  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? First, we introduce 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. Then, we propose a novel framework to find portfolios that
effectively balance the risk contributions from both assets and factors.
Leveraging standard stochastic algorithms, our framework enables the use of a
wide range of risk measures to construct diversified portfolios.
