Hurst obtained a dimensionless statistical exponent by dividing the range by the standard deviation of the observations, so this approach is commonly referred to as rescaled range (R/S) analysis.
HurstIndex(R, ...)
| R | an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns |
|---|---|
| … | any other passthru parameters |
$$H = log(m)/log(n)$$
where \(m = [max(r_i) - min(r_i)]/sigma_p\) and \(n = number of observations\) A Hurst index between 0.5 and 1 suggests that the returns are persistent. At 0.5, the index suggests returns are totally random. Between 0 and 0.5 it suggests that the returns are mean reverting.
H.E. Hurst originally developed the Hurst index to help establish optimal water storage along the Nile. Nile floods are extremely persistent, measuring a Hurst index of 0.9. Peters (1991) notes that Equity markets have a Hurst index in excess of 0.5, with typical values of around 0.7. That appears to be anomalous in the context of the mainstream 'rational behaviour' theories of economics, and suggests existence of a powerful 'long-term memory' causal dependence. Clarkson (2001) suggests that an 'over-reaction bias' could be expected to generate a powerful 'long-term memory' effect in share prices.
Clarkson, R. (2001) FARM: a financial actuarial risk model. In Chapter 12 of Managing Downside Risk in Financial Markets, ed. Sortino, F. and Satchel, S. Woburn MA. Butterworth-Heinemann Finance.
Peters, E.E (1991) Chaos and Order in Capital Markets, New York: Wiley.
Bacon, Carl. (2008) Practical Portfolio Performance Measurement and Attribution, 2nd Edition. London: John Wiley & Sons.