8.8.2. Time-Series Momentum

Time-Series Momentum

Time-series momentum (Moskowitz, Ooi & Pedersen, 2012; Lempérière et al., 2014) is a trend-following strategy that trades each asset based on the sign of its own past return, with positions sized to constant expected volatility, and aggregates the resulting per-asset positions into a diversified long-short portfolio. It is the time-series counterpart to cross-sectional momentum: where cross-sectional momentum compares assets against each other, time-series momentum compares each asset against its own past.

There are two canonical signals:

• MOP-style sign signal (Moskowitz, Ooi & Pedersen): trade asset $j$ at time $t$ in the direction $\mathrm{sign}\left(r_{j,t-12\to t}^{\text{ex}}\right)$, where $r^{\text{ex}}$ is the excess return over the risk-free rate (i.e., long assets with positive excess return and short otherwise).
• EMA divergence signal (Lempérière et al., “Two centuries of trend following”): measures the divergence of a price $p_j$ from its recent average, such that
  $s_{n,j}(t)=\frac{p_j(t-1)-⟨p_j{⟩}_{n,t-1}}{{\sigma }_{n,j}(t-1)},$
  where $⟨p_j{⟩}_{n,t}=(1-e^{-{\lambda }_n})p(t)+e^{-{\lambda }_n}⟨p_j{⟩}_{n,t-1}$ is exponential moving average and and ${\sigma }_{n,j}$ is the EMSD of the price $p_j$ on a timescale of $n$ months with ${\lambda }_n=1/n$.
  • The strategy is to buy when ${\sigma }_{n,j}^{-1}{s}_{n,j}$ is positive, and sell when it is negative (i.e., trade less in riskier bets, and more in safer bets).
  • If the price is above its trend, that’s positive (bullish signal); below its trend, negative (bearish). The denominator just scales it by volatility so the signal is comparable across assets with different price levels and volatilities.

Position sizing targets a fixed forecast volatility per asset (typically $40\%$ annualized, so that diversification across $\sim 50$ contracts produces a sensible portfolio volatility):
$w_{j,t}=\mathrm{sign}(s_{j,t})\cdot \frac{{\sigma }_{\text{tgt}}}{{\hat{\sigma }}_{j,t}}.$

Headline empirical results:

• In Moskowitz, Ooi & Pedersen’s $58$-contract panel ($1965$–$2009$ across equities, bonds, currencies, and commodities), every contract has a positive Sharpe ratio from the $12$-month sign signal, and the diversified TSMOM portfolio achieves a Sharpe ratio above $1$.
• The TSMOM alpha is not absorbed by exposures to size, value, market, cross-sectional momentum, liquidity, or sentiment factors.
• Lempérière et al. extend the strategy back to $1800$ across four asset classes and find a persistent positive Sharpe over $200$ years, including the half-century before any modern academic discovered the effect.

Time-series momentum is universal across asset classes, robust to two centuries of out-of-sample data, and not driven by standard risk factors. Together with cross-sectional momentum and momentum crashes, it forms the empirical core of the momentum section.

Intuition

Time-series momentum is just trend following: if an asset has gone up over the past year, buy it; if it has gone down, sell it. The striking claim is that this naive rule, applied to every liquid futures market with the same parameters, gives a positive risk-adjusted return in every single contract tested, over more than a century. Because nothing is tuned to the sample, the result is hard to dismiss as a lucky fit to the data.

Two design choices make it work. Volatility scaling sizes each position in inverse proportion to that asset’s recent volatility, so a calm bond gets a bigger position than a wild commodity; without this, one wild contract would dominate the portfolio and the cross-asset Sharpe would collapse. Sign-only signal uses only the direction of the past return, not its magnitude — this makes the strategy robust to outliers and to the exact look-back chosen (anything from $1$ to $12$ months works). The cognitive story is the same as for cross-sectional momentum: slow absorption of news creates short-run drift, and late buyers create long-run overshoot. Lempérière et al. add that trend-following is itself amplified by the flows of trend-following funds — once enough professional money is chasing trends, every nascent move gets pushed further along.

Properties

(Why Volatility Scaling Is Load-Bearing) Without inverse-volatility position sizing the cross-asset Sharpe collapses, even though the per-asset signal is unchanged.

The signal is sign-only — it knows nothing about volatility — so an unscaled portfolio implicitly puts dollar-equal weights on every contract. A wild commodity like lean hogs then contributes $5$–$10$ times as much variance as a calm bond future, and the diversification benefit of running $58$ contracts in parallel is wiped out by a handful of dominant names. Scaling each position by ${\sigma }_{\text{tgt}}/{\hat{\sigma }}_{j,t}$ equalizes risk contributions, so the diversified Sharpe above $1$ is jointly produced by the signal and the sizing rule. This is why sign-only signals outperform raw-magnitude signals: magnitude information is already used to set position size, and using it twice double-counts the same volatility.

(Cross-Asset Diversification Floor) The $58$-contract panel’s individual Sharpes range from $\approx 0.1$ to $\approx 0.7$ but the diversified Sharpe exceeds $1$.

The aggregate is larger than any constituent because per-contract returns are only weakly correlated. If TSMOM were driven by a single global risk factor, all per-contract returns would correlate strongly and the diversification gain would vanish. The fact that diversification works is itself evidence against a one-factor risk story — the contracts each have their own private trend dynamics that happen to share a sign-and-hold rule.

(Weak Correlation with Cross-Sectional Momentum) TSMOM and the Carhart UMD factor have correlation around $0.6$ in equities and much lower elsewhere.

The two strategies use overlapping information (both rely on the past $12$-month return) but operate on different dimensions: TSMOM signs each asset against zero, while cross-sectional momentum signs each asset against its peers. In a market that fell $20\%$, a stock that fell only $5\%$ is a cross-sectional winner but a time-series loser. This sign disagreement is why a portfolio of TSMOM + UMD has a higher Sharpe than either alone — they are different trades.

(Long-Run Reversal Mirrors the Cross-Section) Moskowitz, Ooi & Pedersen document partial reversal over months $13$–$36$.

The pattern qualitatively matches the De Bondt & Thaler (1985) reversal in cross-sectional momentum. Any complete behavioral account of momentum must therefore generate the same continuation-then-reversal trajectory in both the cross-section and the time-series — see over- and under-reaction models. A model that produces only cross-sectional reversal does not pass the joint test.

(Failure Modes) TSMOM suffers sharp drawdowns when correlations spike or sustained trends abruptly reverse.

The Sharpe above $1$ at the diversified portfolio level hides individual-month losses of $-10\%$ to $-20\%$ in episodes like late $2008$ and March $2020$. The mechanism is correlation breakdown: in a normal month the $58$ contracts diversify each other, but in a crisis they all reverse together, so the realized portfolio volatility blows past the volatility target the position sizes were calibrated against. Position-size caps, vol-targeting at the portfolio level (not just per-asset), and trend-confirmation filters are the standard mitigations. See momentum crashes for the parallel cross-sectional failure mode.

Examples

Computing the Lempérière signal on a toy price series

Consider a single asset with monthly closing prices $p(t-5),\dots ,p(t-1)$ equal to $100,102,105,109,112$. The exponential moving average with timescale $n=5$ assigns geometrically declining weights to past prices; with smoothing factor $\alpha =2/(n+1)\approx 0.33$, the EMA at $t-1$ is approximately $107$. The exponential moving standard deviation of the same series, also with $n=5$, is approximately $4.5$.

The raw signal is therefore
$s_5(t)=\frac{p(t-1)-⟨p{⟩}_5}{{\sigma }_5}=\frac{112-107}{4.5}\approx \mathrm{1.11.}$
The sign is positive, so the strategy goes long this asset at the start of month $t$. The magnitude of $s_5$ does not enter the position size — only its sign does — but the magnitude can be useful as a confidence indicator for portfolio weighting in more sophisticated implementations.

Why every contract has a positive Sharpe in Moskowitz-Ooi-Pedersen

The MOP $58$-contract panel includes the S&P 500 future, the German Bund, the Japanese yen, WTI crude, gold, lean hogs, and $52$ others. For each contract, MOP run the same rule: at the start of each month, look at the past $12$-month excess return; if positive, go long; if negative, go short. Size the position to a $40\%$ ex-ante annualized volatility target. Hold for one month. Roll.

Out of $58$ contracts, all $58$ have a positive Sharpe ratio from this rule over the $1965$–$2009$ sample. The cross-section of per-contract Sharpe ratios runs from about $0.1$ (worst) to about $0.7$ (best), with a median around $0.3$. After diversifying across all $58$ contracts (after volatility scaling), the aggregate TSMOM portfolio Sharpe ratio is above $1$ — substantially better than any single contract because the per-contract returns are weakly correlated.

The point of the “every single one” framing is statistical: under any standard null in which trend signals are noise, the probability of observing $58/58$ positive Sharpe ratios by chance is essentially zero. The result is not driven by a few outliers.

Trend following in $19$th-century commodities

Lempérière et al. push the strategy back to $1800$ on monthly commodity prices (cotton, sugar, coffee, copper, wheat) for which long historical series exist. They run exactly the same EMA-divergence signal with $n=5$ and the same volatility scaling. The aggregate trend-following Sharpe is approximately $0.7$ in the $1800$–$1900$ subperiod, $0.9$ in $1900$–$1950$, and just over $1.0$ in $1950$–$2013$. The slight increase over time may reflect deeper, more liquid markets, but the qualitative result — persistent positive Sharpe in every $50$-year window — holds for all four asset classes for which they have data.

The implication is that time-series momentum is not an artifact of post-Bretton-Woods macro shocks, post-1980s financialization, or post-publication crowding. It is a feature of asset prices going back at least two centuries.

See Also

• 8.8. Momentum — parent hub: comparison of time-series, cross-sectional, and momentum-crash treatments.
• 8.8.1. Cross-Sectional Momentum — sibling: ranks assets against each other rather than against their own past.
• 8.8.3. Momentum Crashes — sibling: failure mode in panic states.
• 8.9. Reaction Models — leading behavioral explanations.
• 8.3. Challenges to Rational Pricing — momentum sits inside the broader predictability puzzle.