R/Omega.R
In AmurG/PerformanceAnalytics: Econometric tools for performance and risk analysis
Defines functions
Omega
Documented in
Omega
#' calculate Omega for a return series
#'
#' Keating and Shadwick (2002) proposed Omega (referred to as Gamma in their
#' original paper) as a way to capture all of the higher moments of the returns
#' distribution.
#'
#' Mathematically, Omega is: integral[L to b](1 - F(r))dr / integral[a to
#' L](F(r))dr
#'
#' where the cumulative distribution F is defined on the interval (a,b). L is
#' the loss threshold that can be specified as zero, return from a benchmark
#' index, or an absolute rate of return - any specified level. When comparing
#' alternatives using Omega, L should be common.
#'
#' Input data can be transformed prior to calculation, which may be useful for
#' introducing risk aversion.
#'
#' This function returns a vector of Omega, useful for plotting. The steeper,
#' the less risky. Above it's mean, a steeply sloped Omega also implies a very
#' limited potential for further gain.
#'
#' Omega has a value of 1 at the mean of the distribution.
#'
#' Omega is sub-additive. The ratio is dimensionless.
#'
#' Kazemi, Schneeweis, and Gupta (2003), in "Omega as a Performance Measure"
#' show that Omega can be written as: Omega(L) = C(L)/P(L) where C(L) is
#' essentially the price of a European call option written on the investment
#' and P(L) is essentially the price of a European put option written on the
#' investment. The maturity for both options is one period (e.g., one month)
#' and L is the strike price of both options.
#'
#' The numerator and the denominator can be expressed as: exp(-Rf=0) * E[max(x
#' - L, 0)] exp(-Rf=0) * E[max(L - x, 0)] with exp(-Rf=0) calculating the
#' present values of the two, where rf is the per-period riskless rate.
#'
#' The first three methods implemented here focus on that observation. The
#' first method takes the simplification described above. The second uses the
#' Black-Scholes option pricing as implemented in fOptions. The third uses the
#' binomial pricing model from fOptions. The second and third methods are not
#' implemented here.
#'
#' The fourth method, "interp", creates a linear interpolation of the cdf of
#' returns, calculates Omega as a vector, and finally interpolates a function
#' for Omega as a function of L. This method requires library \code{Hmisc},
#' which can be found on CRAN.
#'
#' @param R an xts, vector, matrix, data frame, timeSeries or zoo object of
#' asset returns
#' @param L L is the loss threshold that can be specified as zero, return from
#' a benchmark index, or an absolute rate of return - any specified level
#' @param method one of: simple, interp, binomial, blackscholes
#' @param output one of: point (in time), or full (distribution of Omega)
#' @param Rf risk free rate, as a single number
#' @param \dots any other passthru parameters
#' @author Peter Carl
#' @seealso \code{\link[Hmisc]{Ecdf}}
#' @references Keating, J. and Shadwick, W.F. The Omega Function. working
#' paper. Finance Development Center, London. 2002. Kazemi, Schneeweis, and
#' Gupta. Omega as a Performance Measure. 2003.
###keywords ts multivariate distribution models
#' @examples
#'
#' data(edhec)
#' Omega(edhec)
#' Omega(edhec[,13],method="interp",output="point")
#' Omega(edhec[,13],method="interp",output="full")
#'
#' @export
Omega <-
function(R, L = 0, method = c("simple", "interp", "binomial", "blackscholes"), output = c("point", "full"), Rf = 0, ...)
{ # @author Peter Carl
# DESCRIPTION
# Keating and Shadwick (2002) proposed Omega (referred to as Gamma in their
# original paper) as a way to capture all of the higher moments of the
# returns distribution. Mathematically, Omega is:
# integral[L to b](1 - F(r))dr / integral[a to L](F(r))dr
# where the cumulative distribution F is defined on the interval (a,b).
# L is the loss threshold that can be specified as zero, return from a
# benchmark index, or an absolute rate of return - any specified level.
# When comparing alternatives using Omega, L should be common. Input data
# can be transformed prior to calculation, which may be useful for
# introducing risk aversion.
# This function returns a vector of Omega, useful for plotting. The
# steeper, the less risky. Above it's mean, a steeply sloped Omega also
# implies a very limited potential for further gain.
# Omega has a value of 1 at the mean of the distribution.
# Omega is sub-additive. The ratio is dimensionless.
# Kazemi, Schneeweis, and Gupta (2003), in "Omega as a Performance Measure"
# shows that Omega can be written as:
# Omega(L) = C(L)/P(L)
# where C(L) is essentially the price of a European call option written
# on the investment and P(L) is essentially the price of a European put
# option written on the investment. The maturity for both options is
# one period (e.g., one month) and L is the strike price of both options.
# The numerator and the denominator can be expressed as:
# exp(-Rf) * E[max(x - L, 0)]
# exp(-Rf) * E[max(L - x, 0)]
# with exp(-Rf) calculating the present values of the two, where Rf is
# the per-period riskless rate.
# The first three methods implemented here focus on that observation.
# The first method takes the simplification described above. The second
# uses the Black-Scholes option pricing as implemented in fOptions. The
# third uses the binomial pricing model from fOptions. The second and
# third methods are not implemented here.
# The fourth method, "interp", creates a linear interpolation of the cdf of
# returns, calculates Omega as a vector, and finally interpolates a function
# for Omega as a function of L.
# FUNCTION
method = method[1]
output = output[1]
if (is.vector(R)) {
x = na.omit(R)
switch(method,
simple = {
numerator = exp(-Rf) * mean(pmax(x - L, 0))
denominator = exp(-Rf) * mean(pmax(L - x, 0))
omega = numerator/denominator
},
binomial = {
warning("binomial method not yet implemented, using interp")
method = "interp"
},
blackscholes = {
warning("blackscholes method not yet implemented, using interp")
method = "interp"
},
interp = {
# require("Hmisc")
# stopifnot("package:Hmisc" %in% search() || require("Hmisc",quietly=TRUE))
a = min(x)
b = max(x)
xcdf = Hmisc::Ecdf.default(x, pl=FALSE)
f <- approxfun(xcdf$x,xcdf$y,method="linear",ties="ordered")
if(output == "full") {
omega = as.matrix(cumsum(1-f(xcdf$x))/cumsum(f(xcdf$x)))
names(omega) = xcdf$x
}
else {
# returns only the point value for L
# to get a point measure for omega, have to interpolate
omegafull = cumsum(1-f(xcdf$x))/cumsum(f(xcdf$x)) # ????????
g <- approxfun(xcdf$x,omegafull,method="linear",ties="ordered")
omega = g(L)
}
}
) # end method switch
result = omega
return(result)
}
else {
if(length(Rf)>1) Rf<-mean(Rf)
if(length(L)>1) L<-mean(L)
R = checkData(R, method = "matrix", ... = ...)
if(output=="full")
R = R[,1,drop=FALSE] # constrain to one column
result = apply(R, 2, Omega, L = L, method = method, output = output, Rf = Rf,
... = ...)
if(output!="full") {
dim(result) = c(1,NCOL(R))
rownames(result) = paste("Omega (L = ", round(L*100,1),"%)", sep="")
}
colnames(result) = colnames(R)
return(result)
}
}
###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# Copyright (c) 2004-2014 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: Omega.R 3528 2014-09-11 12:43:17Z braverock $
#
###############################################################################
AmurG/PerformanceAnalytics documentation built on May 5, 2019, 4:55 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Defines functions Omega
Documented in Omega
#' calculate Omega for a return series
#'
#' Keating and Shadwick (2002) proposed Omega (referred to as Gamma in their
#' original paper) as a way to capture all of the higher moments of the returns
#' distribution.
#'
#' Mathematically, Omega is: integral[L to b](1 - F(r))dr / integral[a to
#' L](F(r))dr
#'
#' where the cumulative distribution F is defined on the interval (a,b). L is
#' the loss threshold that can be specified as zero, return from a benchmark
#' index, or an absolute rate of return - any specified level. When comparing
#' alternatives using Omega, L should be common.
#'
#' Input data can be transformed prior to calculation, which may be useful for
#' introducing risk aversion.
#'
#' This function returns a vector of Omega, useful for plotting. The steeper,
#' the less risky. Above it's mean, a steeply sloped Omega also implies a very
#' limited potential for further gain.
#'
#' Omega has a value of 1 at the mean of the distribution.
#'
#' Omega is sub-additive. The ratio is dimensionless.
#'
#' Kazemi, Schneeweis, and Gupta (2003), in "Omega as a Performance Measure"
#' show that Omega can be written as: Omega(L) = C(L)/P(L) where C(L) is
#' essentially the price of a European call option written on the investment
#' and P(L) is essentially the price of a European put option written on the
#' investment. The maturity for both options is one period (e.g., one month)
#' and L is the strike price of both options.
#'
#' The numerator and the denominator can be expressed as: exp(-Rf=0) * E[max(x
#' - L, 0)] exp(-Rf=0) * E[max(L - x, 0)] with exp(-Rf=0) calculating the
#' present values of the two, where rf is the per-period riskless rate.
#'
#' The first three methods implemented here focus on that observation. The
#' first method takes the simplification described above. The second uses the
#' Black-Scholes option pricing as implemented in fOptions. The third uses the
#' binomial pricing model from fOptions. The second and third methods are not
#' implemented here.
#'
#' The fourth method, "interp", creates a linear interpolation of the cdf of
#' returns, calculates Omega as a vector, and finally interpolates a function
#' for Omega as a function of L. This method requires library \code{Hmisc},
#' which can be found on CRAN.
#'
#' @param R an xts, vector, matrix, data frame, timeSeries or zoo object of
#' asset returns
#' @param L L is the loss threshold that can be specified as zero, return from
#' a benchmark index, or an absolute rate of return - any specified level
#' @param method one of: simple, interp, binomial, blackscholes
#' @param output one of: point (in time), or full (distribution of Omega)
#' @param Rf risk free rate, as a single number
#' @param \dots any other passthru parameters
#' @author Peter Carl
#' @seealso \code{\link[Hmisc]{Ecdf}}
#' @references Keating, J. and Shadwick, W.F. The Omega Function. working
#' paper. Finance Development Center, London. 2002. Kazemi, Schneeweis, and
#' Gupta. Omega as a Performance Measure. 2003.
###keywords ts multivariate distribution models
#' @examples
#'
#' data(edhec)
#' Omega(edhec)
#' Omega(edhec[,13],method="interp",output="point")
#' Omega(edhec[,13],method="interp",output="full")
#'
#' @export
Omega <-
function(R, L = 0, method = c("simple", "interp", "binomial", "blackscholes"), output = c("point", "full"), Rf = 0, ...)
{ # @author Peter Carl
# DESCRIPTION
# Keating and Shadwick (2002) proposed Omega (referred to as Gamma in their
# original paper) as a way to capture all of the higher moments of the
# returns distribution. Mathematically, Omega is:
# integral[L to b](1 - F(r))dr / integral[a to L](F(r))dr
# where the cumulative distribution F is defined on the interval (a,b).
# L is the loss threshold that can be specified as zero, return from a
# benchmark index, or an absolute rate of return - any specified level.
# When comparing alternatives using Omega, L should be common. Input data
# can be transformed prior to calculation, which may be useful for
# introducing risk aversion.
# This function returns a vector of Omega, useful for plotting. The
# steeper, the less risky. Above it's mean, a steeply sloped Omega also
# implies a very limited potential for further gain.
# Omega has a value of 1 at the mean of the distribution.
# Omega is sub-additive. The ratio is dimensionless.
# Kazemi, Schneeweis, and Gupta (2003), in "Omega as a Performance Measure"
# shows that Omega can be written as:
# Omega(L) = C(L)/P(L)
# where C(L) is essentially the price of a European call option written
# on the investment and P(L) is essentially the price of a European put
# option written on the investment. The maturity for both options is
# one period (e.g., one month) and L is the strike price of both options.
# The numerator and the denominator can be expressed as:
# exp(-Rf) * E[max(x - L, 0)]
# exp(-Rf) * E[max(L - x, 0)]
# with exp(-Rf) calculating the present values of the two, where Rf is
# the per-period riskless rate.
# The first three methods implemented here focus on that observation.
# The first method takes the simplification described above. The second
# uses the Black-Scholes option pricing as implemented in fOptions. The
# third uses the binomial pricing model from fOptions. The second and
# third methods are not implemented here.
# The fourth method, "interp", creates a linear interpolation of the cdf of
# returns, calculates Omega as a vector, and finally interpolates a function
# for Omega as a function of L.
# FUNCTION
method = method[1]
output = output[1]
if (is.vector(R)) {
x = na.omit(R)
switch(method,
simple = {
numerator = exp(-Rf) * mean(pmax(x - L, 0))
denominator = exp(-Rf) * mean(pmax(L - x, 0))
omega = numerator/denominator
},
binomial = {
warning("binomial method not yet implemented, using interp")
method = "interp"
},
blackscholes = {
warning("blackscholes method not yet implemented, using interp")
method = "interp"
},
interp = {
# require("Hmisc")
# stopifnot("package:Hmisc" %in% search() || require("Hmisc",quietly=TRUE))
a = min(x)
b = max(x)
xcdf = Hmisc::Ecdf.default(x, pl=FALSE)
f <- approxfun(xcdf$x,xcdf$y,method="linear",ties="ordered")
if(output == "full") {
omega = as.matrix(cumsum(1-f(xcdf$x))/cumsum(f(xcdf$x)))
names(omega) = xcdf$x
}
else {
# returns only the point value for L
# to get a point measure for omega, have to interpolate
omegafull = cumsum(1-f(xcdf$x))/cumsum(f(xcdf$x)) # ????????
g <- approxfun(xcdf$x,omegafull,method="linear",ties="ordered")
omega = g(L)
}
}
) # end method switch
result = omega
return(result)
}
else {
if(length(Rf)>1) Rf<-mean(Rf)
if(length(L)>1) L<-mean(L)
R = checkData(R, method = "matrix", ... = ...)
if(output=="full")
R = R[,1,drop=FALSE] # constrain to one column
result = apply(R, 2, Omega, L = L, method = method, output = output, Rf = Rf,
... = ...)
if(output!="full") {
dim(result) = c(1,NCOL(R))
rownames(result) = paste("Omega (L = ", round(L*100,1),"%)", sep="")
}
colnames(result) = colnames(R)
return(result)
}
}
###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# Copyright (c) 2004-2014 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: Omega.R 3528 2014-09-11 12:43:17Z braverock $
#
###############################################################################
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