---
description: "Ranks stocks by past cumulative or risk-adjusted returns and buys winners while selling losers, exploiting the empirical momentum effect in cross-sectional equity returns."
tags: [stocks, momentum]
---

# Price-Momentum

**Section**: 3.1 | **Asset Class**: Stocks | **Type**: Momentum

## Overview
The momentum effect describes the empirical tendency for future stock returns to be positively correlated with past returns. Stocks are ranked by a performance criterion computed over a formation period and a portfolio is constructed by buying the top-ranked stocks (winners) and shorting the bottom-ranked stocks (losers). The portfolio is then held for a predefined holding period before being reconstituted.

## Construction / Signal
Let `P_i(t)` be the fully split- and dividend-adjusted price for stock `i`, with `t` measured in months and `t=0` the most recent time. The monthly return is:

```
R_i(t) = P_i(t) / P_i(t+1) - 1                          (266)
```

Cumulative return over the T-month formation period starting S months ago:

```
R_i^cum = P_i(S) / P_i(S+T) - 1                          (267)
```

Mean monthly return over the formation period:

```
R_i^mean = (1/T) * sum_{t=S}^{S+T-1} R_i(t)              (268)
```

Risk-adjusted mean return (Sharpe-like):

```
R_i^risk.adj = R_i^mean / sigma_i                         (269)

sigma_i^2 = 1/(T-1) * sum_{t=S}^{S+T-1} (R_i(t) - R_i^mean)^2   (270)
```

Stocks are sorted by `R_i^cum`, `R_i^mean`, or `R_i^risk.adj` (descending). The trader buys stocks in the top decile (winners) and shorts stocks in the bottom decile (losers).

## Entry / Exit Rules
- **Entry**: At rebalance, buy top-decile stocks and short bottom-decile stocks according to the chosen ranking criterion.
- **Exit**: Hold for the predefined holding period (typically 1 month, but can be longer). Liquidate before end of holding period if unforeseen events occur (e.g., market crash).
- **Skip period**: Typically skip the most recent S=1 month to avoid short-term mean-reversion / microstructure effects.

## Key Parameters
- **Formation period T**: Typically 12 months
- **Skip period S**: Typically 1 month
- **Holding period**: 1 month (longer periods show diminishing returns as momentum fades)
- **Ranking criterion**: `R_i^cum`, `R_i^mean`, or `R_i^risk.adj`
- **Portfolio construction**: Long-only (sum w_i = 1) or dollar-neutral (sum |w_i| = 1, sum w_i = 0)
- **Weights**: Uniform (1/N), or volatility-suppressed (w_i ∝ 1/sigma_i or 1/sigma_i^2)

## Variations
- **Long-only**: Buy top-decile only; weights w_i ≥ 0, sum w_i = 1.
- **Dollar-neutral (long/short)**: Buy winners, short losers; sum |w_i| = 1, sum w_i = 0. Modulus-uniform weights: w_i = 1/(2*N_L) for longs, w_i = -1/(2*N_S) for shorts.
- **Overlapping portfolios**: Multi-month holding via overlapping 1-month-holding portfolios (Jegadeesh and Titman, 1993).
- **Nonuniform weights**: w_i ∝ 1/sigma_i or w_i ∝ 1/sigma_i^2 to suppress volatile stocks.

## Notes
- The momentum effect is well-documented but fades over longer horizons; holding periods beyond 12 months tend to reverse (value effect).
- Transaction costs can be significant, especially for high-turnover monthly rebalancing.
- The 1-month skip period is empirically motivated by microstructure/liquidity mean-reversion observed at the 1-month horizon.
- Dollar-neutral construction removes broad market beta exposure.
- Typical holding period: 1 month; diminishing returns for longer holds before trading costs.