Risk-budgeting multi-portfolio optimization with portfolio and marginal risk constraints

Multi-portfolio optimization problems and the incorporation of marginal risk contribution constraints have recently received a sustained interest from academia and financial practitioners. We propose a class of new stochastic risk budgeting multi-portfolio optimization models that impose portfolio as well as marginal risk constraints. The models permit the simultaneous and integrated optimization of multiple sub-portfolios in which the marginal risk contribution of each individual security is accounted for. A risk budget defined with a downside risk measure is allocated to each security. We consider the two cases in which the asset universes of the sub-portfolios are either disjoint (diversification of style) or overlap (diversification of judgment). The proposed models take the form of stochastic programming problems and include each a probabilistic constraint with multi-row random technology matrix. We expand a combinatorial modeling framework to represent the feasible set of the chance constraints first as a set of mixed-integer linear inequalities. The new reformulation proposed in this paper is much sparser than previously presented reformulations and allows the efficient solution of problem instances that could not be solved otherwise. We evaluate the efficiency and scalability of the proposed method that is general enough to be applied to general chance-constrained optimization problems. We conduct a cross-validation study via a rolling-horizon procedure to assess the performance of the models, and understand the impact of the parameters and diversification types on the portfolios.