Mean Reversion

Mean reversion is the tendency of a variable to return towards its long-run average after a deviation. The concept is used in the analysis of financial markets, of economic time series, and of interest rates.

Concept

A variable that exhibits mean reversion tends to move back towards its mean after a shock. If the variable is above its mean, it tends to fall; if below, it tends to rise. The speed of reversion determines how quickly the variable returns to its mean.

Mean reversion is distinguished from a random walk, in which the variable moves randomly and does not tend to return to any level. The distinction matters for forecasting, since a mean-reverting variable can be forecast from its current level, while a random walk cannot.

Applications in Finance

Mean reversion is used in the analysis of asset prices, of exchange rates, and of interest rates. The prices of some assets, including commodities and currencies, are sometimes modelled as mean-reverting, on the grounds that deviations from fundamental values are corrected over time.

Interest rates are often modelled as mean-reverting, since they tend to return to a long-run level determined by the fundamentals of the economy. The models are used to price interest rate derivatives and to assess the risk of interest rate positions.

The evidence on mean reversion in asset prices is mixed. Some studies find evidence of reversion over long horizons, while others find that prices follow a random walk. The evidence depends on the sample period, on the assets studied, and on the methods used.

Statistical Testing

The testing of mean reversion is based on the analysis of time series. The standard test is the unit root test, which examines whether the series has a unit root, in which case it follows a random walk, or is stationary, in which case it tends to revert to its mean.

The tests have low power, particularly over short samples, and they may fail to reject the null hypothesis of a unit root even when the series is mean-reverting. The results are therefore sensitive to the sample and to the specification.

The analysis of mean reversion also involves the estimation of the speed of reversion, which is the parameter that determines how quickly the variable returns to its mean. The estimates are used in forecasting and in the pricing of derivatives.

Implications

Mean reversion has implications for investment strategy. If prices are mean-reverting, then a strategy of buying when prices are low and selling when they are high may be profitable. The strategy is used in some hedge fund strategies and in the management of portfolios.

Mean reversion also has implications for the analysis of risk. If a variable is mean-reverting, then its variance is bounded, and the risk of extreme outcomes is limited. If it follows a random walk, the variance grows without bound, and the risk is greater.

Mean Reversion in Interest Rates

The modelling of interest rates often assumes mean reversion, on the grounds that rates tend to return to a long-run level. The models are used to price interest rate derivatives and to manage the risk of interest rate positions.

The Vasicek model and the Cox-Ingersoll-Ross model are the classic mean-reverting models of the short rate. Both assume that the rate is pulled towards a long-run mean, with the strength of the pull determined by the speed of reversion. The models differ in their treatment of volatility and in the possibility of negative rates.

The estimation of the models requires data on the short rate and the application of statistical methods. The estimates are used to assess the persistence of rate movements and to forecast future rates.

Mean Reversion in Exchange Rates

The analysis of exchange rates sometimes assumes mean reversion, on the grounds that deviations from fundamental values are corrected over time. The evidence on mean reversion in exchange rates is, however, mixed.

The purchasing power parity condition implies that real exchange rates should revert to a long-run level, as differences in inflation are offset by changes in nominal rates. The evidence suggests that reversion occurs, but that it is slow and that the speed of reversion is difficult to estimate.

The evidence on nominal exchange rates is less supportive of mean reversion, since nominal rates can drift for long periods. The analysis therefore distinguishes between the short-run behaviour of rates, which may resemble a random walk, and the long-run behaviour, which may exhibit reversion.

Implications for Portfolio Management

The evidence on mean reversion has implications for portfolio management. If asset prices are mean-reverting, then a strategy of buying after price declines and selling after price increases may be profitable, and the risk of holding a position over a long horizon may be lower than the random walk model implies.

The strategy is used in some investment approaches, including value investing and tactical asset allocation. The evidence on the profitability of these strategies is mixed, and the results are sensitive to the sample and to the methods.

The analysis of mean reversion is also relevant to the design of risk management. The horizon over which the risk is measured matters, since mean reversion reduces the risk over long horizons relative to the random walk benchmark.

Summary

Mean reversion is the tendency of a variable to return towards its long-run average, and it is used in the analysis of interest rates, of exchange rates, and of asset prices. The evidence on its presence and on its strength is mixed, and its implications for investment and for risk management depend on the context.

See also

• Statistics
• Financial Economics
• Econometrics