Infinite Factor Structure in Conditional Asset Pricing

The return covariance matrix is widely assumed to have an eigenvalue cliff: a few large eigenvalues, then a bounded bulk, as in the approximate factor structure of Chamberlain and Rothschild (1983). This premise underlies factor tests, principal-component methods, and large-covariance estimation. These methods estimate moments from returns pooled across states, so they necessarily view a conditional world unconditionally. We show that, generically, this view removes the cliff: a conditional model with a few factors is, unconditionally, an infinite-factor model in which infinitely many eigenvalues diverge. This holds for the characteristic-managed portfolios that machine-learning methods trade, and for individual stocks whose loadings differ persistently across firms. Yet as long as conditional expected returns admit no asymptotic arbitrage, the maximal Sharpe ratio stays finite. For generic expected returns, infinitely many components are priced, so no finite set of principal components spans expected returns. Reducing returns to a few principal components therefore discards priced risk.