Four Limits to Portfolio Estimation

Portfolio estimation is described through a single notion of sample size, yet different sources of error respond to different forms of information. This paper organizes estimation risk around four limiting quantities: the convergence rate of the estimator, the ratio of portfolio dimension to observation count, the calendar span of the return history, and the breadth of the strategy search. Each quantity enters a common certainty-equivalent loss framework and is matched to a distinct practical remedy.For Gaussian return increments, the paper derives an exact finite-sample expression for the expected loss of the plug-in mean–variance portfolio. The calculation separatesobservation frequency from calendar span. Increasing sampling frequency over a fixed history improves covariance estimation but cannot eliminate uncertainty about expected returns, leaving a nonzero span floor. When portfolio dimension grows proportionally with the number of observations, covariance estimation produces a positive loss barrier even when the mean is known. The same finite-sample calculation distinguishes correction of inverse covariance bias from the different decision problem of choosing an optimal scalar shrinkage factor.The rate limit is developed using regular variation: moment estimators converge more slowly when the population tail index lies between the existence and centrallimit thresholds. The selection limit is characterized through independent Gaussian candidate signals, yielding an exact selection probability and a sharp boundary between successful signal recovery and domination by search breadth. A single-factor experiment, an oracle cost–value decomposition, and an angular approximation for out-of-sample Sharpe ratios explain when optimization can overtake equal weighting and when its estimated inputs cost more than their economic value.A controlled Student experiment shows that rank-based dependence estimation helps below the covariance moment threshold but requires positive-semidefinite regularization at moderate dimension ratios. The framework is auditable, not a universal ranking of portfolio rules. Its central conclusion is that more data is not one operation: heavier tails require robust functionals, high dimension requires structure or shrinkage, expected returns require longer span or prior information, and strategy search requires multiplicity control.