1.4.3 Time-Series Momentum

Basic Prerequisites: Econometrics & Time Series (forecasting, EWMA/GARCH-style vol) and Probability & Measure Theory (moments).

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1. Intuition & Practical Objective

Time-series momentum (TSMOM) is the Moskowitz–Ooi–Pedersen strategy and the mathematical heart of CTA / trend-following: look at each asset in isolation; if its own trailing 12-month return is positive, go long; if negative, go short; and scale each position inversely to its own ex-ante volatility so every contract carries the same risk.

The bet is absolute, not relative: an asset is traded because it trended, regardless of its peers. This is what makes TSMOM behave differently from cross-sectional momentum - notably, TSMOM does well in crashes (it flips short as markets fall), while XSMOM crashes (see 05).

Verified facts (MOP 2012, 58 futures contracts across commodities, currencies, global equity indexes, and fixed income, 1985–2009):

• All 58 contracts show positive TSMOM predictability; 52 significant at 5%.
• The diversified TSMOM portfolio runs at ~12% annualized volatility (each position is sized to 40% ex-ante vol), delivering large alpha vs standard factors (1.58%/mo) and loading only on the cross-sectional momentum factor.
• TSMOM’s worst month is nowhere near as bad as XSMOM’s - it profits during the extreme 2008 episode.

Two engineering facts make TSMOM a system, not a slogan: volatility scaling (equalize risk across wildly different assets) and sign-based signals (robust to magnitudes, comparable across assets).

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2. Mathematical Ground Truth & Derivations

Signal. For asset $s$ with daily/monthly excess returns, the sign of its own trailing 12-month excess return:

$$S_t^s=\mathrm{sign}\left(r_{t-12,t}^s\right),r_{t-12,t}^s=\prod _{k=1}^{12}(1+r_{t-k}^s)-1.$$

Ex-ante volatility (MOP §2.4). An exponentially weighted variance with a 60-day center of mass, annualized by 261:

$${\sigma }_t^2=261\sum _{i\ge 0}(1-\delta ){\delta }^i(r_{t-1-i}-{\bar{r}}_t{)}^2,\frac{\delta }{1-\delta }=60\text{days}.$$

To avoid look-ahead, the volatility measured at $t-1$ is applied to the time-$t$ return.

Position sizing and TSMOM return. Size each position to an ex-ante annualized volatility ${\sigma }_{\text{tgt}}$ (MOP use 40%):

$$w_t^s=\frac{{\sigma }_{\text{tgt}}}{{\sigma }_{t-1}^s}\mathrm{sign}\left(r_{t-12,t}^s\right),r_{t,t+1}^{\text{TSMOM,s}}=w_t^s{r}_{t,t+1}^s.$$

The diversified factor (equal-weight across the $S_t$ available contracts) is

$$r_{t,t+1}^{\text{TSMOM}}=\frac{1}{S_t}\sum _{s=1}^{S_t}\mathrm{sign}\left(r_{t-12,t}^s\right)\frac{{\sigma }_{\text{tgt}}}{{\sigma }_{t-1}^s}{r}_{t,t+1}^s.$$

Because each position is scaled to the same ex-ante vol, the diversified portfolio has vol $\approx {\sigma }_{\text{tgt}}/\sqrt{S_t}$ (diversification) - in the corpus, ~12% annualized.

Expected return decomposition (MOP eq. 7). With portfolio weights $w_t^{\text{TS}}=(1/N)r_{t-12,t}$,

$$\mathbb{E}\left[r_{t,t+1}^{\text{TS}}\right]=\frac{\mathrm{tr}(\Omega )}{N}+\frac{12{\mu }^{\prime }\mu }{N},$$

so TSMOM profit is driven primarily by own time-series autocovariance $\mathrm{tr}(\Omega )$ - the cleanest statement that TSMOM is a bet on each asset’s own return continuation.

TSMOM vs XSMOM. Regressing TSMOM on XSMOM (MOP §5) gives $\beta =0.66$ ($t=15.2$, $R^2=44\%$) with a positive significant alpha of 76 bp/month ($t=5.9$): related, but not the same strategy.

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3. Computational Implementation - vol-targeted trend engine

Standard library only. Implements the MOP engine: EWMA ex-ante vol, sign-based 12-month signal, inverse-vol position sizing. Verifies (a) vol-targeting (a single scaled contract realizes ~the target vol) and (b) a positive-Sharpe diversified TSMOM portfolio.

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4. Failure Modes & First-Principles Breakdowns

1. The turning-point whipsaw. In range-bound, mean-reverting regimes the sign flips on every cross of the trailing mean, buying tops and shorting bottoms - a slow, persistent drawdown (“death by a thousand papercuts”) over 12–18 months. Vol scaling doesn’t fix a wrong-sign problem; it only fixes risk allocation.
2. Volatility estimator quality drives returns. The strategy is only as good as its ex-ante vol. A noisy vol estimator mis-sizes every position; better estimators (EWMA with sound half-life, or realized-vol extensions per Baltas & Kosowski) cut turnover and lift net Sharpe. Baltas & Kosowski (2013) show transaction costs can fall to ~105 bp without a significant Sharpe loss using more efficient estimators.
3. Burst risk / momentum-in-a-crash cuts both ways. TSMOM profits in the 2008 crash (it is short before the fall) but can be caught short when a long decline reverses sharply upward - the trend-flip whipsaw. It is not crash-proof, only crash-timed differently from XSMOM.
4. Sign + magnitude ambiguity at the flip. Near $S_t=0$ the sign is unstable; small measurement noise flips a large position. Practitioners use a dead-band / threshold rather than raw sign.
5. Capacity & turnover. Scaling by inverse vol concentrates turnover in low-vol, high-turnover instruments; costs and market impact erode the diversified factor (see Baltas & Kosowski turnover analysis).

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5. References

• Moskowitz, Ooi & Pedersen (2012), Time Series Momentum, J. Financial Economics 104(2)
• Baltas & Kosowski (2013), Demystifying Time-Series Momentum Strategies
• Daniel & Moskowitz (2016), Momentum Crashes
• Fung & Hsieh (2001)

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6. Connected Graph Bridges

• Base: Econometrics & Time Series · Probability & Measure Theory
• Sibling: 02 · Cross-Sectional Momentum (TSMOM’s $\beta =0.66$ cousin) · 04 · Value–Momentum
• Practice: 06 · Advanced Extensions (vol scaling, dynamic weighting) · Risk Parity (vol targeting) · Transaction Costs & Turnover