Multi-Objective Optimization with a Closed-Form Solution for Capital Allocation in Environmental Energy Stock Portfolio

Abstract

This study proposes a multi-objective optimization model for capital allocation in an energy stock portfolio. The model integrates two financial objectives (maximizing return and minimizing value-at-risk) and four environmental objectives (minimizing carbon, energy, water, and waste intensities), providing a more comprehensive representation of corporate environmental performance in the energy sector. A closed-form analytical solution is developed to enhance theoretical clarity, analytical tractability, and interpretability without relying on iterative simulations. Methodologically, the model adopts a weighted utility function approach to aggregate multiple objectives into a single unified function, and the optimal solution is derived using the Lagrange multiplier method. The proposed model is then implemented on Indonesian energy stock data selected based on the lowest aggregate scores of financial and environmental attributes. This selection yields four stocks across three different energy subsectors: oil, gas, and coal. This implementation demonstrates that the optimal portfolio solution is simply and efficiently obtained without the need for iterative numerical approaches. Additionally, this implementation also shows a clear, representative, and rational trade-off between financial aspects and environmental impacts. This study makes a theoretical contribution to the sustainable portfolio literature and has practical implications for investors seeking to balance financial and environmental objectives quantitatively.