Bridging inverse participation ratio and portfolio theory

This research seeks to delve deeper into the financial significance of the eigenvector associated with the second-largest eigenvalue of the correlation matrix of returns. The objective is not only to gain a better understanding of this value, but also of the components that determine its behavior. To this end, we rely on daily logarithmic returns of the S&P’s 500 index constituents over the period ranging from 2005 to 2024. We propose the use of the Inverse Participation Ratio (IPR), which is a measure extensively used in Physics, and we introduce the Neff metric to identify the effective stocks within a portfolio; i.e., those capable of preserving the return dynamics of the full portfolio by themselves. Our empirical analysis shows that, in general, only long positions in the utilities sector are effective within this portfolio, which is consistent with the counter behaviour already observed and the traditional defensive nature of this economic sector. We also find that during periods of financial distress, however, short sales in the materials sector and the financial sector also emerge as effective.