Half-Life of Mean Reversion
Half-life quantifies mean reversion speed. It measures the time for a deviation from the mean to decay by half. A shorter half-life indicates faster mean reversion. A longer half-life suggests slower mean reversion. Traders use half-life to estimate optimal holding periods for mean-reverting positions.
The Ornstein-Uhlenbeck (OU) process models mean reversion. Its stochastic differential equation is:
$dX_t = \theta(\mu - X_t)dt + \sigma dW_t$
Here, $X_t$ is the asset price at time $t$. $\mu$ represents the long-term mean. $\theta$ is the speed of reversion. $\sigma$ denotes the volatility. $dW_t$ is a Wiener process.
The half-life ($T_{1/2}$) derives from the speed of reversion $\theta$:
$T_{1/2} = \frac{\ln(2)}{\theta}$
A higher $\theta$ value results in a shorter half-life. This means the asset reverts to its mean more quickly.
Estimating Half-Life
Traders estimate $\theta$ from historical price data. We convert the continuous OU process into a discrete-time autoregressive model (AR(1)):
$X_t - X_{t-1} = \alpha + \beta X_{t-1} + \epsilon_t$
Here, $\alpha = \theta \mu \Delta t$ and $\beta = -\theta \Delta t$. We estimate $\alpha$ and $\beta$ using ordino least squares (OLS) regression. The time step $\Delta t$ is typically 1 day.
From the regression, we get an estimate for $\hat{\beta}$. Then, $\hat{\theta} = -\hat{\beta} / \Delta t$. Finally, calculate half-life using $\hat{T}_{1/2} = \frac{\ln(2)}{\hat{\theta}}$.
Consider a pair of exchange-traded funds (ETFs): SPY (S&P 500) and IVV (iShares Core S&P 500). From January 1, 2020, to December 31, 2022, their log-price spread $Y_t = \ln(SPY_t) - \ln(IVV_t)$ exhibits mean reversion.
Assume we performed the regression and found:
$\hat{\alpha} = 0.00001$ $\hat{\beta} = -0.05$ (daily basis)
Given $\Delta t = 1$ day:
$\hat{\theta} = -(-0.05) / 1 = 0.05$
Then, the half-life:
$\hat{T}_{1/2} = \frac{\ln(2)}{0.05} \approx \frac{0.693}{0.05} \approx 13.86$ days.
This indicates that any deviation in the log-price spread between SPY and IVV halves its magnitude in approximately 14 trading days.
Half-Life and Optimal Holding Period
The optimal holding period for a mean-reverting strategy directly relates to the half-life. Holding a position for too short a time misses full reversion. Holding it too long exposes the trade to new trends or noise.
Academic research and practical experience suggest an optimal holding period of one to two half-lives. This period allows sufficient time for the deviation to revert significantly. It minimizes exposure to potential trend reversals or regime shifts.
If a half-life is 14 days, an optimal holding period might be 14 to 28 days. A trader initiates a position when the spread deviates significantly from its mean. They aim to close the position as the spread approaches its mean within this optimal window.
Consider the SPY/IVV example with a 14-day half-life. A trader observes the log-price spread at +2 standard deviations from its historical mean. They short SPY and long IVV. If the spread reverts as expected, it will be +1 standard deviation in 14 days. It will be +0.5 standard deviations in 28 days.
Closing the position after one half-life captures a substantial portion of the reversion. It also reduces the risk of the spread moving further away. Holding for two half-lives captures more reversion but increases market exposure. The exact optimal period depends on transaction costs, volatility, and the trader's risk tolerance.
Practical Considerations for Half-Life
Half-life is not static. It changes with market conditions. High volatility might shorten half-life. Low liquidity might lengthen it. Traders must re-estimate half-life periodically. Use a rolling window of data for estimation. For example, use the last 252 trading days to estimate half-life every month.
Not all time series exhibit mean reversion. A stationarity test, like the Augmented Dickey-Fuller (ADF) test, should precede half-life calculation. A non-stationo series does not mean revert. Its half-life is infinite. Trading a non-stationo series with a mean-reversion strategy leads to sustained losses.
For example, consider the price of TSLA (Tesla Inc.) from 2020 to 2022. An ADF test would likely fail to reject the null hypothesis of a unit root. This indicates non-stationarity. Applying a half-life calculation to TSLA's raw price would be without purpose. TSLA's price exhibits strong trending behavior, not mean reversion.
Conversely, the spread between KO (Coca-Cola) and PEP (PepsiCo) might exhibit mean reversion. Their similar business models and competitive dynamics often lead to co-movement. A hypothetical half-life for KO/PEP spread might be 45 days. A trader would then consider holding positions for 45 to 90 days.
The half-life provides a quantitative framework for setting trade duration. It helps manage expectations regarding trade profitability and risk. Use it as a guide, not a rigid rule. Combine half-life with other risk management techniques.