  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 (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.
