Some general results on risk budgeting portfolios

Given a reference risk measure, the risk budgeting is the portfolio where each asset contributes a predetermined amount to the total risk. We propose a novel approach, alternative to the ones proposed in the literature, for the calculation of the risk budgeting portfolio. This different perspective on the problem has several interesting consequences. For the calculation of the portfolio, we define a Cauchy sequence within the simplex of R^n, whose limit corresponds to the risk budgeting portfolio. This construction allows for the straightforward implementation of an efficient algorithm, avoiding the need to solve auxiliary, equivalent optimization problems, which may be computationally challenging and hard to interpret in the decision theory context. We compare our algorithm with the standard optimization-based methods proposed in the literature. From a theoretical point of view, starting from the Cauchy sequence, we define a function for which the risk budgeting portfolio is a fixed point. Therefore, sufficient conditions for the existence and uniqueness of the fixed point can be used. The methodology is developed for general risk measures and implemented in detail in the case of standard deviation.