### Outline

We consider long-only momentum returns to be composed of market returns *plus* excess returns.

If excess returns over a specified period, *n-days,* (say, 5- or 10-days) either trends or mean-reverts, then that can be used to trade the momentum portfolio.

One way to check if a time-series is trending is to calculate the Hurst exponent (**H**) over a rolling window (say, 5-years) of *n-day *excess returns . If **H** < 0.5, then the time-series is mean-reverting; if **H** > 0.5, then it is trending, else it is random.

A simple strategy would be, for **H** < 0.5 (mean-reverting), if excess returns is greater than its median, then exit **or** if excess returns is less than its median, then enter.

The problem boils down to specifying the excess-return calculation periods (*n-days*) and the Hurst exponent rolling windows so that it makes sense (avoid data-mining.)

### Setup

We use the Barclays Euro-zone, UK, Japan and US momentum index data-sets to run our experiment. Since they provide both an excess-return index and a total-return index, we can use the former to time entries and exits in the latter.

We ran for two *n-day* configurations: 5 and 10. We set the Hurst (and median) rolling window at 5-years.

We expected to find **H** to be either consistently above or below 0.5. My personal expectation was that **H** would be above 0.5 (trend.)

### Results

Using the Hurst exponent did not improve momentum returns. In both *5-day* and *10-day* configurations, a strategy that went long if *n-day* returns were less than their median out-performed those that incorporated **H**.

The back-test using *5-day* returns mostly worked on Euro-zone and US momentum indices. So we are skeptical that this approach can be generalized and it will likely fall prey to data-mining.

The back-test using *10-day* returns saw buy-and-hold emerge a consistent winner.

The *5-day* **H** toggled between *trending* and *mean-reverting* but spent most of its time *trending*. The *10-day* **H** was consistently *trending*. For the specifications that we tested, **momentum excess returns trends**.

### Code and Charts

The R code is here: 5-day and 10-day config. You can login to pluto and play around with the *lookback* and *statWindow* variables to see how H, median and back-test results change.

Questions? Ask them over at our slack workspace.