Debt Ratio Optimization as an Alternative Approach to Capital Structure Determination Using the Lagrange Method
DOI:
https://doi.org/10.30872/jfor.v27i1/4449Keywords:
Capital Structure, Lagrange Optimization, Debt Ratio, Equity Value, Trade-Off Theory, Asymptotic Behavior, Econometric ModelingAbstract
This study develops a mathematical optimization model for determining the optimal debt ratio in a firm’s capital structure using the Lagrange method. By expressing equity value as a function of leverage, the model identifies a structural asymptote that divides the feasible region into positive and negative equity outcomes, thereby eliminating the possibility of an interior maximum and mathematically refuting the central prediction of the Trade-Off Theory. The first-order condition produces two critical points, but only one lies within the positive region and is shown through second-order analysis to be a local minimum rather than an optimum. Numerical simulation and empirical validation further demonstrate that optimal equity value arises under two extreme leverage conditions: zero leverage for conservative strategies and high leverage approaching the asymptote for aggressive strategies. These results establish a new analytical foundation for capital structure theory, provide a basis for developing more complex econometric models, and offer practical guidance for firms in prioritizing leverage decisions according to their risk preferences and strategic objectives
References
[1] P. Tharavanij, “Optimal Book-Value Debt Ratio,” 2021, doi: 10.1177/2158244020985788.
[2] F. Ahmed, M. Ur, H. Mudassir, M. Imran, A. Dunay, and B. Hossain, “Heliyon Corporate capital structure effects on corporate performance pursuing a strategy of innovation in manufacturing companies,” Heliyon, vol. 10, no. 3, p. e24677, 2024, doi: 10.1016/j.heliyon.2024.e24677.
[3] T. N. Bui and X. H. Nguyen, “The Effect of Capital Structure on Firm Value : A Study of Companies Listed on the Vietnamese Stock Market,” 2023.
[4] R. Arhinful, H. Ismail, M. Amin, and L. Mensah, “Determining an optimal capital structure and its impact on financial performance . Insight from the firms listed on the New York Stock Exchange,” Cogent Econ. Financ., vol. 13, no. 1, p., 2025, doi: 10.1080/23322039.2025.2571401.
[5] N. T. X. Trang, “A Review of Transfer Pricing: From Domestic to International Transfer Pricing 1,” Int. J. Business, Econ. Law, vol. 10, no. 3, pp. 18–23, 2016.
[6] A. Damodaran, Damodaran on Valuation, 2nd ed. John Wiley & Sons, Inc., 2012. doi: 10.1002/9781119201786.
[7] R. Goldstein, N. Ju, and L. Hayne, “An EBIT‐Based Model of Dynamic Capital Structure,” J. Business, vol. 74, no. 4, pp. 483–512, 2001.
[8] A. Kraus and R. H. Litzenberger, “A State-Preference Model of Optimal Financial Leverage,” J. Finance, vol. 28, pp. 911–922, 1973.
[9] S. C. Myers, “The Capital Structure Puzzle,” J. Finance, vol. 39, pp. 575–592, 1984.
[10] M. C. Jensen, “Agency Costs of Free Cash Flow, Corporate Finance, and Takeovers,” Am. Econ. Rev., vol. 76, pp. 323–329, 1986.
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Copyright (c) 2025 Muhammad Ario, Alphero Tanlianto, Muhammad Halim Fathanah, Jesen Jovan Lo, Felisitas Defung
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