Factor Risk Budgeting and Beyond

Portfolio optimization methods have evolved significantly since Markowitz introduced the mean-variance framework in 1952. While the theoretical appeal of this approach is undeniable, its practical implementation poses important challenges, primarily revolving around the intricate task of estimating expected returns. As a result, practitioners and scholars have explored alternative methods that prioritize risk management and diversification. One such approach is Risk Budgeting, where portfolio risk is allocated among assets according to predefined risk budgets. The effectiveness of Risk Budgeting in achieving true diversification can, however, be questioned, given that asset returns are often influenced by a small number of risk factors. From this perspective, one question arises: is it possible to allocate risk at the factor level using the Risk Budgeting approach? This paper introduces a comprehensive framework to address this question by introducing risk measures directly associated with risk factor exposures and demonstrating the desirable mathematical properties of these risk measures, making them suitable for optimization. We also propose a framework to find the portfolio that effectively balances the risk contributions from both assets and factors. Leveraging standard stochastic algorithms, our framework enables the use of a wide range of risk measures.