Risk Budgeting portfolios: Existence and computation

Modern portfolio theory has provided for decades the main framework for optimizing portfolios. Because of its sensitivity to small changes in input parameters, especially expected returns, the mean–variance framework proposed by Markowitz in 1952 has, however, been challenged by new construction methods that are purely based on risk. Among risk-based methods, the most popular ones are Minimum Variance, Maximum Diversification, and Risk Budgeting (especially Equal Risk Contribution) portfolios. Despite some drawbacks, Risk Budgeting is particularly attracting because of its versatility: based on Euler's homogeneous function theorem, it can indeed be used with a wide range of risk measures. This paper presents mathematical results regarding the existence and the uniqueness of Risk Budgeting portfolios for a very wide spectrum of risk measures and shows that, for many of them, computing the weights of Risk Budgeting portfolios only requires a standard stochastic algorithm.