In a previous post, I used the S&P 500 as an example to demonstrate the use of a sophisticated quantitative method, rescaled range analysis, for evaluating whether a time series is random, persistent, or mean reverting. Rescaled range analysis was developed to spot trends hidden in the seeming randomness of African rainfall and its effect on Nile river flooding — but its application to investing yields many interesting insights.

Without much further explanation in my prior post, I stated that the S&P 500 had a Hurst exponent, H, of 0.49, for the time period 3 January 1950 to 15 November 2012. What does that mean? Is there greater granularity in the rescaled range analysis that reveals even more interesting findings? What other associated measures can be used to provide even greater insight?

First, let’s recap the basics. Recall that H takes on values between 0 and 1, and that a reading near 0.5 is representative of a randomly generated time series. Put another way, a data point in this kind of time series does not influence the result of another. Persistent time series are those with a Hurst exponent between 0.51 and 1.0, meaning that subsequent data is likely to take on the sign of preceding data. Data between 0.0 and 0.49 are mean reverting. In other words, subsequent data are likely to have an opposite directional sign from preceding data.

The S&P 500’s Hurst exponent of 0.49 is right near the ’randomness’ center of 0.5 and only slightly on the mean reversion side of things. If we divide the Hurst exponent range into thirds — 0.00 to 0.33 for mean reverting, 0.34 to 0.67 for random, and 0.68 to 1.00 for persistence — a strong statement can be made that the S&P 500 is a random time series.

Index investing would have been a valuable strategy over the many years considered in the time series. Why? Because the time series is not predictable, though, on average the index generates a mean return of 0.03% daily (covered in yet another previous post). So rescaled range analysis seems to have resolved the age old question of active versus passive investing!

Well, not exactly. There is more information to behold in the data. Look at the chart below to see why we cannot quite crown value investing as the winning strategy.

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