Designing Trend-Following Systems from Autocorrelation and Return Spectra

Summary

This document develops a framework for designing and comparing trend-following systems, including European, American, and time-series momentum approaches. For European systems, it derives a relationship between P&L, autocorrelation, and drift in volatility-normalized returns. Analyses using fractional ARFIMA processes indicate that positive long-term autocorrelation can support profitability even when short-term returns mean-revert. A frequency-domain view interprets this as trend-following benefiting from sufficiently strong low-frequency spectral mass; drift can further aid longer lookbacks.

The work derives Sharpe ratio expressions, incorporates innovation kurtosis, and examines net performance and cost-optimal spans under trading costs. It also predicts positive skewness for aggregated returns under white noise, with simulations supporting the theoretical results. Tests on liquid contracts find that the systems are strongly correlated and that the framework can assist performance attribution. These findings depend on the model assumptions and reported empirical setting; they do not guarantee profitability in other markets or after unmodeled costs.

Key ideas

• Trend-following performance can be related to autocorrelation, drift, and the spectrum of volatility-normalized returns.
• Positive long-term autocorrelation may support trend following even when short-term mean reversion is present.
• Low-frequency spectral mass is associated with trend-following returns in the frequency-domain analysis.
• Trading costs affect the net Sharpe ratio and the preferred strategy span.
• The analysis finds structural positive skewness in trend-following returns under several modeled conditions.

Tags

# The Science and Practice of Trend-Following Systems


# The Science and Practice of Trend-Following Systems









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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