## What Is Rescaled Range Analysis?

Rescaled range analysis is a statistical technique used to analyze trends in time series. It was developed by British hydrologist Harold Edwin Hurst to predict flooding on the river Nile. Investors have used it to look for cycles, patterns, and trends in stock and bond prices that might repeat or reverse in the future.

### Key Takeaways

• Rescaled range analysis looks at a data series and determines the persistence or mean-reverting tendencies within that data.
• The rescaled range can be used to compute the Hurst exponent, which can extrapolate a future value or average for the data.
• The Hurst exponent fluctuates between zero and one.
• When the Hurst exponent is greater than 0.5, the data is exhibiting a strong long-term trend, and when H is less than 0.5, a trend reversal is more likely.

## Understanding Rescaled Range Analysis

Rescaled range analysis can be used to detect and evaluate the amount of persistence, randomness, or mean reversion in financial markets time series data. Exchange rates and stock prices do not follow a random walk, or unpredictable path, like they would if price changes were independent of each other. Markets, in other words, are not perfectly efficient, which means there are opportunities for investors to capitalize on.

If a strong trend exists in the data, it will be captured by the Hurst exponent (H exponent), which can also be used to rate mutual funds. The H exponent, which is also known as the index of long-range dependence, can extrapolate a future value or average for the data.

The Hurst exponent ranges between zero and one, and it measures persistence, randomness, or mean reversion. Time series that display a random stochastic process have H exponents close to 0.5. When H is greater than 0.5, the data is exhibiting a strong long-term trend, and when H is less than 0.5, it is likely to reverse trend over the time frame considered.

H exponents below 0.5 are also known as the Joseph effect, in reference to the biblical story of seven years of plenty being followed by seven years of famine. Low values are likely to be followed by high values, or vice versa.

## Rescaled Range and the Hurst Exponent

Rescaled range analysis assesses how the variability of times series data changes with the length of the time period being considered. The rescaled range is calculated by dividing the range (maximum value minus minimum value) of the cumulative mean adjusted data points (sum of each data point minus the mean of the data series) by the standard deviation of the values over the same portion of the time series.

As the number of observations in a time series increases, the rescaled range increases. By plotting these increases as the logarithm of R/S versus the logarithm of n, one can determine the slope of this line, which is the Hurst exponent, H.

## Examples of How to Use Rescaled Range Analysis

The Hurst exponent can be used in trend trading investment strategies. An investor would be looking for stocks that show strong persistence. These stocks would have an H greater than 0.5. An H less than 0.5 could be paired with technical indicators to spot price reversals. For example, in order to time their investment, a value investor might look for stocks with H less than 0.5 whose prices have been declining for some time.

Mean reversion trading looks to capitalize on extreme changes in the price of a security, based on the assumption that it will revert to its previous state. The H exponent is used by algorithmic traders to speculate on mean-reverting time series strategies like pairs trading, where the spread between two assets is mean-reverting.

The following chart shows a 15-period moving average (MA) of the Hurst Exponent based on the SPDR S&P 500 (SPY) price chart. The MA can be adjusted, with a longer MA smoothing out fluctuations.

For traders wanting to buy during an uptrend in the price, they could look for opportunities where the H is above 0.5 and the price is moving up. Used in this way, the indicator wouldn't necessarily provide trade signals, but it could aid in providing confirmation for other trade signals based on the trend.