Cardinality-constrained risk parity portfolios

The risk parity optimization problem produces portfolios where each asset contributes an equal amount to the overall portfolio risk. While most work has investigated the problem using all assets, minimal work has investigated the cardinality constrained variant, which reduces the associated portfolio overhead. In this work, we present the first formulations that can be solved to global optimality by off-the-shelf solvers. Specifically, we propose two new quadratically constrained quadratic integer programs, a non-convex and a convex one, that can be solved to global optimality without the need of specialized algorithms, heuristics or approximations. We strengthen our formulations by adding tighter variable bounds and valid constraints. Computational experiments on real-world financial data indicate the effectiveness of our formulations at producing portfolios with equal risk contributions of chosen cardinality size. Specifically, the convex formulation is shown to be very efficient in terms of both speed and accuracy, while producing portfolios with great out-of-sample performance.