Construction of the efficient frontier for portfolios combining risky and risk-free assets: an MPT-Based optimization model and visualization analysis

Research Article
Open access

Construction of the efficient frontier for portfolios combining risky and risk-free assets: an MPT-Based optimization model and visualization analysis

Jiarui Zhou 1*
  • 1 Xi’an Jiaotong-Liverpool University, Suzhou, China    
  • *corresponding author H3376351035@163.com
JFBA Vol.2 Issue 2
ISSN (Print): 3049-5776
ISSN (Online): 3049-5768

Abstract

The volatility of financial markets has driven the diversification of investment instruments, encouraging investors to keep improving their portfolio-picking techniques. While numerous studies based on Modern Portfolio Theory (MPT) have developed accepted methods for determining optimal portfolios, not enough research has been done on the visual graphical analysis of risk preferences to accommodate diverse investors. Furthermore, by including risk-free assets in the analysis, this study presents an innovative methodology. This study's main goal is to find and analyze the risk portfolio frontier while examining the complementing of risk-free investments. Under idealized assumptions, portfolio returns and risks are formulated and solved as equality-constrained optimization problems, yielding frontier portfolios. The risk-efficient frontier and preference levels are depicted graphically, with explicit discussion of unfettered short-selling possibilities. The inclusion of risk-free assets further broadens the model’s practical applicability. The framework's viability is empirically validated using historical data from twelve stocks. These results illustrate investors can use the efficient frontier as a foundation to match portfolio selections to their own risk tolerances, providing practical guidance for adaptable wealth management.

Keywords:

Modern Portfolio Theory, risk portfolio optimization, efficient frontier, risk preference analysis, frontier portfolio with risk-free assets

Zhou,J. (2025). Construction of the efficient frontier for portfolios combining risky and risk-free assets: an MPT-Based optimization model and visualization analysis. Journal of Fintech and Business Analysis,2(2),1-9.
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References

[1]. Joemon, B., Ghazanfar, M. A., Azam, M. A., Jhanjhi, N. Z., & Khan, A. A. (2023). Novel heuristics for Stock portfolio optimization using machine learning and Modern Portfolio Theory. 2023 International Conference on Business Analytics for Technology and Security (ICBATS), Business Analytics for Technology and Security (ICBATS), 2023 International Conference On, 1-6. https://doi.org/10.1109/ICBATS57792.2023.10111321

[2]. Dubach, P. (2022). A Python integration of practical asset allocation based on modern portfolio theory and its advancements.

[3]. Yang, R. (2011). Optimizing the Real Estate Portfolio Decision Model Based on Modern Portfolio Theory. 2011 Fourth International Joint Conference on Computational Sciences and Optimization, Computational Sciences and Optimization (CSO), 2011 Fourth International Joint Conference On, 1232–1235. https://doi-org-s.elink.xjtlu.edu.cn:443/10.1109/CSO.2011.195

[4]. Cinciulescu, D. (2024). The Impact of Tail Risk and Black Swan Events on Modern Portfolio Theory. A Reassessment of Risk Assumptions in Extreme Market Conditions. Young Economists Journal / Revista Tinerilor Economisti, 21(43), 83–96.

[5]. Silva Filho, M. C., Monteiro, C. C., Inácio, P. R. M., & Freire, M. M. (2024). A Distributed Virtual-Machine Placement and Migration Approach Based on Modern Portfolio Theory. Journal of Network and Systems Management, 32(1). https://doi-org-s.elink.xjtlu.edu.cn:443/10.1007/s10922-023-09775-8

[6]. V., J. B., & H. R., T. (2024). Construction of Optimum Portfolio Using Modern Portfolio Theory and Sharpe’s Single Index Model. Prajnān, 52(4), 335–351.

[7]. Ngo, V. M., Nguyen, H. H., & Van Nguyen, P. (2023). Does reinforcement learning outperform deep learning and traditional portfolio optimization models in frontier and developed financial markets? Research in International Business and Finance, 65. https://doi-org-s.elink.xjtlu.edu.cn:443/10.1016/j.ribaf.2023.101936

[8]. Maqbool, F., & Husnain, M. (2022). Optimising the Investor’s Portfolio through Modern Portfolio Theory: Empirical Evidence from Pakistan. NUML International Journal of Business & Management, 17(2), 1–15. https://doi-org-s.elink.xjtlu.edu.cn:443/10.52015/nijbm.v17i2.138

[9]. Chakraborty, S., & Patel, A. K. (2018). Construction of Optimal Portfolio Using Sharpe’s Single Index Model and Markowitz Model: An Empirical Study on Nifty 50 Stock. Journal of General Management Research, 5(1), 86–103.

Cite this article

Zhou,J. (2025). Construction of the efficient frontier for portfolios combining risky and risk-free assets: an MPT-Based optimization model and visualization analysis. Journal of Fintech and Business Analysis,2(2),1-9.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Journal：Journal of Fintech and Business Analysis

Volume number: Vol.2
Issue number: Issue 2
ISSN：3049-5768(Print) / 3049-5776(Online)

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References

[1]. Joemon, B., Ghazanfar, M. A., Azam, M. A., Jhanjhi, N. Z., & Khan, A. A. (2023). Novel heuristics for Stock portfolio optimization using machine learning and Modern Portfolio Theory. 2023 International Conference on Business Analytics for Technology and Security (ICBATS), Business Analytics for Technology and Security (ICBATS), 2023 International Conference On, 1-6. https://doi.org/10.1109/ICBATS57792.2023.10111321

[2]. Dubach, P. (2022). A Python integration of practical asset allocation based on modern portfolio theory and its advancements.

[3]. Yang, R. (2011). Optimizing the Real Estate Portfolio Decision Model Based on Modern Portfolio Theory. 2011 Fourth International Joint Conference on Computational Sciences and Optimization, Computational Sciences and Optimization (CSO), 2011 Fourth International Joint Conference On, 1232–1235. https://doi-org-s.elink.xjtlu.edu.cn:443/10.1109/CSO.2011.195

[4]. Cinciulescu, D. (2024). The Impact of Tail Risk and Black Swan Events on Modern Portfolio Theory. A Reassessment of Risk Assumptions in Extreme Market Conditions. Young Economists Journal / Revista Tinerilor Economisti, 21(43), 83–96.

[5]. Silva Filho, M. C., Monteiro, C. C., Inácio, P. R. M., & Freire, M. M. (2024). A Distributed Virtual-Machine Placement and Migration Approach Based on Modern Portfolio Theory. Journal of Network and Systems Management, 32(1). https://doi-org-s.elink.xjtlu.edu.cn:443/10.1007/s10922-023-09775-8

[6]. V., J. B., & H. R., T. (2024). Construction of Optimum Portfolio Using Modern Portfolio Theory and Sharpe’s Single Index Model. Prajnān, 52(4), 335–351.

[7]. Ngo, V. M., Nguyen, H. H., & Van Nguyen, P. (2023). Does reinforcement learning outperform deep learning and traditional portfolio optimization models in frontier and developed financial markets? Research in International Business and Finance, 65. https://doi-org-s.elink.xjtlu.edu.cn:443/10.1016/j.ribaf.2023.101936

[8]. Maqbool, F., & Husnain, M. (2022). Optimising the Investor’s Portfolio through Modern Portfolio Theory: Empirical Evidence from Pakistan. NUML International Journal of Business & Management, 17(2), 1–15. https://doi-org-s.elink.xjtlu.edu.cn:443/10.52015/nijbm.v17i2.138

[9]. Chakraborty, S., & Patel, A. K. (2018). Construction of Optimal Portfolio Using Sharpe’s Single Index Model and Markowitz Model: An Empirical Study on Nifty 50 Stock. Journal of General Management Research, 5(1), 86–103.

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