Scenario-based stochastic model and efficient cross-entropy algorithm for the risk-budgeting problem

Risk budgeting is one of the most recent and successful approaches for the portfolio selection problem. Considering mean-standard-deviation as a risk measure, this paper addresses the risk budgeting problem under the uncertainty of the covariance matrix and the mean vector, assuming that a finite set of scenarios is possible. The problem is formulated as a scenario-based stochastic programming model, and its stability is examined over real-world instances. Then, since investing in all available assets in the market is practically impossible, the stochastic model is extended by incorporating the cardinality constraint so that all selected assets have the same risk contribution while maximizing the expected portfolio return. The extended problem is formulated as a bi-level programming model, and an efficient hybrid algorithm based on the cross-entropy is adopted to solve it. To calibrate the algorithm's parameters, an effective mechanism is introduced. Numerical experiments on real-world datasets confirm the efficiency of the proposed models and algorithm.