The Half-Life of Mean Reversion

Knowing that a spread "mean-reverts" isn't enough to trade it. A spread that snaps back to its average in two days and a spread that takes eight months need completely different position sizes, holding periods, and stop-loss rules, even if both are mathematically mean-reverting in exactly the same statistical sense. The number that turns "it reverts" into "here's how to trade it" is the half-life: how long, on average, it takes a stretched spread to close half the gap back to its long-run mean.

Where the number comes from

A spread that mean-reverts can usually be described by a discrete-time relationship where tomorrow's deviation from the mean is today's deviation shrunk by some factor $\phi$ (a number between 0 and 1 for a stationary series):

$$x_{t+1} - \mu = \phi \,(x_t - \mu)$$

In words: whatever distance the spread sits from its average today, tomorrow it will have closed part of that gap, and $\phi$ is the fraction of the gap that's left. The half-life is the number of steps it takes for that shrinkage to cut the gap in half:

$$\text{half-life} = \frac{\ln(0.5)}{\ln(\phi)}$$

In words: take the natural log of one-half, divide by the natural log of the persistence factor, and the result is how many time steps — days, if $\phi$ was estimated on daily data — it takes for the average deviation to shrink by 50%.

Drag the mean-reversion speed in the explorer above and watch how a faster speed produces a tighter, choppier path around the mean while a slower speed lets the path wander further before turning back — that visual difference is the difference between a short and a long half-life.

Worked example

A spread's daily deviations are found (by regressing today's deviation on yesterday's) to have a persistence factor $\phi = 0.95$ — each day, 95% of yesterday's deviation from the mean survives into today. The half-life is $\ln(0.5) / \ln(0.95) = -0.6931 / -0.0513 \approx 13.5$ days. If the spread is currently 2 standard deviations away from its mean, it should on average sit around 1 standard deviation away in about 13–14 trading days, and around 0.5 standard deviations away roughly 27 days after that. A trader sizing a stat-arb book around this pair would plan for holding periods on the order of two to four weeks per round trip, not two days and not two quarters.

What this means in practice

Half-life feeds directly into two practical decisions: how wide to set entry and exit bands (a short half-life justifies tighter bands, since the spread reverts quickly enough to make small moves worth trading) and how much capital to allocate per trade relative to the total book, since a longer half-life ties up capital for longer and needs a correspondingly smaller position to hit the same annualized risk budget.