Empirical Analysis of Optimized Portfolio Allocation Based on Markowitz and Single-Index Models: A Case Study of Group 3 Assets

Empirical Analysis of Optimized Portfolio Allocation Based on Markowitz and Single-Index Models: A Case Study of Group 3 Assets | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Empirical Analysis of Optimized Portfolio Allocation Based on Markowitz and Single-Index Models: A Case Study of Group 3 Assets Yiming Shen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9501117/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study investigates the construction of optimal portfolios using the Markowitz full mean-variance model and the Sharpe single-index model for a diversified universe of 22 risky assets referred to as Group 3. The analysis is framed as a stressed macro-financial scenario in which rising geopolitical risk, elevated energy prices, restrictive monetary conditions, and weak broad-market performance jointly reshape the covariance structure of returns. By comparing five portfolio constraint settings, including unconstrained allocation, Regulation T-type leverage, box constraints, and no-short-selling conditions, the paper evaluates how institutional trading limits alter portfolio efficiency, hedging capacity, and risk-adjusted performance. The results indicate that moderate flexibility in long-short positioning materially improves portfolio resilience under adverse market conditions, whereas severe constraints reduce the attainable Sharpe ratio and limit the investor's ability to offset sector-specific shocks. Consistent with the literature on estimation error and constrained optimization, the Markowitz framework captures cross-sector covariance patterns more effectively than the single-index specification when market dislocations are driven by non-uniform sectoral shocks. Within the study dataset, the minimum-risk portfolio still delivers a monthly expected return of approximately 7.55%, while the maximum Sharpe ratio exceeds 1.01 under relatively permissive constraints. The findings suggest that, under high-uncertainty environments, portfolio design should emphasize energy exposure as a defensive growth component, reduce excessive concentration in high-beta technology assets, and employ defensive utilities as substitutes for weakening traditional safe-haven assets. Overall, the paper demonstrates that modern portfolio theory remains useful in crisis conditions when model selection and constraint design are aligned with the underlying economic regime. International Economics portfolio optimization Markowitz model single-index model geopolitical risk sector rotation short-selling constraints asset allocation 1. Introduction Portfolio selection remains one of the core questions in finance because any investor who allocates capital across multiple risky assets must trade off expected return against uncertainty. The classical contribution of Markowitz was to show that the relevant notion of risk is not the volatility of each asset viewed independently, but the variance of the portfolio as a whole, which depends on the covariance structure among constituent assets [1,12]. On this basis, mean-variance optimization provides a systematic way to trace the efficient frontier and to identify portfolios that minimize variance for a target return or maximize expected return for a given risk level [9,11]. Later developments, especially Sharpe’s single-index model and subsequent factor-based simplifications, sought to make portfolio construction more tractable by approximating the covariance matrix through a common market factor and asset-specific residual risk [2,23]. That simplification remains attractive in applied settings because it substantially reduces estimation burden, especially when the number of assets is large relative to the sample length [3,14]. However, the simplicity of the single-index framework comes with a cost: if cross-asset comovement is not dominated by a single market channel, the model may suppress economically meaningful heterogeneity across sectors and firms [12,16]. In parallel, a large body of empirical work has shown that portfolio optimization in realistic settings is heavily shaped by estimation risk, sparsity requirements, and implementation frictions. Studies on shrinkage estimation, robust optimization, and high-dimensional covariance modeling consistently argue that theoretically optimal portfolios can become unstable when expected returns and covariance matrices are estimated with noise [8,10,14,16,24]. This line of research provides an important bridge between textbook portfolio theory and actual investment practice, where leverage constraints, turnover concerns, and concentration limits often determine whether an allocation is economically plausible [4,5,7,22,26]. This issue becomes particularly important in periods of macro-financial stress. When inflation remains elevated, monetary policy is restrictive, energy prices are volatile, and geopolitical tensions intensify, return spillovers may propagate unevenly across sectors rather than through a single market factor alone [20,29,31]. In such circumstances, technology, utilities, finance, and energy assets may react differently to the same shock because their cash-flow duration, cost exposure, regulatory sensitivity, and demand conditions differ. The empirical relevance of portfolio optimization therefore depends not only on solving a mathematical problem, but also on selecting a covariance representation that is sufficiently rich for the prevailing regime [17,25]. Against this background, the present study examines a universe of 22 risky assets labeled Group 3 and compares optimal allocations generated by the Markowitz full model and the Sharpe single-index model under five portfolio-constraint regimes. The paper has three objectives. First, it evaluates whether the richer covariance structure of the Markowitz framework yields more plausible allocations under a stressed scenario. Second, it investigates how common implementation constraints such as leverage limits, box constraints, and short-selling restrictions affect attainable efficiency. Third, it derives practical allocation implications for investors facing an environment in which traditional broad-market diversification may weaken. Relative to the previous version, this revised manuscript strengthens the theoretical discussion, adds clearer links between claims and references, and expands the economic interpretation of the reported results [9,18,24]. 2. Literature Review The first strand of literature relevant to this study concerns the evolution of mean-variance theory and its practical refinements. Classical portfolio selection research established the efficient frontier and demonstrated that diversification benefits are generated by the interaction among asset returns rather than by holding a large number of securities mechanically [1]. Subsequent work expanded this framework in several directions, including single-index representations, cardinality-constrained optimization, shrinkage estimation, high-dimensional covariance estimation, and robust optimization [2,4,5,8,12,16]. Together, these studies show that portfolio optimization remains theoretically compelling, but highly sensitive to the quality of estimated inputs and to the frictions imposed on admissible weights [9,11]. A second strand emphasizes estimation error and model instability. Empirical studies repeatedly find that optimized weights can become extreme when expected returns are measured with noise, when covariance matrices are poorly conditioned, or when the sample is short relative to the number of assets [3,6,10,14]. This insight helps explain why unconstrained mean-variance solutions often appear fragile in practice. It also motivates the use of non-negativity constraints, leverage limits, box constraints, shrinkage estimators, robust formulations, and minimum-variance-oriented approaches [7,8,15,18,24,28]. In this literature, constraints are not merely institutional barriers; they sometimes act as regularization devices that reduce overfitting and improve the economic plausibility of portfolio weights [9,16,26]. A third strand addresses the computational and practical implementation of optimized portfolios. Research on cardinality constraints, heuristic algorithms, and fast large-scale optimization shows that the usefulness of a portfolio model depends not only on theoretical correctness, but also on whether the optimization can be solved efficiently and translated into implementable positions [4,5,7,22,26]. This concern is especially relevant when researchers compare unrestricted and restricted portfolios, because the observed performance difference may partly reflect the stabilizing role of admissibility constraints rather than the superiority of a purely mathematical objective [11,17]. A fourth strand links portfolio choice to macro-financial regimes. Recent research on geopolitical risk, oil-price shocks, and time-varying dependence suggests that the covariance structure of returns can shift materially when inflation, conflict, and liquidity stress interact [20,29,31]. Defensive sectors may offer stabilization in one regime yet lose their hedging role in another, while assets with high growth sensitivity may underperform when discount rates rise sharply. Related work on time-varying minimum-variance portfolios and decision-focused covariance estimation further indicates that the value of a covariance model should be judged by its portfolio consequences rather than by statistical fit alone [25,30]. Overall, the existing literature supports two inferences that motivate the present paper. First, there is no single universally superior portfolio model; performance depends on the interaction between data quality, market regime, and the investor’s constraint set [9,16,24]. Second, when shocks are heterogeneous and sector-specific, richer covariance information should be more useful than a one-factor approximation. The present study builds on this insight by comparing the Markowitz and Sharpe frameworks in a common stressed scenario and by emphasizing how regulatory and institutional constraints reshape the attainable efficient set [12,20,23]. 3. Methodology Consider a vector of asset returns r_t=(r_1t, r_2t, ..., r_Nt)' for N = 22 risky assets and a vector of portfolio weights w=(w_1, w_2, ..., w_N)'. In the Markowitz framework, the expected portfolio return is E(r_p) = w'µ and portfolio variance is σ_p^2 = w'Σw, where µ is the vector of expected returns and Σ is the full variance-covariance matrix [1,12]. The optimization problem may then be written either as minimizing w'Σw subject to a target return and budget condition, or as maximizing a risk-adjusted objective such as the Sharpe ratio subject to the same admissibility constraints [9,11]. Because Σ preserves all pairwise correlations, the Markowitz specification is capable of reflecting sectoral spillovers, relative hedging opportunities, and non-uniform shock transmission across the asset universe. In the Sharpe single-index model, each asset return is decomposed into a systematic market component and an idiosyncratic term: r_it = α_i + β_i r_mt + ε_it, where r_mt denotes the market return, β_i measures market sensitivity, and ε_it is asset-specific noise [2,23]. The covariance matrix is then approximated by β_iβ_jσ_m^2 for the systematic component plus diagonal residual variances for the idiosyncratic component. This representation dramatically reduces the number of parameters that must be estimated, which is advantageous when data are limited [3,14]. Its weakness is that residual cross-correlations are suppressed by construction; therefore, any co-movement not explained by the market factor may be understated in the final allocation [12,25]. To make the empirical comparison economically meaningful, five constraint regimes are examined: an unconstrained benchmark, a Regulation T-type regime allowing moderate leverage, a box-constrained regime limiting the magnitude of individual positions, a no-short-selling regime restricting all weights to non-negative values, and a conservative capital-preservation regime that favors lower-volatility exposures. These settings approximate common institutional environments ranging from highly flexible trading desks to more tightly supervised mandates [5,7,9]. The role of the constraint set is crucial because the same return and covariance inputs can produce very different efficient frontiers once hedging capacity, leverage, or concentration are restricted [4,17]. Portfolio performance is interpreted through three metrics: expected return, standard deviation, and the Sharpe ratio. Particular attention is given to the global minimum-variance portfolio and the tangency portfolio because they summarize two important investment logics: capital preservation and maximum risk-adjusted performance [10,18,28]. In addition, the comparative framework adopted here is intentionally scenario-based: rather than claiming universal dominance for one model, the study evaluates whether a richer covariance structure produces more stable and economically interpretable allocations under a stress regime characterized by uneven sector responses [20,24]. Since the original assignment reports selected headline outcomes rather than a full matrix of estimated weights and frontier points, the present article retains the reported quantitative anchors and develops a qualitative interpretation around them. This approach allows the paper to discuss model comparison and economic meaning without overstating numerical precision that is not fully documented in the available dataset description. At the same time, the methodology section explicitly acknowledges that full reproducibility would require the complete asset list, the sample window, the return frequency, the market proxy used for the single-index model, and the definition of the risk-free rate in the Sharpe-ratio calculation [18,23,30]. 4. Results and Discussion The results suggest that portfolio flexibility becomes especially valuable when the market is driven by asymmetric sectoral shocks rather than by a single uniform downturn. In the stress scenario underlying this study, energy-related assets benefit from commodity-price strength and therefore contribute both return enhancement and partial hedging against the inflationary pressure weighing on other sectors [20,29]. By contrast, high-beta growth and technology assets are more exposed to restrictive monetary conditions because their valuations are sensitive to discount-rate increases. Utilities, with relatively defensive cash-flow characteristics, play an important stabilizing role. This sector narrative helps explain why a covariance-rich specification can generate more informative allocations than a simpler one-factor representation [12,25,31]. Across the five constraint regimes, the no-short-selling case emerges as the least favorable from a risk-adjusted perspective. Once the investor is forbidden to take negative positions, direct hedging against sectors expected to suffer under the assumed macro regime becomes impossible. The portfolio must then absorb more downside exposure passively, which weakens the attainable Sharpe ratio and compresses the efficient set. This result is economically intuitive: constraints that appear conservative at the security level may become inefficient at the portfolio level when they eliminate offsetting positions that would otherwise reduce total variance [5,7,17]. By contrast, the Regulation T-type setting illustrates the value of moderate flexibility. Allowing limited leverage and selective short positions enables the optimizer to combine pro-cyclical and defensive exposures in a more balanced way. In the present context, such flexibility supports overweight positions in sectors benefiting from the crisis scenario while simultaneously reducing exposure to sectors that remain vulnerable to discount-rate or cost shocks. The improved Sharpe ratio reported for the more permissive cases therefore does not simply reflect aggressive risk-taking; it reflects better use of cross-sectional information embedded in the covariance structure [9,18,24]. The comparison between the Markowitz and single-index approaches also carries a methodological implication. The Markowitz model appears more suitable when sector divergence is economically meaningful because it preserves the full pattern of pairwise comovements [1,12]. The single-index model remains useful as a parsimonious benchmark and may perform reasonably well when broad market beta dominates return dynamics [2,23]. Nevertheless, under a regime in which energy, utilities, finance, and technology respond differently to inflation, conflict, and policy tightening, a single market factor is unlikely to capture the entire dependence structure. The superior performance of the Markowitz approach in this study is therefore consistent with the idea that richer covariance information is most valuable precisely when the economic environment is most uneven [20,25,30]. The reported numerical outcomes reinforce this interpretation. Even in a stressed setting, the minimum-risk portfolio still produces an expected monthly return of about 7.55%, while the maximum Sharpe ratio exceeds 1.01 under the more permissive constraint sets. These values should not be read as universal benchmarks; instead, they indicate that disciplined reallocation can preserve efficiency even when aggregate sentiment is weak. From an investment perspective, the results imply that energy may serve as a defensive growth component, that excessive concentration in long-duration technology assets should be treated cautiously under restrictive monetary conditions, and that utilities may function as partial substitutes when traditional safe-haven assets lose hedging effectiveness [20,29,31]. At the same time, the findings should be interpreted with caution. Because the available manuscript version does not yet report the full asset list, sample window, return-construction method, or out-of-sample validation, the present evidence is best viewed as internally coherent scenario analysis rather than a fully reproducible empirical test. This limitation does not invalidate the results, but it does indicate that the next stage of the research should prioritize transparency of inputs, robustness checks, and graphical presentation of the efficient frontier and portfolio weights. Recent studies on out-of-sample Sharpe-ratio estimation and decision-focused covariance learning suggest particularly useful avenues for strengthening the empirical design in subsequent revisions [28,30]. 5. Conclusions This study compares the Markowitz full mean-variance model with the Sharpe single-index model for a 22-asset universe under five portfolio-constraint regimes and shows that the interaction between model choice and implementation constraints is central to portfolio quality. The evidence suggests that when macro-financial stress affects sectors unevenly, the full covariance structure of the Markowitz framework provides a more informative basis for allocation than a one-factor approximation, while moderate flexibility in leverage and short-selling improves hedging capacity and risk-adjusted performance relative to highly restrictive regimes. Substantively, the paper indicates that energy exposure can function as a defensive growth component in the assumed crisis setting, that excessive concentration in high-beta technology should be avoided when discount rates remain elevated, and that utilities can partially stabilize portfolios when traditional safe-haven relationships weaken. At the same time, the study should be interpreted as a structured empirical scenario rather than a definitive out-of-sample test, because fuller disclosure of asset composition, sample design, and reproducible calculations is still needed. Future research can therefore build on these results by reporting complete frontier statistics, adding robustness checks and graphical evidence, and integrating robust or multifactor extensions to improve stability under parameter uncertainty. Declarations Acknowledgments: Not applicable. Author Contributions: Conceptualization, Y.S.; methodology, Y.S.; formal analysis, Y.S.; writing—original draft preparation, Y.S. The author has read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data supporting the findings of this study are available from the corresponding author upon reasonable request. Conflicts of Interest: The author declares no conflict of interest. References Fabozzi, F.J.; Huang, D.; Zhou, G. Robust portfolios: Contributions from operations research and finance. Ann. Oper. Res. 2010, 176, 191–220. Scutellà, M.G.; Recchia, R. Robust portfolio asset allocation and risk measures. 4OR-Q. J. Oper. Res. 2010, 8, 113–139. Lai, T.L.; Xing, H.; Chen, Z. Mean-variance portfolio optimization when means and covariances are unknown. Ann. Appl. Stat. 2011, 5, 798–823. Woodside-Oriakhi, M.; Lucas, C.; Beasley, J.E. Heuristic algorithms for the cardinality constrained efficient frontier. Eur. J. Oper. Res. 2011, 213, 538–550. Cesarone, F.; Scozzari, A.; Tardella, F. A new method for mean-variance portfolio optimization with cardinality constraints. Ann. Oper. Res. 2013, 205, 213–234. Fan, J.; Liao, Y.; Mincheva, M. Large covariance estimation by thresholding principal orthogonal complements. J. R. Stat. Soc. Ser. B 2013, 75, 603–680. Gao, J.; Li, D. Optimal cardinality constrained portfolio selection. Oper. Res. 2013, 61, 745–761. DeMiguel, V.; Martin-Utrera, A.; Nogales, F.J. Size matters: Optimal calibration of shrinkage estimators for portfolio selection. J. Bank. Finance 2013, 37, 3018–3034. Kolm, P.N.; Tütüncü, R.; Fabozzi, F.J. 60 years of portfolio optimization: Practical challenges and current trends. Eur. J. Oper. Res. 2014, 234, 356–371. Palczewski, A.; Palczewski, J. Theoretical and empirical estimates of mean-variance portfolio sensitivity. Eur. J. Oper. Res. 2014, 234, 402–410. Mansini, R.; Ogryczak, W.; Speranza, M.G. Twenty years of linear programming based portfolio optimization. Eur. J. Oper. Res. 2014, 234, 518–535. Zhang, Y.; Li, X.; Guo, S. Portfolio selection problems with Markowitz’s mean-variance framework: A review of literature. Fuzzy Optim. Decis. Mak. 2018, 17, 125–158. Dmytriv, S.; Bodnar, T.; Parolya, N.; Schmid, W. Tests for the weights of the global minimum variance portfolio in a high-dimensional setting. IEEE Trans. Signal Process. 2019, 67, 4479–4493. Sun, R.; Ma, T.; Liu, S.; Sathye, M. Improved covariance matrix estimation for portfolio risk measurement: A review. J. Risk Financ. Manag. 2019, 12, 1–33. Cai, T.T.; Hu, J.; Li, Y.; Zheng, X. High-dimensional minimum variance portfolio estimation based on high-frequency data. J. Econometrics 2020, 214, 482–494. Xidonas, P.; Steuer, R.; Hassapis, C. Robust portfolio optimization: A categorized bibliographic review. Ann. Oper. Res. 2020, 292, 533–552. Çela, E.; Hafner, S.; Mestel, R.; Pferschy, U. Mean-variance portfolio optimization based on ordinal information. J. Bank. Finance 2021, 122, 105989. Kircher, F.; Rösch, D. A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks. J. Bank. Finance 2021, 133, 106281. Lyle, M.R.; Yohn, T.L. Fundamental analysis and mean-variance optimal portfolios. Account. Rev. 2021, 96, 303–327. Caldara, D.; Iacoviello, M. Measuring geopolitical risk. Am. Econ. Rev. 2022, 112, 1194–1225. Du, J. Mean-variance portfolio optimization with deep learning based forecasts for cointegrated stocks. Expert Syst. Appl. 2022, 201, 117005. Leung, M.F.; Wang, J.; Ng, S.C.; Cheung, H.; Leung, Y. Cardinality-constrained portfolio selection based on evolutionary multitasking. Neural Netw. 2022, 145, 68–82. Mistry, J.; Khatwani, R.A. Examining the superiority of the Sharpe single-index model of portfolio selection: A study of the Indian mid-cap sector. Humanit. Soc. Sci. Commun. 2023, 10, 1–9. Petukhina, A.; Klochkov, Y.; Hardle, W.K.; Zhivotovskiy, N. Robustifying Markowitz. J. Econometrics 2024, 239, 105387. Fan, Q.; Wu, R.; Yang, Y.; Zhong, W. Time-varying minimum variance portfolio. J. Econometrics 2024, 239, 105339. Deng, W.; Ma, J.; Wang, L. A unified framework for fast large-scale portfolio optimization. Data Sci. Sci. 2024. doi:10.1080/26941899.2023.2295539. Cornuéjols, G.; Elçi, O.; Patil, V. Addressing estimation errors through robust portfolio optimization. Optim. Online 2024. Available online: https://optimization-online.org/2024/12/addressing-estimation-errors-through-robust-portfolio-optimization/ (accessed on 20 April 2026). Meng, X.; Cao, J.; Wang, T. Estimation of out-of-sample Sharpe ratio for high-dimensional portfolios. J. Am. Stat. Assoc. 2025. doi:10.1080/01621459.2025.2535757. Yilmazkuday, H. Geopolitical risk and stock prices. FIU Working Paper 2407, 2024. Kim, J.; Tae, I.; Lee, Y. Estimating covariance for global minimum variance portfolio: A decision-focused learning approach. Proc. ICAIF 2025. doi:10.1145/3768292.3770378. Enescu, A.G.; Manta, O.; Belascu, L. Geopolitical risks and global stock market dynamics. Int. J. Financ. Stud. 2026, 14, 85. Tables Table 1. Comparison of the Markowitz full model and the single-index model used in this study. Dimension Markowitz Full Model Single-Index Model Risk structure Uses the full variance-covariance matrix and preserves all pairwise asset correlations. Approximates covariance through market beta and residual variance. Estimation burden Higher dimensionality; more sensitive to input estimation error. Lower dimensionality; easier to implement for large universes. Crisis suitability Better suited to sector-specific or non-uniform shocks. More appropriate when common market risk dominates cross-asset comovement. Economic interpretation Emphasizes diversification through complete correlation structure. Separates systematic market risk from idiosyncratic risk. Expected behavior in this study Expected to outperform when energy, utilities, and technology respond differently to the stress scenario. May understate sector divergence because one market factor cannot capture all relative-price adjustments. Table 2. Economic interpretation of the five tested portfolio-constraint scenarios. Scenario Constraint logic Expected portfolio effect Interpretation in this study Unconstrained benchmark Allows long and short positions with no major allocation frictions. Highest theoretical efficiency but also highest sensitivity to parameter error. Useful as the reference frontier for judging the cost of regulatory constraints. Regulation T-type leverage Permits moderate leverage subject to margin discipline. Improves hedging flexibility while avoiding unlimited leverage. Reported to preserve a comparatively strong Sharpe ratio under market stress. Box-constrained allocation Caps extreme position sizes and limits concentration in individual assets. Reduces weight instability and improves practical implementability. Appropriate when estimation risk is non-trivial and portfolio weights would otherwise become extreme. No-short-selling Restricts all weights to non-negative values. Protects against excessive leverage but removes direct hedging capacity. Identified as the most damaging regime for risk-adjusted performance in the original analysis. Capital-preservation / defensive regime Prioritizes lower-volatility exposures and conservative allocation patterns. Enhances stability but may sacrifice upside and frontier efficiency. Supports the use of utilities and other defensive assets when traditional hedges weaken. Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9501117","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":628064452,"identity":"ae047ea0-7ea2-48bd-9541-984df56eaffd","order_by":0,"name":"Yiming Shen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+ElEQVRIiWNgGAWjYDACZhBhwJDAwMDY+OBDhQ0PP38D0VqYmw1nnEmTkZxxgDjLgFrY26R52w7bGDQk4Feq2857+DVPQV0ev3QjSMt5HgOGA4wfPubg1mJ2mC/NmsfgcLHknIPNlnPO3eYxZ25glpy5DZ8WHjNjHoMDiRtuJDbeeFN2m8ey4QAbMy9hLXWJ+28kNkjwsJ0Dak8gqMX4MY8Bc+IGicQmSZ62A0RpMWOcY3A4ccaNRFAgJ/NIzjjYjN8v588Yf3jzpy6xf0b6Q2BU2tnz8zcf/PARjxYgYJPiQRVgbMCrHgiYP/4gpGQUjIJRMApGNgAADIpXN+GsskAAAAAASUVORK5CYII=","orcid":"","institution":"Lingnan University","correspondingAuthor":true,"prefix":"","firstName":"Yiming","middleName":"","lastName":"Shen","suffix":""}],"badges":[],"createdAt":"2026-04-23 01:52:09","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9501117/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9501117/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108180920,"identity":"662c73ac-b695-4acd-8b41-bf8495d45458","added_by":"auto","created_at":"2026-04-30 08:55:14","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":173222,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9501117/v1/1d20a005-d1f2-4934-bb1d-80adf782a8f4.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eEmpirical Analysis of Optimized Portfolio Allocation Based on Markowitz and Single-Index Models: A Case Study of Group 3 Assets\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003ePortfolio selection remains one of the core questions in finance because any investor who allocates capital across multiple risky assets must trade off expected return against uncertainty. The classical contribution of Markowitz was to show that the relevant notion of risk is not the volatility of each asset viewed independently, but the variance of the portfolio as a whole, which depends on the covariance structure among constituent assets [1,12]. On this basis, mean-variance optimization provides a systematic way to trace the efficient frontier and to identify portfolios that minimize variance for a target return or maximize expected return for a given risk level [9,11].\u003c/p\u003e \u003cp\u003eLater developments, especially Sharpe\u0026rsquo;s single-index model and subsequent factor-based simplifications, sought to make portfolio construction more tractable by approximating the covariance matrix through a common market factor and asset-specific residual risk [2,23]. That simplification remains attractive in applied settings because it substantially reduces estimation burden, especially when the number of assets is large relative to the sample length [3,14]. However, the simplicity of the single-index framework comes with a cost: if cross-asset comovement is not dominated by a single market channel, the model may suppress economically meaningful heterogeneity across sectors and firms [12,16].\u003c/p\u003e \u003cp\u003eIn parallel, a large body of empirical work has shown that portfolio optimization in realistic settings is heavily shaped by estimation risk, sparsity requirements, and implementation frictions. Studies on shrinkage estimation, robust optimization, and high-dimensional covariance modeling consistently argue that theoretically optimal portfolios can become unstable when expected returns and covariance matrices are estimated with noise [8,10,14,16,24]. This line of research provides an important bridge between textbook portfolio theory and actual investment practice, where leverage constraints, turnover concerns, and concentration limits often determine whether an allocation is economically plausible [4,5,7,22,26].\u003c/p\u003e \u003cp\u003eThis issue becomes particularly important in periods of macro-financial stress. When inflation remains elevated, monetary policy is restrictive, energy prices are volatile, and geopolitical tensions intensify, return spillovers may propagate unevenly across sectors rather than through a single market factor alone [20,29,31]. In such circumstances, technology, utilities, finance, and energy assets may react differently to the same shock because their cash-flow duration, cost exposure, regulatory sensitivity, and demand conditions differ. The empirical relevance of portfolio optimization therefore depends not only on solving a mathematical problem, but also on selecting a covariance representation that is sufficiently rich for the prevailing regime [17,25].\u003c/p\u003e \u003cp\u003eAgainst this background, the present study examines a universe of 22 risky assets labeled Group 3 and compares optimal allocations generated by the Markowitz full model and the Sharpe single-index model under five portfolio-constraint regimes. The paper has three objectives. First, it evaluates whether the richer covariance structure of the Markowitz framework yields more plausible allocations under a stressed scenario. Second, it investigates how common implementation constraints such as leverage limits, box constraints, and short-selling restrictions affect attainable efficiency. Third, it derives practical allocation implications for investors facing an environment in which traditional broad-market diversification may weaken. Relative to the previous version, this revised manuscript strengthens the theoretical discussion, adds clearer links between claims and references, and expands the economic interpretation of the reported results [9,18,24].\u003c/p\u003e"},{"header":"2. Literature Review","content":"\u003cp\u003eThe first strand of literature relevant to this study concerns the evolution of mean-variance theory and its practical refinements. Classical portfolio selection research established the efficient frontier and demonstrated that diversification benefits are generated by the interaction among asset returns rather than by holding a large number of securities mechanically [1]. Subsequent work expanded this framework in several directions, including single-index representations, cardinality-constrained optimization, shrinkage estimation, high-dimensional covariance estimation, and robust optimization [2,4,5,8,12,16]. Together, these studies show that portfolio optimization remains theoretically compelling, but highly sensitive to the quality of estimated inputs and to the frictions imposed on admissible weights [9,11].\u003c/p\u003e \u003cp\u003eA second strand emphasizes estimation error and model instability. Empirical studies repeatedly find that optimized weights can become extreme when expected returns are measured with noise, when covariance matrices are poorly conditioned, or when the sample is short relative to the number of assets [3,6,10,14]. This insight helps explain why unconstrained mean-variance solutions often appear fragile in practice. It also motivates the use of non-negativity constraints, leverage limits, box constraints, shrinkage estimators, robust formulations, and minimum-variance-oriented approaches [7,8,15,18,24,28]. In this literature, constraints are not merely institutional barriers; they sometimes act as regularization devices that reduce overfitting and improve the economic plausibility of portfolio weights [9,16,26].\u003c/p\u003e \u003cp\u003eA third strand addresses the computational and practical implementation of optimized portfolios. Research on cardinality constraints, heuristic algorithms, and fast large-scale optimization shows that the usefulness of a portfolio model depends not only on theoretical correctness, but also on whether the optimization can be solved efficiently and translated into implementable positions [4,5,7,22,26]. This concern is especially relevant when researchers compare unrestricted and restricted portfolios, because the observed performance difference may partly reflect the stabilizing role of admissibility constraints rather than the superiority of a purely mathematical objective [11,17].\u003c/p\u003e \u003cp\u003eA fourth strand links portfolio choice to macro-financial regimes. Recent research on geopolitical risk, oil-price shocks, and time-varying dependence suggests that the covariance structure of returns can shift materially when inflation, conflict, and liquidity stress interact [20,29,31]. Defensive sectors may offer stabilization in one regime yet lose their hedging role in another, while assets with high growth sensitivity may underperform when discount rates rise sharply. Related work on time-varying minimum-variance portfolios and decision-focused covariance estimation further indicates that the value of a covariance model should be judged by its portfolio consequences rather than by statistical fit alone [25,30].\u003c/p\u003e \u003cp\u003eOverall, the existing literature supports two inferences that motivate the present paper. First, there is no single universally superior portfolio model; performance depends on the interaction between data quality, market regime, and the investor\u0026rsquo;s constraint set [9,16,24]. Second, when shocks are heterogeneous and sector-specific, richer covariance information should be more useful than a one-factor approximation. The present study builds on this insight by comparing the Markowitz and Sharpe frameworks in a common stressed scenario and by emphasizing how regulatory and institutional constraints reshape the attainable efficient set [12,20,23].\u003c/p\u003e"},{"header":"3. Methodology","content":"\u003cp\u003eConsider a vector of asset returns r_t=(r_1t, r_2t, ..., r_Nt)' for N\u0026thinsp;=\u0026thinsp;22 risky assets and a vector of portfolio weights w=(w_1, w_2, ..., w_N)'. In the Markowitz framework, the expected portfolio return is E(r_p)\u0026thinsp;=\u0026thinsp;w'\u0026micro; and portfolio variance is σ_p^2\u0026thinsp;=\u0026thinsp;w'Σw, where \u0026micro; is the vector of expected returns and Σ is the full variance-covariance matrix [1,12]. The optimization problem may then be written either as minimizing w'Σw subject to a target return and budget condition, or as maximizing a risk-adjusted objective such as the Sharpe ratio subject to the same admissibility constraints [9,11]. Because Σ preserves all pairwise correlations, the Markowitz specification is capable of reflecting sectoral spillovers, relative hedging opportunities, and non-uniform shock transmission across the asset universe.\u003c/p\u003e \u003cp\u003eIn the Sharpe single-index model, each asset return is decomposed into a systematic market component and an idiosyncratic term: r_it\u0026thinsp;=\u0026thinsp;α_i\u0026thinsp;+\u0026thinsp;β_i r_mt\u0026thinsp;+\u0026thinsp;ε_it, where r_mt denotes the market return, β_i measures market sensitivity, and ε_it is asset-specific noise [2,23]. The covariance matrix is then approximated by β_iβ_jσ_m^2 for the systematic component plus diagonal residual variances for the idiosyncratic component. This representation dramatically reduces the number of parameters that must be estimated, which is advantageous when data are limited [3,14]. Its weakness is that residual cross-correlations are suppressed by construction; therefore, any co-movement not explained by the market factor may be understated in the final allocation [12,25].\u003c/p\u003e \u003cp\u003eTo make the empirical comparison economically meaningful, five constraint regimes are examined: an unconstrained benchmark, a Regulation T-type regime allowing moderate leverage, a box-constrained regime limiting the magnitude of individual positions, a no-short-selling regime restricting all weights to non-negative values, and a conservative capital-preservation regime that favors lower-volatility exposures. These settings approximate common institutional environments ranging from highly flexible trading desks to more tightly supervised mandates [5,7,9]. The role of the constraint set is crucial because the same return and covariance inputs can produce very different efficient frontiers once hedging capacity, leverage, or concentration are restricted [4,17].\u003c/p\u003e \u003cp\u003ePortfolio performance is interpreted through three metrics: expected return, standard deviation, and the Sharpe ratio. Particular attention is given to the global minimum-variance portfolio and the tangency portfolio because they summarize two important investment logics: capital preservation and maximum risk-adjusted performance [10,18,28]. In addition, the comparative framework adopted here is intentionally scenario-based: rather than claiming universal dominance for one model, the study evaluates whether a richer covariance structure produces more stable and economically interpretable allocations under a stress regime characterized by uneven sector responses [20,24].\u003c/p\u003e \u003cp\u003eSince the original assignment reports selected headline outcomes rather than a full matrix of estimated weights and frontier points, the present article retains the reported quantitative anchors and develops a qualitative interpretation around them. This approach allows the paper to discuss model comparison and economic meaning without overstating numerical precision that is not fully documented in the available dataset description. At the same time, the methodology section explicitly acknowledges that full reproducibility would require the complete asset list, the sample window, the return frequency, the market proxy used for the single-index model, and the definition of the risk-free rate in the Sharpe-ratio calculation [18,23,30].\u003c/p\u003e"},{"header":"4. Results and Discussion","content":"\u003cp\u003eThe results suggest that portfolio flexibility becomes especially valuable when the market is driven by asymmetric sectoral shocks rather than by a single uniform downturn. In the stress scenario underlying this study, energy-related assets benefit from commodity-price strength and therefore contribute both return enhancement and partial hedging against the inflationary pressure weighing on other sectors [20,29]. By contrast, high-beta growth and technology assets are more exposed to restrictive monetary conditions because their valuations are sensitive to discount-rate increases. Utilities, with relatively defensive cash-flow characteristics, play an important stabilizing role. This sector narrative helps explain why a covariance-rich specification can generate more informative allocations than a simpler one-factor representation [12,25,31].\u003c/p\u003e \u003cp\u003eAcross the five constraint regimes, the no-short-selling case emerges as the least favorable from a risk-adjusted perspective. Once the investor is forbidden to take negative positions, direct hedging against sectors expected to suffer under the assumed macro regime becomes impossible. The portfolio must then absorb more downside exposure passively, which weakens the attainable Sharpe ratio and compresses the efficient set. This result is economically intuitive: constraints that appear conservative at the security level may become inefficient at the portfolio level when they eliminate offsetting positions that would otherwise reduce total variance [5,7,17].\u003c/p\u003e \u003cp\u003eBy contrast, the Regulation T-type setting illustrates the value of moderate flexibility. Allowing limited leverage and selective short positions enables the optimizer to combine pro-cyclical and defensive exposures in a more balanced way. In the present context, such flexibility supports overweight positions in sectors benefiting from the crisis scenario while simultaneously reducing exposure to sectors that remain vulnerable to discount-rate or cost shocks. The improved Sharpe ratio reported for the more permissive cases therefore does not simply reflect aggressive risk-taking; it reflects better use of cross-sectional information embedded in the covariance structure [9,18,24].\u003c/p\u003e \u003cp\u003eThe comparison between the Markowitz and single-index approaches also carries a methodological implication. The Markowitz model appears more suitable when sector divergence is economically meaningful because it preserves the full pattern of pairwise comovements [1,12]. The single-index model remains useful as a parsimonious benchmark and may perform reasonably well when broad market beta dominates return dynamics [2,23]. Nevertheless, under a regime in which energy, utilities, finance, and technology respond differently to inflation, conflict, and policy tightening, a single market factor is unlikely to capture the entire dependence structure. The superior performance of the Markowitz approach in this study is therefore consistent with the idea that richer covariance information is most valuable precisely when the economic environment is most uneven [20,25,30].\u003c/p\u003e \u003cp\u003eThe reported numerical outcomes reinforce this interpretation. Even in a stressed setting, the minimum-risk portfolio still produces an expected monthly return of about 7.55%, while the maximum Sharpe ratio exceeds 1.01 under the more permissive constraint sets. These values should not be read as universal benchmarks; instead, they indicate that disciplined reallocation can preserve efficiency even when aggregate sentiment is weak. From an investment perspective, the results imply that energy may serve as a defensive growth component, that excessive concentration in long-duration technology assets should be treated cautiously under restrictive monetary conditions, and that utilities may function as partial substitutes when traditional safe-haven assets lose hedging effectiveness [20,29,31].\u003c/p\u003e \u003cp\u003eAt the same time, the findings should be interpreted with caution. Because the available manuscript version does not yet report the full asset list, sample window, return-construction method, or out-of-sample validation, the present evidence is best viewed as internally coherent scenario analysis rather than a fully reproducible empirical test. This limitation does not invalidate the results, but it does indicate that the next stage of the research should prioritize transparency of inputs, robustness checks, and graphical presentation of the efficient frontier and portfolio weights. Recent studies on out-of-sample Sharpe-ratio estimation and decision-focused covariance learning suggest particularly useful avenues for strengthening the empirical design in subsequent revisions [28,30].\u003c/p\u003e"},{"header":"5. Conclusions","content":"\u003cp\u003eThis study compares the Markowitz full mean-variance model with the Sharpe single-index model for a 22-asset universe under five portfolio-constraint regimes and shows that the interaction between model choice and implementation constraints is central to portfolio quality. The evidence suggests that when macro-financial stress affects sectors unevenly, the full covariance structure of the Markowitz framework provides a more informative basis for allocation than a one-factor approximation, while moderate flexibility in leverage and short-selling improves hedging capacity and risk-adjusted performance relative to highly restrictive regimes. Substantively, the paper indicates that energy exposure can function as a defensive growth component in the assumed crisis setting, that excessive concentration in high-beta technology should be avoided when discount rates remain elevated, and that utilities can partially stabilize portfolios when traditional safe-haven relationships weaken. At the same time, the study should be interpreted as a structured empirical scenario rather than a definitive out-of-sample test, because fuller disclosure of asset composition, sample design, and reproducible calculations is still needed. Future research can therefore build on these results by reporting complete frontier statistics, adding robustness checks and graphical evidence, and integrating robust or multifactor extensions to improve stability under parameter uncertainty.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgments:\u0026nbsp;\u003c/strong\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions:\u003c/strong\u003e Conceptualization, Y.S.; methodology, Y.S.; formal analysis, Y.S.; writing\u0026mdash;original draft preparation, Y.S. The author has read and agreed to the published version of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding:\u0026nbsp;\u003c/strong\u003eThis research received no external funding.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eInstitutional Review Board Statement:\u003c/strong\u003e Not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eInformed Consent Statement:\u003c/strong\u003e Not applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability Statement:\u0026nbsp;\u003c/strong\u003eThe data supporting the findings of this study are available from the corresponding author upon reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflicts of Interest:\u0026nbsp;\u003c/strong\u003eThe author declares no conflict of interest.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eFabozzi, F.J.; Huang, D.; Zhou, G. Robust portfolios: Contributions from operations research and finance. Ann. Oper. Res. 2010, 176, 191\u0026ndash;220.\u003c/li\u003e\n \u003cli\u003eScutell\u0026agrave;, M.G.; Recchia, R. Robust portfolio asset allocation and risk measures. 4OR-Q. J. Oper. Res. 2010, 8, 113\u0026ndash;139.\u003c/li\u003e\n \u003cli\u003eLai, T.L.; Xing, H.; Chen, Z. Mean-variance portfolio optimization when means and covariances are unknown. Ann. Appl. Stat. 2011, 5, 798\u0026ndash;823.\u003c/li\u003e\n \u003cli\u003eWoodside-Oriakhi, M.; Lucas, C.; Beasley, J.E. Heuristic algorithms for the cardinality constrained efficient frontier. Eur. J. Oper. Res. 2011, 213, 538\u0026ndash;550.\u003c/li\u003e\n \u003cli\u003eCesarone, F.; Scozzari, A.; Tardella, F. A new method for mean-variance portfolio optimization with cardinality constraints. Ann. Oper. Res. 2013, 205, 213\u0026ndash;234.\u003c/li\u003e\n \u003cli\u003eFan, J.; Liao, Y.; Mincheva, M. Large covariance estimation by thresholding principal orthogonal complements. J. R. Stat. Soc. Ser. B 2013, 75, 603\u0026ndash;680.\u003c/li\u003e\n \u003cli\u003eGao, J.; Li, D. Optimal cardinality constrained portfolio selection. Oper. Res. 2013, 61, 745\u0026ndash;761.\u003c/li\u003e\n \u003cli\u003eDeMiguel, V.; Martin-Utrera, A.; Nogales, F.J. Size matters: Optimal calibration of shrinkage estimators for portfolio selection. J. Bank. Finance 2013, 37, 3018\u0026ndash;3034.\u003c/li\u003e\n \u003cli\u003eKolm, P.N.; T\u0026uuml;t\u0026uuml;nc\u0026uuml;, R.; Fabozzi, F.J. 60 years of portfolio optimization: Practical challenges and current trends. Eur. J. Oper. Res. 2014, 234, 356\u0026ndash;371.\u003c/li\u003e\n \u003cli\u003ePalczewski, A.; Palczewski, J. Theoretical and empirical estimates of mean-variance portfolio sensitivity. Eur. J. Oper. Res. 2014, 234, 402\u0026ndash;410.\u003c/li\u003e\n \u003cli\u003eMansini, R.; Ogryczak, W.; Speranza, M.G. Twenty years of linear programming based portfolio optimization. Eur. J. Oper. Res. 2014, 234, 518\u0026ndash;535.\u003c/li\u003e\n \u003cli\u003eZhang, Y.; Li, X.; Guo, S. Portfolio selection problems with Markowitz\u0026rsquo;s mean-variance framework: A review of literature. Fuzzy Optim. Decis. Mak. 2018, 17, 125\u0026ndash;158.\u003c/li\u003e\n \u003cli\u003eDmytriv, S.; Bodnar, T.; Parolya, N.; Schmid, W. Tests for the weights of the global minimum variance portfolio in a high-dimensional setting. IEEE Trans. Signal Process. 2019, 67, 4479\u0026ndash;4493.\u003c/li\u003e\n \u003cli\u003eSun, R.; Ma, T.; Liu, S.; Sathye, M. Improved covariance matrix estimation for portfolio risk measurement: A review. J. Risk Financ. Manag. 2019, 12, 1\u0026ndash;33.\u003c/li\u003e\n \u003cli\u003eCai, T.T.; Hu, J.; Li, Y.; Zheng, X. High-dimensional minimum variance portfolio estimation based on high-frequency data. J. Econometrics 2020, 214, 482\u0026ndash;494.\u003c/li\u003e\n \u003cli\u003eXidonas, P.; Steuer, R.; Hassapis, C. Robust portfolio optimization: A categorized bibliographic review. Ann. Oper. Res. 2020, 292, 533\u0026ndash;552.\u003c/li\u003e\n \u003cli\u003e\u0026Ccedil;ela, E.; Hafner, S.; Mestel, R.; Pferschy, U. Mean-variance portfolio optimization based on ordinal information. J. Bank. Finance 2021, 122, 105989.\u003c/li\u003e\n \u003cli\u003eKircher, F.; R\u0026ouml;sch, D. A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks. J. Bank. Finance 2021, 133, 106281.\u003c/li\u003e\n \u003cli\u003eLyle, M.R.; Yohn, T.L. Fundamental analysis and mean-variance optimal portfolios. Account. Rev. 2021, 96, 303\u0026ndash;327.\u003c/li\u003e\n \u003cli\u003eCaldara, D.; Iacoviello, M. Measuring geopolitical risk. Am. Econ. Rev. 2022, 112, 1194\u0026ndash;1225.\u003c/li\u003e\n \u003cli\u003eDu, J. Mean-variance portfolio optimization with deep learning based forecasts for cointegrated stocks. Expert Syst. Appl. 2022, 201, 117005.\u003c/li\u003e\n \u003cli\u003eLeung, M.F.; Wang, J.; Ng, S.C.; Cheung, H.; Leung, Y. Cardinality-constrained portfolio selection based on evolutionary multitasking. Neural Netw. 2022, 145, 68\u0026ndash;82.\u003c/li\u003e\n \u003cli\u003eMistry, J.; Khatwani, R.A. Examining the superiority of the Sharpe single-index model of portfolio selection: A study of the Indian mid-cap sector. Humanit. Soc. Sci. Commun. 2023, 10, 1\u0026ndash;9.\u003c/li\u003e\n \u003cli\u003ePetukhina, A.; Klochkov, Y.; Hardle, W.K.; Zhivotovskiy, N. Robustifying Markowitz. J. Econometrics 2024, 239, 105387.\u003c/li\u003e\n \u003cli\u003eFan, Q.; Wu, R.; Yang, Y.; Zhong, W. Time-varying minimum variance portfolio. J. Econometrics 2024, 239, 105339.\u003c/li\u003e\n \u003cli\u003eDeng, W.; Ma, J.; Wang, L. A unified framework for fast large-scale portfolio optimization. Data Sci. Sci. 2024. doi:10.1080/26941899.2023.2295539.\u003c/li\u003e\n \u003cli\u003eCornu\u0026eacute;jols, G.; El\u0026ccedil;i, O.; Patil, V. Addressing estimation errors through robust portfolio optimization. Optim. Online 2024. Available online: https://optimization-online.org/2024/12/addressing-estimation-errors-through-robust-portfolio-optimization/ (accessed on 20 April 2026).\u003c/li\u003e\n \u003cli\u003eMeng, X.; Cao, J.; Wang, T. Estimation of out-of-sample Sharpe ratio for high-dimensional portfolios. J. Am. Stat. Assoc. 2025. doi:10.1080/01621459.2025.2535757.\u003c/li\u003e\n \u003cli\u003eYilmazkuday, H. Geopolitical risk and stock prices. FIU Working Paper 2407, 2024.\u003c/li\u003e\n \u003cli\u003eKim, J.; Tae, I.; Lee, Y. Estimating covariance for global minimum variance portfolio: A decision-focused learning approach. Proc. ICAIF 2025. doi:10.1145/3768292.3770378.\u003c/li\u003e\n \u003cli\u003eEnescu, A.G.; Manta, O.; Belascu, L. Geopolitical risks and global stock market dynamics. Int. J. Financ. Stud. 2026, 14, 85.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Tables","content":"\u003cp\u003eTable 1. Comparison of the Markowitz full model and the single-index model used in this study.\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 125px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eDimension\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eMarkowitz Full Model\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eSingle-Index Model\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 125px;\"\u003e\n \u003cp\u003eRisk structure\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eUses the full variance-covariance matrix and preserves all pairwise asset correlations.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eApproximates covariance through market beta and residual variance.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 125px;\"\u003e\n \u003cp\u003eEstimation burden\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eHigher dimensionality; more sensitive to input estimation error.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eLower dimensionality; easier to implement for large universes.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 125px;\"\u003e\n \u003cp\u003eCrisis suitability\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eBetter suited to sector-specific or non-uniform shocks.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eMore appropriate when common market risk dominates cross-asset comovement.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 125px;\"\u003e\n \u003cp\u003eEconomic interpretation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eEmphasizes diversification through complete correlation structure.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eSeparates systematic market risk from idiosyncratic risk.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 125px;\"\u003e\n \u003cp\u003eExpected behavior in this study\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eExpected to outperform when energy, utilities, and technology respond differently to the stress scenario.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 238px;\"\u003e\n \u003cp\u003eMay understate sector divergence because one market factor cannot capture all relative-price adjustments.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eTable 2. Economic interpretation of the five tested portfolio-constraint scenarios.\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 132px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eScenario\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 170px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eConstraint logic\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 159px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eExpected portfolio effect\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 163px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eInterpretation in this study\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 132px;\"\u003e\n \u003cp\u003eUnconstrained benchmark\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 170px;\"\u003e\n \u003cp\u003eAllows long and short positions with no major allocation frictions.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 159px;\"\u003e\n \u003cp\u003eHighest theoretical efficiency but also highest sensitivity to parameter error.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 163px;\"\u003e\n \u003cp\u003eUseful as the reference frontier for judging the cost of regulatory constraints.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 132px;\"\u003e\n \u003cp\u003eRegulation T-type leverage\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 170px;\"\u003e\n \u003cp\u003ePermits moderate leverage subject to margin discipline.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 159px;\"\u003e\n \u003cp\u003eImproves hedging flexibility while avoiding unlimited leverage.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 163px;\"\u003e\n \u003cp\u003eReported to preserve a comparatively strong Sharpe ratio under market stress.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 132px;\"\u003e\n \u003cp\u003eBox-constrained allocation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 170px;\"\u003e\n \u003cp\u003eCaps extreme position sizes and limits concentration in individual assets.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 159px;\"\u003e\n \u003cp\u003eReduces weight instability and improves practical implementability.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 163px;\"\u003e\n \u003cp\u003eAppropriate when estimation risk is non-trivial and portfolio weights would otherwise become extreme.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 132px;\"\u003e\n \u003cp\u003eNo-short-selling\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 170px;\"\u003e\n \u003cp\u003eRestricts all weights to non-negative values.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 159px;\"\u003e\n \u003cp\u003eProtects against excessive leverage but removes direct hedging capacity.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 163px;\"\u003e\n \u003cp\u003eIdentified as the most damaging regime for risk-adjusted performance in the original analysis.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 132px;\"\u003e\n \u003cp\u003eCapital-preservation / defensive regime\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 170px;\"\u003e\n \u003cp\u003ePrioritizes lower-volatility exposures and conservative allocation patterns.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 159px;\"\u003e\n \u003cp\u003eEnhances stability but may sacrifice upside and frontier efficiency.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 163px;\"\u003e\n \u003cp\u003eSupports the use of utilities and other defensive assets when traditional hedges weaken.\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Lingnan University","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"portfolio optimization, Markowitz model, single-index model, geopolitical risk, sector rotation, short-selling constraints, asset allocation","lastPublishedDoi":"10.21203/rs.3.rs-9501117/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9501117/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study investigates the construction of optimal portfolios using the Markowitz full mean-variance model and the Sharpe single-index model for a diversified universe of 22 risky assets referred to as Group 3. The analysis is framed as a stressed macro-financial scenario in which rising geopolitical risk, elevated energy prices, restrictive monetary conditions, and weak broad-market performance jointly reshape the covariance structure of returns. By comparing five portfolio constraint settings, including unconstrained allocation, Regulation T-type leverage, box constraints, and no-short-selling conditions, the paper evaluates how institutional trading limits alter portfolio efficiency, hedging capacity, and risk-adjusted performance. The results indicate that moderate flexibility in long-short positioning materially improves portfolio resilience under adverse market conditions, whereas severe constraints reduce the attainable Sharpe ratio and limit the investor's ability to offset sector-specific shocks. Consistent with the literature on estimation error and constrained optimization, the Markowitz framework captures cross-sector covariance patterns more effectively than the single-index specification when market dislocations are driven by non-uniform sectoral shocks. Within the study dataset, the minimum-risk portfolio still delivers a monthly expected return of approximately 7.55%, while the maximum Sharpe ratio exceeds 1.01 under relatively permissive constraints. The findings suggest that, under high-uncertainty environments, portfolio design should emphasize energy exposure as a defensive growth component, reduce excessive concentration in high-beta technology assets, and employ defensive utilities as substitutes for weakening traditional safe-haven assets. Overall, the paper demonstrates that modern portfolio theory remains useful in crisis conditions when model selection and constraint design are aligned with the underlying economic regime.\u003c/p\u003e","manuscriptTitle":"Empirical Analysis of Optimized Portfolio Allocation Based on Markowitz and Single-Index Models: A Case Study of Group 3 Assets","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-27 10:12:00","doi":"10.21203/rs.3.rs-9501117/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"92072cb4-0b03-4665-9d0d-586fba739650","owner":[],"postedDate":"April 27th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":66849592,"name":"International Economics"}],"tags":[],"updatedAt":"2026-04-27T10:12:01+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-27 10:12:00","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9501117","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9501117","identity":"rs-9501117","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}