Deriving a General Formula for the Global Minimum-Variance Point of a Two-Asset Markowitz Portfolio

Author(s)	Mr. Aryaveer Jain
Country	India
Abstract	This paper derives a general formula for the Global Minimum Variance (GMV) point of a two-security portfolio within the Markowitz mean–variance framework. Using the method of Lagrange multipliers together with a matrix representation of the optimisation problem, the paper obtains closed-form expressions for the optimal portfolio weights and for the efficient frontier. The model, using five-year average data (2019–2024), is then applied to a case study of Tata Consultancy Services (TCS) and Indian Tobacco Company (ITC). Using approximate weights of 43.7% in TCS and 56.3% in ITC, the efficient frontier suggests a GMV expected return of about 16%. Additionally, the paper evaluates the practical limitations of the Markowitz model in real financial markets and the impact of numerical approximations in the calculations.
Keywords	Markowitz portfolio theory, global minimum variance, efficient frontier, Lagrange multipliers, mean-variance optimisation, two-asset portfolio, portfolio selection, risk and return
Field	Mathematics > Economy / Commerce
Published In	Volume 7, Issue 6, November-December 2025
Published On	2025-11-29
DOI	https://doi.org/10.36948/ijfmr.2025.v07i06.61997
Short DOI	https://doi.org/hbdspj

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