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30 January 2013

Rescaled Range Analysis: A Method for

Detecting Persistence, Randomness, or Mean

Reversion in Financial Markets

By Jason Voss, CFA

Categories: Quantitative Methods

Editor’s note: Thanks to the diligence of Armin Grueneich

this post has been amended to reflect the addition of step

#5, below, in the calculation of the rescaled range.

Rescaled range analysis is a statistical technique

designed to assess the nature and magnitude of

variability in data over time. In investing rescaled range

analysis has been used to detect and evaluate the

amount of persistence, randomness, or mean reversion

in financial markets time series data. Insight of this kind

into financial data naturally suggests investment

strategies.

Originally invented for the field of hydrology by Harold

Edwin Hurst, the technique was developed to predict Nile River flooding in advance of the

construction of the Aswan High Dam. The dam needed to fulfill multiple and divergent

purposes, including serving as both a store of water to protect against drought for farmers

down river, and as flood protection for those same farmers during typical annual flooding.

Rainfall levels in Central Africa were seemingly random each year, yet the Nile River flows

seemed to show autocorrelation. That is, rainfall in one time period seemed to influence

rainfall in subsequent periods. Hurst needed to be able to see if there was a hidden

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long-term trend — statistically known as a long-memory process — in the Nile River data

that might guide him in building a better dam for Egypt.

Does this sound familiar? A time series of varying levels that is seemingly random but in

which it is suspected that there might also be a long-term, hidden trend. Not surprisingly

rescaled range analysis had its moment in the financial analysis sun in the mid-1990s,

when chaos theory, as applied to financial markets was a hot topic. Chaos theory is a

branch of science that studies the interconnectedness of events that otherwise, on the

surface, seem random.

Closely associated with rescaled range analysis is the Hurst exponent, indicated by H,

also known as the “index of dependence” or the “index of long-range dependence.” A

Hurst exponent ranges between 0 and 1, and measures three types of trends in a time

series: persistence, randomness, or mean reversion.

If a time series is persistent with H ≥ 0.5, then a future data point is likely to be like

a data point preceding it. So an equity with H of 0.77 that has been up for the past

week is more likely to be up next week as well, because its Hurst exponent is

greater than 0.5.

If the Hurst exponent of a time series is H < 0.5, then it is likely to reverse trend

over the time frame considered. Thus, an equity with H = 0.26 that was up last

month is more likely than chance to be down next month.

Time series that have Hurst exponents near to 0.5 display a random (i.e., a

stochastic) process, in which knowing one data point does not provide insight into

predicting future data points in the series.

So what are the steps to conducting a rescaled range analysis and to estimating the Hurst

exponent? As an instructional example, please reference the spreadsheet of the rescaled

range analysis of daily return data for the S&P 500 Index from 3 January 1950 through 15

November 2012.

Rescaled Range Analysis Steps

1. Choose your time series. Do you want to analyze fluctuations in the yield curve? West

Texas sweet crude? Apple (AAPL) or Google (GOOG) stock? Or the Dow Jones Industrial

Average (DJIA)? Here I am going to select the S&P 500’s daily returns.

2. Choose your ranges. Rescaled range analysis depends on multiple lengths of time

(i.e., ranges) to be analyzed and chosen arbitrarily by the analyst. In the example of the

S&P 500, there are 15,821 daily returns. So I chose the following ranges, all powers of


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two:

a. Size of range is the entire data series = one range of 15,821 daily returns.

b. Size of each range is 1/2 of the entire data series = 15,821 ÷ 2 = two ranges of

either 7,911 or 7,910 daily returns.

c. Size of each range is 1/4 of the entire data series = 15,821 ÷ 4 = four ranges of

either 3,956 or 3,955 daily returns.

d. Size of each range is 1/8 of the entire data series = 15,821 ÷ 8 = eight ranges of

either 1,978 or 1,977.

e. Size of each range is 1/16 of the entire data series = 15,821 ÷ 16 = sixteen ranges

of either 989 or 988 daily returns.

f. Size of each range is 1/32 of the entire data series = 15,821 ÷ 32 = thirty-two

ranges of either 495 or 494 daily returns.

3. Calculate the mean for each range . For each of the ranges, calculate a mean per the

formula below. Note: In the above example of the S&P 500 there are 1 + 2 + 4 + 8 + 16 +

32 = 63 means calculated, one for each range.

Where:

s = series (Series 1 is whole data series for S&P 500, or 15,821 daily returns;

series 5 is 16 ranges of either 989 or 988 daily returns.)

n = the size of the range for which you are calculating the mean

X = the value of one element in the range

4. Create a series of deviations for each range. Create another time series of

deviations using the mean for each range. Note: In the case of the S&P 500, there will be

six new “deviations from the mean” ranges, given the six categories of ranges chosen in

Step 2 above (i.e. ranges a, b, c, d, e, and f).


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Where:

Y = the new time series adjusted for deviations from the mean

X = the value of one element in the range

m = the mean for the range calculated in Step 3 above

5. Create a series which is the running total of the d eviations from the mean. Now

that you have a series of deviations from the mean for each range, you need to calculate

a running total for each range’s deviations from the mean.

Where:

y = the running total of the deviations from the mean for each series

Y = the time series adjusted for deviations from the mean

6. Calculate the widest difference in the series of de viations. Find both the maximum

and minimum values in the series of deviations for each range. Take the difference

between the maximum and minimum in order to calculate the widest difference. Note: For

the S&P 500 example, there are 63 calculations, one for each of the 63 ranges.

Where:

R = the widest spread in each range

Y = the value of one element in the “deviations from the mean” range

7. Calculate the standard deviation for each range. Note: There will be 63 standard


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deviations, one for each range.

8. Calculate the rescaled range for each range in the time series. This step creates a

new measure for each range in the time series that shows how wide is the range

measured in standard deviations.

Where:

R/S = the rescaled range for each range in the time series

R = the range created in step 5 above

σ = the standard deviation for the range under consideration

9. Average the rescaled range values for each region t o summarize each range. For

each region, average the rescaled range (R/S) values. Using the S&P 500 data as an

example, we have the following R/S values for each of the four ranges of ~3,955 in size:

“Range 1/4”, part 1, R/S: 83.04

“Range 1/4”, part 2, R/S: 63.51

“Range 1/4”, part 3, R/S: 84.16

“Range 1/4”, part 4, R/S: 88.09

Average of the four R/S values for “Range 1/4” = (83.04 + 63.51 + 84.16 + 88.09) ÷ 4

= 79.70

For the S&P 500 we have the following values for the rescaled ranges:


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Now that you have rescaled each range in the time series, you can calculate the Hurst

exponent, H, that will summarize in one number the degree of persistence, randomness,

or mean reversion in your time series.

Calculating the Hurst Exponent Steps

1. Calculate the logarithmic values for the size of ea ch region and for each region’s

rescaled range. For example, consider the above S&P 500 data:

2. Plot the logarithm of the size ( x axis) of each series versus the logarithm of the

rescaled range ( y axis). This results in a graph that looks something like this one for the

S&P 500:

Rescaled Range Analysis of the S&P 500 (3 January 1 950 to 15 November 2012)


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30 January 2013 at 12:20 Reply ↓

3. Calculate the slope of the data to find the Hurst e xponent. H is the slope of the plot

of each range’s log (R/S) versus each range’s log (size). For the S&P 500 for 3 January

1950 to 15 November, H is 0.49. Recall that this means that the S&P 500 demonstrates

randomness.

Knowing the H, suggests some hypothetical trading strategies. For example, stocks with H

≥ 0.5 — that is, persistence — and positive price appreciation would be attractive to a

growth manager wanting future capital appreciation. Whereas, stocks with H < 0.5 with

prices declining for some time suggest an eventual price trend reversal to a value investor.

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.

Photo credit: ©iStockphoto.com/ugurhan

Tags: equities, quantitative methods, S&P 500

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25 comments on “Rescaled Range Analysis: A Method f orDetecting Persistence, Randomness, or Mean Reversio n inFinancial Markets”

Hisham Shariff said:

I have read your write-up with great interest. Have you carried out the hurst

exponent calculations, and if so which stocks and markets. How can I use this

for shares listed in the Colombo Stock Exchange. Just one more question is this

something like mean reversion?

Jason A. Voss, CFA said:

Hello Hisham,

Thanks for your comment. Yes, the Hurst exponent calculations for the S&P 500

will appear in a subsequent post here on CFA Institute’s The Enterprising Investor

95J’aimeJ’aime


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Documents

Rescaled Range Analysis