Title: Diversity and diversification (joint work with Maria Laura Torrente and Pierpaolo Uberti)

Abstract: Measuring and maximizing diversification is a central topic in portfolio theory. Various classes of measures of diversification have been considered in the literature, of which the simplest one are the so-called weight-based measures of diversification. A prominent role is played here by the exponential of Rényi entropies, also known as diversities of order $s$ in ecology and as effective number of constituents in the financial literature. We introduce a partial order of diversification based on the pointwise comparison of diversity profiles, derive necessary and sufficient conditions and compare it with the classical majorization order. We generalize the previous framework by taking into account additional features of the assets represented by a similarity matrix, mutuating the ideas introduced in Leinster and Cobbold (2012), where similarity-based diversity measures have been first considered in ecology. We discuss the connection with recent related results based on Rao Quadratic Entropy and discuss the applicability of the maximum diversity paradigm in portfolio optimization.