On a general class of portfolio diversification measures induced by risk measures

This paper introduces a novel and effective methodology for constructing portfolio diversification measures derived from any reference risk measure. The central contribution lies in leveraging the extensive theoretical developments in risk measurement to systematically inform and enhance the design of diversification metrics. The link between risk and diversification is exploited through an optimization framework in which the objective function is defined as a weighted Euclidean distance dependent on risk. We prove that the resulting objective function satisfies key axiomatic properties typically required to diversification measures, and that the corresponding optimization problem admits a unique solution that is inherently related to the intuitive concept of geometric diversification, thereby providing theoretical support for it. The key economic interpretation relies on determining the point in the allocation space that is equally distant – under a risk-sensitive metric – from the vertices of the simplex, i.e. the fully concentrated portfolios. An important economic insight of our approach is its applicability within a general long-short investment framework–a significant advancement, given that most classical diversification measures are restricted to long-only portfolios. Finally, to support the robustness of our findings, we present a comprehensive empirical analysis across multiple real-world financial datasets, highlighting meaningful comparisons between our proposed measure and several widely used diversification metrics.