Professor Dacheng Xiu, from the University of Chicago Booth School of Business, delivered a keynote at the Inquire Europe Spring Seminar 2025 on The Statistical Limit of Arbitrage, exploring the gap between theoretical and practically achievable Sharpe ratios in quantitative trading strategies powered by machine learning.
Xiu opened by reflecting on the rapid rise in the use of machine learning techniques in financial markets over the past decade. Despite their promise, he emphasized a key limitation: “machine learning cannot give you an infinite Sharpe ratio.” Drawing from recent research conducted with Bryan Kelly, Xiu posed the central question: Is there a statistical ceiling on arbitrage opportunities, even in a world of advanced machine learning? His answer: yes — and that ceiling is lower than many in the empirical literature might assume.
The talk introduced the concept of statistical limits to arbitrage, rooted not in liquidity constraints, transaction costs, or short-sale restrictions — the typical frictions in the literature — but in the uncertainty inherent in statistical learning itself. As Xiu explained, “just because you can find alpha doesn’t mean it translates to arbitrage. Alphas are not arbitrage opportunities.” The distinction is crucial in high-dimensional settings, such as hedge funds trading thousands of stocks while aiming to remain factor-neutral.
At the heart of the talk was a stylized model where returns are decomposed into alpha and idiosyncratic noise. Under rational expectations, where arbitrageurs know the true parameters of the data-generating process, optimal strategies can achieve high Sharpe ratios. But in reality, investors must estimate these parameters, typically with limited data, which introduces noise and limits performance. As Xiu noted, “the act of learning alpha statistically introduces mistakes — and those mistakes generate losses.”
A vivid analogy grounded the concept: imagine coins on the ground, some real (profitable) and some fake (loss-generating), with arbitrageurs tasked with selecting the real ones. “There are too many coins and not enough time,” Xiu said. “Even the best strategies will pick up some fakes.” This intuition led to the paper’s main theoretical result: a persistent and measurable gap between feasible Sharpe ratios (those achievable with learned parameters) and infeasible Sharpe ratios (those requiring full information).
The feasible Sharpe ratio depends on the strength and sparsity of alphas in the data — in other words, how large and how rare they are. Empirically, Xiu’s findings suggest that alphas in individual stocks are both small and sparse. Using a dataset of over 10,000 U.S. equities and a 27-factor model including 11 style factors and 16 industry factors, the study finds that few stocks exhibit statistically significant alpha. “In practice, most t-stats for alpha are below 2.0,” he said, “and only a tiny fraction rise above 3.0.”
Even with optimal, machine learning–based feasible strategies, the achievable Sharpe ratio is capped at about 0.7. In contrast, a model assuming full information and no learning frictions might suggest a Sharpe ratio of 4.8 — a stark contrast. “This tells us that Arbitrage Pricing Theory (APT) works well,” Xiu concluded, “because in reality, you can’t extract all the alpha that theory assumes is there.”
Xiu also evaluated commonly used trading strategies — such as raw average alpha, significance-based selection (e.g., Benjamini-Hochberg cutoffs), and LASSO shrinkage — and found that none fully close the gap. Highly selective strategies work only when alphas are very strong, while naive approaches perform well when alphas are weak. Shrinkage techniques come close to optimal but are sensitive to tuning and assumptions.
A key insight was that econometric tests used to detect alpha, such as the GRS test, often report large, statistically significant results. But these reflect infeasible Sharpe ratios and overstate what arbitrageurs can capture in practice. “What matters,” Xiu emphasized, “is not whether alpha exists, but whether it can be profitably harvested under uncertainty.”
The talk concluded with empirical simulations comparing the performance of various trading strategies, both for individual stocks and portfolios. Even under ideal conditions, Sharpe ratios remained low — often below 1.0 — for feasible strategies. Once transaction costs and other real-world frictions are considered, the returns may be even lower, potentially null. As Xiu put it, “statistical limits alone are enough to collapse the arbitrage opportunity — even before other frictions come into play.”
In summary, Professor Xiu’s research challenges the common assumption that statistical significance in asset pricing translates into economic opportunity. “We need to stop interpreting large t-stats as a green light for trading,” he concluded. “Because once you account for the statistical cost of learning, the alpha you can actually earn is far more modest.”
View the research in full via : https://www.inquire-europe.org/event/joint-spring-seminar-2025-brussels/