*By Jason Voss CFA*

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 0.35, for the time period January 3, 1950 to November 15, 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 0.35 is 30% [= (0.50 - 0.35) รท 0.50] of the distance away from the randomness center of 0.5 and on the mean reversion side of things. Even 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 - the strongest statement that can be made about the S&P 500 is that the index is very weakly random and borders on mean reversion.

Value investing would have been a valuable strategy over the many years considered in the time series. Why? Because after price declines, the index usually returns back to generating its mean return of 0.03% daily (covered in yet another previous post), so having knowledge of valuations that are below the mean resulted in excess returns. Rescaled range analysis has resolved the age old question of growth versus value 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.

**Rescaled Range Analysis of the S&P 500 (3 January 1950 to 15 November 2012)**

While there is a very high r-squared for the regression line - 0.98 - the trend line is a match only in the smaller ranges of the rescaled range analysis. You see rescaled range analysis and the Hurst exponent are ways of looking at the self-similarity, or autocorrelation, of data. When rescaled range analysis is conducted, data are divided into ever smaller ranges. Those smaller ranges are then evaluated to see if the relationships detected for the whole data series are present in smaller time scales, too. Incidentally, this is why this type of analysis is favored by chaos theoreticians, who also focus on self-similarity and scalability of phenomenon in hard science data.

As you can see from the graph, mean reversion does not hold for all of the time scales. So what do they say? They say that for very long time scales, daily returns for the S&P 500 have a Hurst exponent of 0.60 - in the realm between random and persistent.

How long are these time scales since we are looking at lognormal data? The time frames are precisely 7,911 trading days (31.39 years!) up to the full 15,821 trading days examined. Put another way, over the very long run growth investors would have benefited as subsequent periods are of the same sign as preceding periods: up.

Finally, in Euclidean geometry *a straight line* - i.e., a time series with no variation in our case - is considered to have a topological dimension of 1, whereas a square is a 2-dimensional surface and a cube is a three-dimensional object. Yet financial market data do not unfold in a straight line, and in fact the rougher they are the more closely they approach being a surface of two dimensions. A measure for the smoothness/roughness of data is the *fractal dimension* and is calculated as D = 2 - Hurst Exponent.

For the S&P 500, the fractal dimension for the entire time series is 2 - 0.35 = 1.65. In other words, the time series line of daily returns for the S&P 500 is fairly rough. Yet, at shorter time frames, as discussed above, the fractal dimension shrinks to around 1.40, calculated as 2 - 0.60.

All of this is a quantitative way to communicate a qualitative message: The S&P 500 is a volatile time series. Caveat emptor!

**Disclaimer**: Please note that the content of this site should not be construed as investment advice, nor do the opinions expressed necessarily reflect the views of CFA Institute.