Half-Life of Mean Reversion

The half-life is derived from the Ornstein-Uhlenbeck (OU) process, the standard stochastic model for mean-reverting behavior. The OU process is defined as dX_t = θ(μ - X_t)dt + σdW_t, where θ is the speed of mean reversion. The half-life is calculated as ln(2) / θ. This represents the time required for the expected value of the spread to close half the distance to its long-term mean μ. A larger θ implies a shorter half-life and a faster-reverting spread.

The half-life directly dictates the optimal holding period for a mean-reversion trade. A strategy should be designed to capture the bulk of the reversion move without overstaying the position. Common heuristics include:

• Entry Timing: Entering a trade when the spread deviation exceeds 1.5 to 2 standard deviations.
• Exit Timing: Setting a profit target at the equilibrium mean or holding for a duration equal to 1-2 half-lives, after which the expected marginal reversion decays significantly.
• Stop-Loss Logic: A stop-loss should trigger if the spread continues to diverge beyond a threshold calibrated to the half-life, indicating a potential structural break in the relationship.

Half-life is a property of cointegration, not correlation. While correlation measures the short-term directional alignment of returns, cointegration identifies a long-term equilibrium relationship between non-stationary price series. A pair of assets can be highly correlated but have an infinite half-life if their spread is not stationary. Conversely, a cointegrated pair with a stable half-life may exhibit low short-term correlation. Cointegration is a necessary condition for a finite half-life to exist.

The half-life is not a static parameter; it is estimated from historical data and is highly sensitive to market regime changes. A structural break in the underlying economic relationship between two assets can permanently alter or destroy the mean-reversion property. Common causes include:

• Mergers and Acquisitions: A corporate action fundamentally changes the equity relationship.
• Sector Rotation: A persistent shift in investor capital flows can break historical spread relationships.
• Monetary Policy Shocks: Changes in interest rates can permanently alter the cost-of-carry relationship in futures spreads. Continuous monitoring and walk-forward re-estimation are required to detect when a half-life has extended to infinity, signaling a broken pair.

The half-life is a direct input into the Kelly Criterion and other dynamic position-sizing models for mean-reversion strategies. A shorter half-life implies a higher frequency of trading opportunities and a faster convergence of expected value, allowing for more aggressive capital allocation per signal. Conversely, a long half-life exposes capital to divergence risk for an extended period, requiring smaller position sizes to withstand prolonged adverse excursions. The half-life also defines the decay rate of the alpha signal, directly influencing the strategy's information ratio.

In a portfolio of multiple cointegrated pairs, the aggregate half-life is not a simple average. The dollar-weighted average half-life of the positions dictates the overall portfolio's sensitivity to mean-reversion decay. A portfolio manager must balance fast-reverting pairs (short half-life, high turnover, low margin per trade) with slow-reverting pairs (long half-life, low turnover, higher divergence risk) to achieve a stable aggregate strategy Sharpe ratio. Diversifying across half-life regimes helps smooth the equity curve when specific sector relationships temporarily break down.

The foundational statistical property that makes mean-reversion trading possible. A cointegrated portfolio is a linear combination of non-stationary asset prices that produces a stationary residual spread. Unlike simple correlation, cointegration implies a long-run equilibrium relationship that is not spurious. The Johansen test and Engle-Granger two-step method are standard tests for identifying cointegrated pairs. Without cointegration, the concept of a half-life of mean reversion is meaningless, as there is no stable mean to which the spread can revert.

The stochastic differential equation used to model the evolution of a mean-reverting spread. The process is defined as dX_t = θ(μ - X_t)dt + σ dW_t, where θ is the speed of mean reversion and μ is the long-term mean. The half-life is derived directly from θ using the formula Half-Life = ln(2) / θ. This model assumes a constant rate of reversion and normally distributed increments, making it a tractable but simplified representation of real market dynamics.

A recursive Bayesian algorithm used to estimate the dynamic hedge ratio between cointegrated assets in real time. Unlike a static linear regression, a Kalman filter allows the relationship between two securities to evolve smoothly over time, adapting to structural market shifts. It outputs a time-varying state estimate that forms the basis for a dynamic spread. This is critical because a fixed hedge ratio can break down, causing a strategy to trade on a spread that is no longer mean-reverting.

The Augmented Dickey-Fuller test is the primary statistical hypothesis test for verifying the stationarity of a time series. In pairs trading, it is applied to the residual spread to confirm the presence of a unit root (non-stationarity) in the null hypothesis. A low p-value rejects the null, providing evidence that the spread is stationary and mean-reverting. The test is a prerequisite before calculating the half-life; applying the half-life formula to a non-stationary series yields a spurious, unreliable result.

The optimal holding period for a trade, which is directly dictated by the half-life of mean reversion. A strategy should be designed so that its expected holding time aligns with the signal's predictive power. If the half-life is 5 days, a high-frequency intraday strategy is inappropriate; conversely, a half-life of 30 minutes requires automated execution. Mismatching the trading horizon and the half-life leads to exiting trades too early (leaving profit on the table) or too late (suffering adverse moves after reversion is complete).

The normalized deviation of the spread from its mean, used to generate entry and exit signals. A spread is typically measured as Z = (Spread - Mean) / StdDev. Common entry thresholds are ±2.0 standard deviations, with an exit at 0.0 (the mean). The half-life informs the urgency of the trade; a very short half-life allows for tighter thresholds and faster round-trips, while a long half-life requires wider thresholds to avoid being whipsawed by noise before reversion occurs.