Portfolio Diversification and Concentration under Dependence Uncertainty: A Majorization Approach

Modern portfolio theory identifies diversification as the primary tool for risk reduction. However, under model uncertainty, this cornerstone may no longer remain optimal. This paper investigates the tension between portfolio diversification and concentration under dependence uncertainty. In the absence of model uncertainty, we employ the framework of the majorization order and doubly stochastic matrices to formalize the degree of diversification, and prove that quasi-convexity is a necessary and sufficient property for a risk functional to be weakly consistent with the majorization order. We further derive worst-case risk measure inequalities and solve robust portfolio selection problems for a broad class of risk measures, including VaR, ES, Range-VaR (RVaR), and standard deviation (SD). Our results reveal a ''concentration paradox'' for many widely-used risk functionals: when the dependence structure is fully ambiguous, robust optimization often recommends concentrating investment in a single asset to hedge against the worst-case dependence scenario. As an application, we propose a weighted robustness formulation that interpolates between a reference dependence structure and the worst-case structure. The formulation is structurally analogous to the constrained/unconstrained Expected Shortfall blend in the Fundamental Review of the Trading Book (FRTB) and provides a theoretical foundation for balancing diversification against robustness in the presence of model uncertainty.