Mean Reversion

Mean reversion is the tendency of a price, rate or other financial quantity to move back towards its long-run average after it has moved away from it. A quantity that mean-reverts does not wander without limit: high values tend to be followed by falls and low values by rises. The term also covers the family of trading strategies that bet on this behaviour by selling what has risen far above its average and buying what has fallen far below.

Mean reversion is the opposite of momentum, in which recent moves tend to continue, and of a random walk, in which past moves say nothing about future ones. Evidence for it varies by market and time scale. It is strong in volatility, in interest rate spreads and in the price difference between closely related securities, and weaker and more debated in the prices of individual shares and indices.

The standard mathematical description is the Ornstein-Uhlenbeck process, in which the pull back towards the average is proportional to the distance from it:

• $x_t$ is the value of the series at time $t$.
• $\mu$ is the long-run mean.
• $\theta$ is the speed of reversion.
• $\sigma$ is the size of the random shocks, and $dW_t$ is the random shock itself.

A useful summary is the half-life, the expected time for a deviation from the mean to shrink by half:

If $\theta$ is estimated at 0.1 per day, the half-life is $0.693 / 0.1 = 6.9$ days. A spread that is 4 points above its mean would be expected to be about 2 points above it a week later. The half-life tells a trader how long positions are likely to be held and what length of history to use when computing averages. In discrete time the same idea appears as an autoregression with a coefficient below one.

A series can only be expected to revert to a mean if it is stationary, meaning its statistical properties do not drift over time. Stationarity is tested with the augmented Dickey-Fuller test, whose null hypothesis is that the series behaves like a random walk. Most individual prices fail this test. A combination of prices can pass it even when each one fails: if two shares are driven by the same underlying factors, the difference between them, suitably weighted, may be stationary. This property is called cointegration, and it is the basis of most practical mean-reversion trading.

• Pairs and baskets. Pairs trading buys one security and sells a related one when the spread between them is unusually wide. Extending the idea to many securities gives statistical arbitrage.
• Short-horizon reversal. Shares that have fallen most over the past few days or weeks are bought and the biggest risers are sold, a pattern known as short-term reversal or, across a whole universe, cross-sectional mean reversion.
• Indicator-based rules. Technical traders buy when a price touches the lower of the Bollinger Bands or when the relative strength index signals an oversold market, and sell on the opposite signals.
• Volatility. Volatility tends to spike and then subside, and many option strategies are built on its reversion.

Entry and exit rules are commonly based on a z-score, the number of standard deviations by which the series sits away from its mean. If a spread has a mean of 1.0 and a standard deviation of 0.6 and now stands at 2.5, its z-score is $(2.5 - 1.0) / 0.6 = 2.5$. A typical rule sells the spread when the z-score rises above 2 and closes the trade when it returns towards zero.

Academic studies in the 1980s reported that shares that had performed worst over three to five years went on to outperform, and that long-horizon index returns were negatively related to past returns. These findings were taken as evidence of overreaction by investors, though later work questioned their statistical strength. At short horizons of days to a month, reversal in individual share returns is well documented and is usually explained as a payment to those who supply liquidity to impatient sellers and buyers. At horizons of three to twelve months the opposite pattern, momentum, dominates. Valuation ratios have also tended to return towards historical ranges, but over periods of many years.

The payoff profile of mean-reversion trading is many small gains and occasional large losses. A position is opened because a spread is wide, and if it widens further the loss grows while the signal says to add more. The mean itself can shift: a merger, a change in a company's business or a new regulation can permanently alter a relationship that held for years. A statistical test passed on historical data offers no protection against such a break.

Because the trades usually involve borrowing and short selling, losses can force a trader to close at the worst moment. The collapse of the hedge fund Long-Term Capital Management in 1998 is the standard example: its convergence trades moved further apart during a market panic before many of them eventually converged. Risk controls therefore include limits on position size, a stop-loss order or a time limit on each trade, and attention to costs, since frequent trading on small edges is sensitive to commissions and spreads.