### DEFINITION of Rescaled Range Analysis

Rescaled range analysis is a statistical technique used to analyze trends in time series, 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.

### BREAKING DOWN 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 that 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 point. It ranges between 0 and 1, and measures persistence, randomness or mean reversion. Time series that display a random stochastic process have H exponents close to 0.5. When H is ≥ 0.5, the data is exhibiting a strong long-term trend, and when H is < 0.5, it is likely to reverse trend over the time frame considered. H exponents above 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.

### Trading the Hurst Exponent

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

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.

### 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 of the values of a portion of the times series, by the standard deviation of the values over the same portion of the time series. For example, consider a time series {1, 4, 3, 6, 8, 13, 5, 2} which has a range, R, of 13 - 1 = 12. Its standard deviation, s, is 3.85, so the rescaled range is R/s = 3.12.

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.

### Calculating Rescaled Range

The Rescaled Range is calculated for a time series,

, as follows:

Calculate the mean for each range

Create a mean adjusted series

Create a series which is the running total of the deviations from the mean

Create a range series R, which is the widest difference in the series of deviations

Create a standard deviation series S;

Where

*m(t)*

is the mean for the time series values through time

Calculate the rescaled range series (R/S)