Half-Life from AR(1) and OU Process

"Half-life" appears in two distinct contexts in QuanterLab: the mean-reversion half-life of a price series (in stochastic methods, SC001STCB) and the factor decay half-life of a cross-sectional IC (in FM103 diagnostics). Both share the same math — the AR(1) coefficient or the OU mean-reversion speed maps to a time-to-half via the same formula. This article documents the derivation.

The AR(1) model

An AR(1) process for a series x_t is:

where 0 < φ < 1 for a mean-reverting process. Iterating from x₀:

An expected value that decays geometrically by φ per step. Half-life is the t for which E[x_t] = x₀ / 2:

For small (1 − φ) the Taylor approximation ln(φ) ≈ −(1 − φ) gives:

This is the common approximation used throughout the platform.

The OU process

Ornstein–Uhlenbeck is the continuous-time mean-reversion process:

where θ is the mean-reversion speed and μ is the long-run mean. The conditional expectation:

The deviation from the mean decays exponentially at rate θ. Half-life:

Mapping between the two

Discretising the OU process at interval Δt produces an AR(1) with:

So OU's ln(2) / θ is the same as AR(1)'s ln(2) / |ln(φ)| in the appropriate units. The two are isomorphic; choose the formulation that fits the discretisation of your data.

Application: factor IC decay

For factor IC at horizons t = 1, 2, 3, ... days from rebalance, the decay model fits:

which is OU-style. Estimate λ via log-linear regression of ln(IC(t)) vs. t; then half-life is ln(2) / λ.

Application: price mean-reversion half-life

For an asset's log-price spread (from cointegration, e.g., a pairs trade), fit AR(1) on the spread:

Half-life of the spread is ln(2) / (1 − φ). A spread half-life of 5–40 trading days is the typical "tradable mean-reversion" sweet spot (Chan 2013).

What the half-life value means

• Half-life < 1 unit: Process reverts within one observation; the AR(1) model is near-zero (random) at this frequency — the signal lives on a finer time scale.
• Half-life ~ 5–40 units: Sweet spot for active strategies — deviations last long enough to trade but revert reliably.
• Half-life ~ 100+ units: Very slow reversion; either buy-and-hold or look for faster signal elsewhere.
• Half-life infinite (φ ≥ 1): Non-stationary — the model has broken down. The series is trending or a random walk.

Estimation caveats

• Small sample bias. φ estimated from short series tends to be biased low (toward stronger reversion) — Marriott & Pope (1954) bias correction is standard for serious work.
• Regime sensitivity. Half-life within a regime can be very different from the full-period estimate. Half-life on the spread of a cointegrated pair often shifts at regime breaks.
• Stationarity precondition. AR(1) and OU both assume mean-reversion. If the series is non-stationary (e.g., a trending price), the half-life estimate is meaningless. Test with KPSS or ADF before fitting.