Otimização de portfólios online com métodos de projeção para controle de risco

Abstract

The objective of this work is to investigate the benefits of incorporating econometric risk metrics into the performance of online learning algorithms for solving the dynamic portfolio allocation problem. The problem is formulated as an online decision process where variants of the Online Gradient Descent (OGD) method are studied. Three modifications to the OGD algorithm were proposed, each using a distinct form of convex projection. The first modification includes a timevarying beta estimation constraint during the projection step, aiming to control the systematic risk of the portfolio. Recognizing that beta alone does not capture all aspects of risk, a second modification introduces Value at Risk (VaR) as a more comprehensive risk measure. This approach allows for more effective control of portfolio risk exposure, considering both systematic and unsystematic risk. The third modification utilizes a convex projection that minimizes Conditional Value at Risk (CVaR). This approach demonstrated more consistent results in terms of balancing risk and return, providing a more robust and effective risk management strategy. Before implementing the proposed modifications, mathematical proof of the projections convexity for each method was provided to ensure theoretical validity and computational efficiency of the developed algorithms. The impact of the choice of probability distribution (normal and Student’s t) on VaR minimization during the projection step and its influence on performance and risk profile of the optimized portfolios is also investigated. The proposed methodologies are empirically evaluated on different Brazilian datasets (IBOVESPA and Small Caps), considering both high and low capitalization stocks in different market periods. The results show that incorporating risk metrics, especially CVaR, into the online optimization process allowed for better risk control while maintaining competitive returns compared to market benchmarks and other optimization strategies. The main contribution of this work is the development and evaluation of methodologies that integrate widely used risk metrics in financial theory with online optimization algorithms, providing an adaptive approach for dynamic portfolio management in volatile markets. Future studies using data from other markets and new types of return distributions may further enhance the robustness of the developed methodologies.