## What is rescaled range analysis?

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

Key takeaways

- Rescaled range analysis examines a data series and determines persistence or mean reversion trends within that data.
- The rescaled range can be used to calculate the Hurst exponent, which can extrapolate a future or average value for the data.
- Hurst’s exponent fluctuates between zero and one.
- When the Hurst exponent is greater than 0.5, the data shows 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 assess the amount of persistence, randomness, or mean reversal in time series data from financial markets. Exchange rates and stock prices do not follow a random or unpredictable path, as 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.

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

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

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

## Rescaled range and Hurst exponent

Rescaled range analysis assesses how the variability of the time 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 of fitted data points (sum of each data point minus the mean of the data series) by the standard deviation of the values on the same portion of the time series.

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

## Examples of how to use rescaled range analysis

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

Mean reversion trading seeks to capitalize on extreme changes in the price of a security, based on the assumption that it will return to its previous state. Algorithmic traders use the exponent H to speculate on mean-reverting time series strategies, such as pair trading, where the spread between two assets is mean-reverting.

The chart below 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 who want to buy during an uptrend in price, they could look for opportunities where H is above 0.5 and the price is rising. Used in this way, the indicator would not necessarily provide trading signals, but could help provide confirmation for other trend-based trading signals.

The indicator will not always provide good signals. It is also important to note that high values of H when the price is declining indicate further declines in price, which can make the indicator a bit confusing when first used.

## The difference between rescaled rank analysis and regression analysis

Rescaled range analysis examines a data series and determines persistence or mean reversion trends within that data. Linear regression analyzes two variables, such as price and time, and finds the midpoint or line of best fit for the data series. Then standard deviation channels can be added to show when the security is potentially overbought or oversold based on the data series. Linear regression is part of the broader field of regression analysis.

## Limitations of Rescaled Range Analysis

For business purposes, a rescaled range is the fitted range divided by the standard deviation. These calculations are based on past data and are not inherently predictive. It is up to the trader to interpret the information provided by the rescaled range or the Hurst exponent.

For trading purposes, the Hurst indicator, which is derived from the rescaled range, can work sometimes, but it doesn’t work all the time. A strong price trend could be drastically reversed, which the indicator did not foresee. The reversals signaled by the indicator may also not develop.