KDE distributionally robust portfolio optimization with higher moment coherent risk

In this paper, distributionally robust mean-HMCR (higher moment coherent risk) portfolio optimization model based on kernel density estimation (KDE) and $$\phi $$ ϕ -divergence is proposed. In order to overcome the so-called “curse of dimensionality”, we consider the one-dimensional probability distribution of the portfolio return, rather than the joint probability distribution of the assets return vector. The two issues of “the distribution dependent on the decision variables” and “the metric-based distributional uncertainty set for the continuous distribution” are effectively addressed by using the finite dimensional KDE based probability distribution. Under the mild conditions of the kernel function and $$\phi $$ ϕ -divergence function, the tractable reformulation of the corresponding distributionally robust optimization model is derived by Fenchel’s Duality Theorem. Moreover, the convergence of optimal value and solution set of the KDE mean-HMCR distributionally robust portfolio optimization problem to those of the corresponding stochastic optimization model with the real distribution is proved. We conduct some empirical tests with the rolling horizon approach and compare the performance of the optimal portfolio strategy obtained by the proposed model to other three strategies by four performance criteria and their cumulative wealth curves. Empirical test results show that the quality of the portfolio strategy obtained by the proposed model is better at most cases. We also conduct empirically sensitivity analysis of model parameters.