1.4.1 Momentum from Zero

Basic Prerequisites: Econometrics & Time Series (autocorrelation, stationarity).

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1. Intuition & Practical Objective

This page builds the why of momentum with no prior knowledge needed. The objective is one idea: past returns contain information about future returns, and that information can be harvested by ranking assets or by following each asset’s own trend.

Start with the dumbest question: why should an asset that went up keep going up? In an efficient market, past prices encode no future signal - price is a martingale and today’s price already discounts everything knowable. Momentum is the empirical finding that this is wrong at intermediate horizons: there is positive serial correlation in returns over 1–12 months. The two leading families of explanations:

1. Behavioral - underreaction & slow diffusion. Investors anchor to stale information, underweight new news (representativeness, conservatism), and information spreads gradually through a network of investors (attention is scarce). So a price move is not fully digested for months; part of the move “leaks” into the future. Winners keep winning because their news is still being priced in.
2. Risk-based - compensation. Momentum’s large, crash-prone, negatively-skewed payoffs (see 05) may be a premium for bearing a systematic risk that is priced. The evidence is mixed: momentum loads on no standard factor and keeps its alpha even when crash-hedged, which tilts toward behavioral/limits-of-arbitrage explanations.

The key quantitative fact that makes it harvestable: the first-order autocorrelation of asset returns is positive at daily–monthly–quarterly horizons, but turns negative at very short (days, one month) and very long (3–5 year) horizons. Momentum lives in the middle band. Get the horizon wrong and you trade away the premium.

A first “aha”: the one-month and 12-month horizons are opposite trades. The most recent month reverses (short-term reversal: Jegadeesh 1990, Lehmann 1990), while the trailing 12 months (ex-month-1) continue. That is precisely why the standard signal skips the most recent month (the $12\text{-}1$ convention).

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2. Mathematical Ground Truth & Derivations

Return continuation as positive autocorrelation. Consider a stationary return process with first-order autocorrelation ${\rho }_1>0$:

$$r_t=\mu +{\rho }_1(r_{t-1}-\mu )+{\epsilon }_t,{\epsilon }_t\sim \text{iid}(0,{\sigma }^2).$$

The conditional expectation of tomorrow’s return given today’s deviation is

$$\mathbb{E}[r_{t+1}\mid r_t]=\mu +{\rho }_1(r_t-\mu ).$$

When ${\rho }_1>0$, an above-average past return raises the expected future return - continuation. A simple long-short bet $+\text{sign}(r_t-\mu )$ earns $\mathbb{E}[\text{sign}(r_t-\mu )(r_{t+1}-\mu )]={\rho }_1\mathbb{E}|r_t-\mu |>0$ per period, provided ${\rho }_1>0$.

The horizon structure. Empirical autocorrelation is not constant:

• $\rho$ negative at horizons of days-to-one-month → short-term reversal;
• $\rho$ positive over roughly 2–12 months → the momentum premium;
• $\rho$ negative over 3–5 years → long-term reversal (De Bondt & Thaler).

So the exploitable signal is a band-passed version of the past - include months 2–12, exclude month 1.

Cross-sectional ranking signal. For a universe of $N$ assets, define the skip-month cumulative return

$$R_i^{(12\text{-}1)}=\prod _{k=2}^{12}(1+r_{i,t-k})-1,$$

then rank and map ranks to dollar-neutral weights

$$w_i=\frac{{\text{rank}}_i-\frac{N+1}{2}}{{\sum }_j|{\text{rank}}_j-\frac{N+1}{2}|},\text{with}\sum _i{w}_i=0,\sum _i|w_i|=1.$$

The zero-sum weights make the portfolio dollar-neutral (no net market exposure by construction) and unit-gross (100% gross leverage). The sub-page 02 · Cross-Sectional Momentum works this machinery, and 03 · Time-Series Momentum builds the absolute/own-asset version.

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3. Computational Implementation - momentum is serial correlation you can harvest

Standard library only. Generates a synthetic universe whose returns obey the continuation process above (${\rho }_1>0$), then shows that ranking by the trailing 12-month return (skipping the last month) predicts next-month returns: past winners keep winning, past losers keep losing.

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4. Failure Modes & First-Principles Breakdowns

1. “Momentum means buying what just went up” is half-true and dangerous. The exploitable signal is a band-passed, peer-relative or own-absolute measure. Naively buying last month’s winner loads the reversing month and destroys the edge - the entire reason for the $12\text{-}1$ skip.
2. Horizon errors are the #1 beginner failure. Use 1-week or 5-year lookbacks and you trade long-term/short-term reversal instead of momentum. The premium is specific to the 1–12 month band.
3. Confusing cross-sectional with time-series momentum. They are related but not the same (in the corpus, TSMOM regresses on XSMOM with $\beta =0.66$, $R^2=44\%$, yet retains a significant alpha). XSMOM is relative and dollar-neutral; TSMOM is absolute and vol-scaled. They fail in different regimes (see 03 and 05).

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5. References

• Jegadeesh & Titman (1993), Returns to Buying Winners and Selling Losers, J. Finance
• Jegadeesh (1990) and Lehmann (1990)
• De Bondt & Thaler (1985)
• Asness, Frazzini, Israel & Moskowitz (2014), Fact, Fiction, and Momentum Investing

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6. Connected Graph Bridges

• Base: Econometrics & Time Series (autocorrelation, stationarity) · Probability & Measure Theory
• Continue: 02 · Cross-Sectional · 03 · Time-Series · Index Hub
• Mirror image: Statistical Arbitrage & Pairs (momentum’s mean-reversion counterpart)