Time-Series Momentum (TSMOM, 时间序列动量)
Continuation of an asset's own past returns, as opposed to Cross-Sectional Momentum. The empirical foundation of CTA Strategy / trend-following strategies.
Definition
Time-series momentum (TSMOM) is the phenomenon that the past return of an asset, over some lookback window, positively predicts its own next-period return. The classic specification (Moskowitz, Ooi & Pedersen, 2012) uses a 12-month lookback:
r^{TSMOM}_{t,t+12} = \text{sign}\!\left(\sum_{\tau=1}^{12} r_{t-\tau}\right) \cdot r_{t,t+12}
A positive past 12-month return implies a long position; a negative past return implies a short. Positions are scaled inversely to a rolling volatility estimate to risk-balance across contracts.
Signal formulation (general)
From Moskowitz, Sabbatucci, Tamoni & Uhl (2025), a generic TSMOM signal is:
s_t = \frac{1}{k_1} \sum_{\tau=0}^{T-1} w_\tau \frac{r_{t-\tau}}{\hat\sigma_t}
where w_\tau \geq 0 are weights (typically w_\tau = 1/T, giving k_1 = \sqrt{T}), and \hat\sigma_t is a rolling volatility estimate (e.g., 260 trading days for daily data). The dollar position in a futures contract is then:
\text{pos}_{i,t} = f(s_{i,t}) \cdot \frac{\sigma_{target}}{\hat\sigma_{i,t}}
where f(\cdot) maps the signal to a portfolio weight. The choice of f is the central design decision in any TSMOM strategy.
Three weighting schemes (the core comparison)
| Scheme | Function f(s) | Source |
|---|---|---|
| Linear | f(s) = s | Levine & Pedersen (2016) |
| Binary | f(s) = \text{sign}(s) | Moskowitz, Ooi & Pedersen (2012) |
| Nonlinear (FS) | f(s) = \lambda s / \sigma_\varepsilon^2, concave for \lvert s\rvert > 1 | Ferson & Siegel (2001) |
| Nonlinear (Empirical) | f learned by ANN trained to maximize Sharpe | Moskowitz et al. (2025) |
The Ferson-Siegel Optimal Weighting page covers the theoretical underpinning; Nonlinear Signal Mapping covers the broader design space.
Key empirical properties
- Universality: documented in equity index, commodity, rates, and FX futures (Moskowitz et al., 2012; Hurst, Ooi & Pedersen, 2017).
- Convexity / tail hedge: TSMOM returns are convex in equity-market returns — best when equities are extreme up or extreme down. This is its main selling point to traditional stock-bond portfolios.
- Momentum crashes: in rapid rebounds from bear markets, TSMOM can suffer large drawdowns (Daniel & Moskowitz, 2016).
- Decay over time: Huang et al. (2020) show aggregate TSMOM profitability is not statistically robust across all asset classes individually; Moskowitz et al. (2025) replicate weaker recent performance in 2015–2024.
Why nonlinear weighting beats binary/linear
A binary rule treats |s| = 0.01 and |s| = 5 identically — both produce a full long or short. A linear rule scales both linearly. A theory-consistent Nonlinear Signal Mapping mutes extreme signals because at high |s|, the variance penalty dominates the expected-return gain. The empirical result: nonlinear weighting is best in down markets and best at shorter lookbacks (1M, 3M). At 12M lookback the binary rule catches up.
Lookback / frequency
Common configurations:
- Daily data, 12-month lookback — canonical Moskowitz et al. (2012) spec, daily-rescaled.
- Monthly data, 12-month lookback — simpler, lower frequency.
- 1M / 3M / 12M blend — "intermediate-speed" strategies (Goulding, Harvey & Mazzoleni, 2023) for better turning-point detection.
Related pages
- Cross-Sectional Momentum — relative-performance momentum; weaker than TSMOM at the same horizon (Moskowitz et al., 2025).
- CTA Strategy — the dominant real-world implementation of TSMOM.
- Ferson-Siegel Optimal Weighting — the theoretical nonlinear weight.
- Nonlinear Signal Mapping — broader design space for f.
- Volatility Targeting — risk-parity sizing layer beneath any f.
- Moskowitz Ooi Pedersen 2012 Time Series Momentum — foundational paper.
- Moskowitz Sabbatucci Tamoni Uhl 2025 Nonlinear TSMOM — extension to nonlinear f.
References
- Moskowitz, T.J., Ooi, Y.H., & Pedersen, L.H. (2012). "Time Series Momentum." Journal of Financial Economics, 104(2), 228–250.
- Hurst, B., Ooi, Y.H., & Pedersen, L.H. (2017). "A Century of Evidence on Trend-Following Investing." Journal of Portfolio Management, 44(1).
- Huang, D., Li, J., Wang, L., & Zhou, G. (2020). "Time series momentum: Is it there?" Journal of Financial Economics, 135(3).
- Moskowitz, T.J., Sabbatucci, R., Tamoni, A., & Uhl, B. (2025). "Nonlinear Time Series Momentum." Working paper.