-title":"The Sharpe Ratio","volume":"21","author":"Sharpe","year":"1994","journal-title":"J. Portf. Manag."},{"key":"ref_45","first-page":"425","article-title":"Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk","volume":"19","author":"Sharpe","year":"1964","journal-title":"J. Financ."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jfineco.2013.10.005","article-title":"Betting Against Beta","volume":"111","author":"Frazzini","year":"2014","journal-title":"J. Financ. Econ."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jfineco.2013.01.003","article-title":"The Other Side of Value: The Gross Profitability Premium","volume":"108","year":"2013","journal-title":"J. Financ. Econ."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1111\/0022-1082.00335","article-title":"Market Liquidity and Trading Activity","volume":"56","author":"Chordia","year":"2001","journal-title":"J. Financ."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"43","DOI":"10.2469\/faj.v42.n3.43","article-title":"Liquidity and Stock Returns","volume":"42","author":"Amihud","year":"1986","journal-title":"Financ. Anal. J."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1086\/374184","article-title":"Liquidity Risk and Expected Stock Returns","volume":"111","author":"Pastor","year":"2003","journal-title":"J. Political Econ."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.patcog.2012.07.021","article-title":"An Extensive Comparative Study of Cluster Validity Indices","volume":"46","author":"Arbelaitz","year":"2013","journal-title":"Pattern Recognit."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/847\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:42:50Z","timestamp":1760031770000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/6\/847"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,29]]},"references-count":51,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["sym17060847"],"URL":"https:\/\/doi.org\/10.3390\/sym17060847","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,29]]}}}