The Hurst Exponent

Hurst Exponent

Today’s post looks at the Hurst Exponent, which is a measure of how strongly a data series is trending.

Hurst Exponent

Many of the most common active trading strategies are based on momentum (a continuation of the relatively recent past) or mean-reversion (to a long-term average).

  • The Hurst Exponent helps us to (automatically) decide which kind of strategy is the most appropriate for a particular market.

I came across the Hurst Exponent (HE) from a couple of articles on Medium. (( References below ))

  • The focus of these notes was how to code the HE in Python, but we won’t be getting into that today.

The posts also contained a fair bit of maths, which we will also sidestep.

  • The featured image above shows the formula for the HE from Wikipedia, and below I show the formula used in one of the articles.

Hurst exponent 2

Luckily, we don’t need to understand these formulae.

  • All we need to know is whether the HE is saying that a market is trending, mean-reverting, or just random.

The Hurst Exponent was invented by the British hydrologist Edwin Hurst in the middle of the last century.

  • It was developed to estimate the volume of water in a river but was later adapted to financial markets by people like Benoit Mandelbrot, the king of fractals.

The HE is bounded in the range of zero to 1:

  • When HE > 0.5, the time series tends to move in a single direction
  • When HE = 0.5, the series oscillates
  • When HE < 0.5, the series is random.

HE is calculated by measuring the rescaled range – a statistical measure of variability – across multiple time periods, and then plotting the data on a log-log plot.

  • The slope of the line is the HE (usually written as H).

You can also use a technique called detrended fluctuation analysis (DFA).

Ignoring the math, we look at the change in price over a series of time differences (called lags).

  • The log of the standard deviation of the lags is plotted against the log of the lags.

Hurst plots

The log-log plots are on the left.

  • The random and trending series are clear enough, but the mean-reverting one is messy.

The key point is that the slope is positive and less than 0.5.


Hurst FX

The chart above is the Hurst for USD/CHF.

  • Currency pairs in general are mean-reverting and particularly those between two strong currencies.

But the dots are below the trend line at the short end.

Hurst FX 2

The H never quite gets to trending levels, but it does become more mean-reverting at around 60+ days, so mean-reversion strategies longer than that should work best.


Hurst BTC

Bitcoin is a momentum play (in both directions).

Hurst BTC 2021

Raposa notes that in 2021 (after a strong bull market in 2020), BTC went into a mean-reverting phase.

Hurst BTC 2018

The same thing happened in 2018-2019 after the big bull run in 2017.


Hurst stocks

I’ve saved the most important asset class for last.

  • The price chart for the S&P 500 looks like a strong upward trend, but the Hurst exponent tells us that this is a mean-reverting market.
See also:  The Market Environment for Trend

What is more interesting is the time structure:

Hurst stocks 2

The index is mean-reverting over the short-term, then strongly-trending in the medium term, and then mean-reverting over the longer term (beyond 400 days).

  • This matches up well with the folk wisdom for individual stocks, though the time frames are a little longer. (( Stocks are understood to revert over one month, trend over a year and then revert again ))

Hurst stocks 3

Raposa repeated this analysis over a variety of timescales.

  • Unfortunately, the results are inconsistent.

Between the dot come boom and the 2008 crisis, there were strong trends.

  • Since then, the trends concentrate around 100 to 300 days.

Raposa notes that this matches Turtle Trade Jerry Parker’s observation that 200+ days works best for trend following.


The Hurst Exponent is an interesting way of classifying markets, but unfortunately, it’s not stable over time in the market of most interest (stocks).

  • There’s plenty of support for the idea that 200-day trends exist, but if we want to use the Hurst as a filter to protect against the times when trend isn’t working, we’ll need to dig into the maths and work out how to code and calculate the HE.

That’s a topic for another day.

  • Until next time.
  • Find Your Best Market to Trade With the Hurst Exponent – Raposa Technologies, Medium
  • Introduction to the Hurst exponent — with code in Python – Eryl Lewinson, Towards Data Science, Medium

Mike is the owner of 7 Circles, and a private investor living in London. He has been managing his own money for 39 years, with some success.

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The Hurst Exponent

by Mike Rawson time to read: 2 min