{"id":5307,"date":"2013-02-18T06:31:55","date_gmt":"2013-02-18T11:31:55","guid":{"rendered":"http:\/\/www.jasonapollovoss.local\/?p=5307"},"modified":"2018-09-21T02:04:13","modified_gmt":"2018-09-21T06:04:13","slug":"is-the-sp-500-mean-reverting-rescaled-range-analysis-provides-the-answer","status":"publish","type":"post","link":"https:\/\/jasonapollovoss.com\/web\/2013\/02\/18\/is-the-sp-500-mean-reverting-rescaled-range-analysis-provides-the-answer\/","title":{"rendered":"Is the S&P 500 Mean Reverting? Rescaled Range Analysis Provides the Answer"},"content":{"rendered":"

Editor\u2019s note: Thanks to the diligence of Armin Grueneich this post has been amended to reflect\u00a0an additional\u00a0step\u00a0in the calculation of the rescaled range which has affected the prior reported results.<\/em><\/span><\/p>\n

In a previous post<\/a>, I used the S&P 500 as an example to demonstrate the use of a sophisticated quantitative method, rescaled range analysis<\/a>, 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 \u2014 but its application to investing yields many interesting insights.<\/span><\/p>\n

Without much further explanation in my prior post, I stated that the S&P 500 had a Hurst exponent<\/a>, 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?<\/span><\/p>\n

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.<\/span><\/p>\n

The S&P 500\u2019s Hurst exponent of 0.49 is\u00a0right near the\u00a0‘randomness’ center of 0.5 and only slightly on\u00a0the mean reversion side of things. If we divide the Hurst exponent range into thirds \u2014\u00a00.00 to 0.33 for mean reverting, 0.34 to 0.67 for random, and 0.68 to 1.00 for persistence \u2014\u00a0a strong statement\u00a0can be made\u00a0that the S&P 500 is\u00a0a random time series.<\/span><\/p>\n

Index\u00a0investing 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\u00a0the index\u00a0generates\u00a0a mean return of 0.03% daily (covered in yet another previous post<\/a>).\u00a0 So rescaled range analysis seems\u00a0to have resolved the age old question of active versus passive investing!<\/span><\/p>\n

Well, not exactly. There is more information to behold in the data.\u00a0Look at the chart below to see why we cannot quite crown value investing as the winning strategy.<\/span><\/p>\n

\n

Rescaled Range Analysis of the S&P 500 (3 January 1950 to 15 November 2012)<\/strong><\/span>
\n\"Rescaled-Range-Analysis-of-the-SP-500\"<\/a><\/span><\/p>\n

\n

While there is a very high r-squared for the regression line\u00a0\u2014 0.98\u00a0\u2014 the\u00a0actual data\u00a0‘wiggle.’ 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.<\/span><\/p>\n

As you can see from the graph,\u00a0randomness\u00a0does not hold for all of the time scales. So what do they say? They say that for the first half of its history the S&P 500\u00a0showed persistence with\u00a0daily returns for the S&P 500 having a Hurst exponent of 0.61 \u2014\u00a0in the realm between random and persistent.\u00a0 Whereas, in the second half of its existence the S&P 500 is actually more\u00a0random with a Hurst exponent of 0.46.\u00a0 This suggests two very different types of investment environments over the long history of the S&P 500.\u00a0 It would be interesting to corroborate the experience of traders from many decades ago with the preceding result and to compare them with veterans of a more recent era.<\/span><\/p>\n

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.<\/span><\/p>\n

Finally, in Euclidean geometry<\/a> a straight line<\/em>\u00a0\u2014 i.e., a time series with no variation in our case \u2014\u00a0is 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<\/em><\/a> and is calculated as D = 2 \u2013 Hurst Exponent.<\/span><\/p>\n

For the S&P 500, the fractal dimension for the entire time series is 2 \u2013 0.49 = 1.51. 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\u00a0fluctuates between\u00a01.39, calculated as 2 \u2013 0.61, and 1.54, calculated as 2 – 0.46.<\/span><\/p>\n

All of this is a quantitative way to communicate a qualitative message: The S&P 500 is a volatile time series. Caveat emptor!<\/span><\/p>\n

 <\/p>\n

\n

Photo credit: \u00a9iStockphoto.com\/luchschen<\/span><\/p>\n

 <\/p>\n

Originally published on CFA Institute\u2019s \u00a0Enterprising Investor<\/a>.<\/em><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Editor\u2019s note: Thanks to the diligence of Armin Grueneich this post has been amended to reflect\u00a0an additional\u00a0step\u00a0in the calculation of the rescaled range which has affected the prior reported results. 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 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[3],"tags":[170,165,164,133],"class_list":["post-5307","post","type-post","status-publish","format-standard","hentry","category-the-blog","tag-mean-reversion","tag-quantitative-methods","tag-rescaled-range-analysis","tag-sp-500"],"_links":{"self":[{"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/posts\/5307","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/comments?post=5307"}],"version-history":[{"count":0,"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/posts\/5307\/revisions"}],"wp:attachment":[{"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/media?parent=5307"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/categories?post=5307"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jasonapollovoss.com\/web\/wp-json\/wp\/v2\/tags?post=5307"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}