The Science and Practice of Trend-Following Systems

2026-07-21

AI Summary

This study addresses the absence of a unified theoretical framework for trend-following systems, which has hindered clear understanding of their profitability and performance drivers. The authors propose a cohesive design framework encompassing European, U.S., and time-series momentum strategies, demonstrating that their alpha originates from excess spectral power in low-frequency components of volatility-normalized returns. Leveraging autocorrelation, drift, and spectral properties, the work derives a closed-form solution for the Sharpe ratio under transaction costs and establishes that positive skewness arises from structural mechanisms rather than偶然 phenomena. Methodologically, the analysis integrates ARFIMA modeling, frequency-domain Poisson kernel techniques, Monte Carlo simulations, and empirical attribution. Both simulated and real-world evidence confirm high cross-strategy correlation, enabling systematic design, simulation, and performance attribution based on trend persistence, mean reversion, drift, and skewness.

Abstract

We present a unified approach to designing trend-following (TF) systems and classify them into European, American, and Time Series Momentum categories. For European TF systems, we derive an exact relationship between profit-and-loss, autocorrelation, and drift in volatility-normalized returns. We analyze the expected return under fractional ARFIMA processes and show that TF systems are profitable when the long-term autocorrelation is positive, even under short-term mean reversion. In the frequency domain, the expected return is represented as a Poisson-kernel reading of the analytical or empirical spectrum of the volatility-normalized returns: the system profits at zero drift when the kernel-weighted spectral mass exceeds one, so trend-following alpha is excess spectral mass at low frequencies. Longer lookbacks benefit in addition from the squared drift of the return process. We derive the closed-form Sharpe ratio, with the excess kurtosis of the innovations entering through a single loading, and the net Sharpe ratio and cost-optimal span under trading costs. Under white noise, we derive the closed-form skewness of aggregated TF returns, which is positive at every horizon and peaks near half the filter span. Monte Carlo experiments confirm the analytical results. We show that the positive skewness of TF returns is structural under various model assumptions. Empirically, we evaluate the systems on liquid contracts, and show that all TF systems are strongly correlated and our analytical results can be applied for their performance attribution. Our results enable design, simulation, and performance attribution of TF systems from trend persistence, mean reversion, drift, and skewness.

Problem

• trend-following
• autocorrelation
• spectral analysis
• return skewness
• momentum

Innovation

• trend-following
• spectral analysis
• autocorrelation
• Sharpe ratio
• return skewness