R/HurstIndex.R

In PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis

#' calculate the Hurst Index
#' The Hurst index can be used to measure whether returns are mean reverting,
#' totally random, or persistent.  
#' 
#' 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. 
#' 
#' \deqn{H = log(m)/log(n)} 
#' 
#' where
#' \eqn{m = [max(r_i) - min(r_i)]/sigma_p} and
#' \eqn{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.
#' 
#' @param R an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns
#' @param \dots any other passthru parameters
#' @references
#' 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. 
#' 
#' @export
HurstIndex <-
function (R, ...) {
    



    R = checkData(R)

    rs <- function(R) {
        R = na.omit(R)
        m = (max(R) - min(R))/sd.xts(R)
        n = length(R)
        result = log(m)/log(n)
        return(result)
    }

    result = apply(R, MARGIN = 2, rs)
    dim(result) = c(1,NCOL(R))
    colnames(result) = colnames(R)
    rownames(result) = "Hurst Index"
    return (result)
}

RescaledRange <- 
function (R, ...) {
    HurstIndex(R, ...)
}

###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# Copyright (c) 2004-2020 Peter Carl and Brian G. Peterson
#
# This R package is distributed under the terms of the GNU Public License (GPL)
# for full details see the file COPYING
#
# $Id$
#
###############################################################################