Selection Confidence Sets for Equally Weighted Portfolios

D. Ferrari${}^a$, A. Fulci${}^b$ and S. Paterlini${}^b$

${}^a$Faculty of Economics and Management, Free University of Bozen-Bolzano, ${}^b$Department of Economics and Management, University of Trento

Given a universe of $N$ assets, investors often form equally weighted portfolios (EWPs) by selecting subsets of assets. EWPs are simple, robust, and competitive out-of-sample, yet the uncertainty about which subset truly performs best is largely ignored. Traditional approaches typically rely on a single selected portfolio, thus failing to consider alternative investment strategies that may perform just as well when accounting for statistical uncertainty or model instability. To address this selection uncertainty, we introduce the Selection Confidence Set (SCS) for EWPs: the set of all portfolios that, under a given loss function and at a specified confidence level, contain the unknown set of optimal portfolios under repeated sampling. The SCS quantifies selection uncertainty by identifying a range of plausible portfolios, challenging the idea of a uniquely optimal choice. Like a confidence set, its size reflects uncertainty – growing with noisy or limited data, and shrinking as the sample size increases. Theoretically, we show that the SCS achieves asymptotic coverage of any fixed population-optimal selection and characterize how its size depends on underlying uncertainty, corroborating these findings through Monte Carlo experiments. Applications to the French 17-Industry Portfolio and Layer-1 Cryptocurrency data underscore the importance of accounting for selection uncertainty when comparing equally weighted strategies.

Keywords: Equally Weighted Portfolios, Selection Confidence Set, Selection Uncertainty, Subset Selection, Wald Test.