Selection Confidence Sets for Equally Weighted Portfolios

Abstract

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, but this fails to consider alternative investment strategies that may perform just as well when accounting for statistical uncertainty. 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, contains 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 establish that the SCS covers the unknown optimal selection with high probability and characterize how its size grows with underlying uncertainty, corroborating these results through Monte Carlo experiments. Applications to the French 17-Industry Portfolios and Layer-1 cryptocurrencies underscore the importance of accounting for selection uncertainty when comparing equally weighted strategies.