### Hurst Exponent

William Hurst was a hydrologist working on what seems like a straight forward problem: how tall should a damn on the Nile river be to contain all possible floods? Hurst encountered a unique problem where the mathematics to answer the question did not yet exist.

Hurst’s work found broad applications in many unrelated areas of sciene. Software engineers use it to measure how prone a network is to congestion. Geologists modeling oil field deposits look at the tendency for oil and gas fields to cluster together. Financial engineers use it to analyze how the tendency of a given time series to form trends.

The exponent itself is a lot simpler than all of the mathematics used to find it. Most people are used to working with data where the Hurst exponent is 0.5. My favorite analogy, that of tossing a coin, falls in the 0.5 Hurst category. Any process that is Gaussian or follows a normal bell curve also has a Hurst exponent of 0.5.

The total available range for the Hurst exponent is 0 to 1. A value close to zero should look like a flat line. Whenever deviations from the average occur, they are extremely prone to returning back to the average. A Hurst exponent close to 1 indicates a strong propensity to trend.

The value of Hurst exponent increases as the charting time period increases. I say this based on my experience reviewing the Hurst exponent and its relationship to various forex pairs, particularly the EUR/USD. Tick and one minute data show Hurst values consistently near 0.5. Moving to the daily chart nudges the needle more towards something like 0.55-0.6. Moving out to weekly charts tends to create values closer to 0.7. Keep in mind that this is not the result of any formal study. It’s based on my general experience.

## Estimate the Hurst Exponent

The great insight that led Hurst to the exponent concept stesm from his idea of rescaled range analysis. The idea is to look at how segmenting a given data set into larger collections changes the overall range of values.

Rescaled range analysis is commonly referred to as R over S. The R stands for the range component and is the hardest to calculate. There are 3 steps involved for each segment.

To calculate R/S from P0 to P1, the first step is to calculate the average vaule from P0 to P1. Step two calculates the deviate series from the average. The deviate series at P0, for example, equals P0 minus the series average. This is repeated across all points in the set to create the deviate series.

Step 3 is the cumulative deviate series. This is a fancy way of saying to go through the deviate series and to pick out the smallest and largest values. The difference between the smallest and largest values in the deviate series is R.

Finding S is much easier. It stands for the standard deviation of the segment. The formula for this can be found on Wikipedia and thousands of other pages across the internet.

## Conclusion

The Hurst exponent can be a great tool for categorizing the general behavior of particular instruments. It can also be used to get a feel for how changing the charting period of a strategy may explain any degradations or improvements in the strategy returns.

I have not found any applicability to predicting financial markets using the Hurst exponent. It does not indicate whether the price will move up or down. I have not found that just because the Hurst reads 0.4 that it has to return to 0.5 any time in the near future. And, if it does, that still does not help with direction. All it says is that perhaps the markets will be less mean reverting than they have been.

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