Nothing
R/ES.R
In PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis
###############################################################################
# $Id$
###############################################################################
#' calculates Expected Shortfall(ES) (or Conditional Value-at-Risk(CVaR) for
#' univariate and component, using a variety of analytical methods.
#'
#' Calculates Expected Shortfall(ES) (also known as) Conditional Value at
#' Risk(CVaR) or Expected Tail Loss (ETL) for univariate, component,
#' and marginal cases using a variety of analytical methods.
#'
#'
#' @export
#' @aliases ES CVaR ETL
#' @rdname ES
#' @param R a vector, matrix, data frame, timeSeries or zoo object of asset
#' returns
#' @param p confidence level for calculation, default p=.95
#' @param method one of "modified","gaussian","historical", see
#' Details.
#' @param clean method for data cleaning through \code{\link{Return.clean}}.
#' Current options are "none", "boudt", "geltner", or "locScaleRob".
#' @param portfolio_method one of "single","component","marginal" defining
#' whether to do univariate, component, or marginal calc, see Details.
#' @param weights portfolio weighting vector, default NULL, see Details
#' @param mu If univariate, mu is the mean of the series. Otherwise mu is the
#' vector of means of the return series, default NULL, see Details
#' @param sigma If univariate, sigma is the variance of the series. Otherwise
#' sigma is the covariance matrix of the return series, default NULL, see
#' Details
#' @param m3 If univariate, m3 is the skewness of the series. Otherwise m3 is
#' the coskewness matrix (or vector with unique coskewness values) of the
#' returns series, default NULL, see Details
#' @param m4 If univariate, m4 is the excess kurtosis of the series. Otherwise
#' m4 is the cokurtosis matrix (or vector with unique cokurtosis values) of the
#' return series, default NULL, see Details
#' @param invert TRUE/FALSE whether to invert the VaR measure, see Details.
#' @param operational TRUE/FALSE, default TRUE, see Details.
#' @param \dots any other passthru parameters
#' @param SE TRUE/FALSE whether to ouput the standard errors of the estimates of the risk measures, default FALSE.
#' @param SE.control Control parameters for the computation of standard errors. Should be done using the \code{\link{RPESE.control}} function.
#' @note The option to \code{invert} the ES measure should appease both
#' academics and practitioners. The mathematical definition of ES as the
#' negative value of extreme losses will (usually) produce a positive number.
#' Practitioners will argue that ES denotes a loss, and should be internally
#' consistent with the quantile (a negative number). For tables and charts,
#' different preferences may apply for clarity and compactness. As such, we
#' provide the option, and set the default to TRUE to keep the return
#' consistent with prior versions of PerformanceAnalytics, but make no value
#' judgement on which approach is preferable.
#'
#' @section Background:
#'
#' This function provides several estimation methods for
#' the Expected Shortfall (ES) (also called Expected Tail Loss (ETL)
#' or Conditional Value at Risk (CVaR)) of a return series and the Component ES
#' (ETL/CVaR) of a portfolio.
#'
#' At a preset probability level denoted \eqn{c}, which typically is between 1
#' and 5 per cent, the ES of a return series is the negative value of the
#' expected value of the return when the return is less than its
#' \eqn{c}-quantile. Unlike value-at-risk, conditional value-at-risk has all
#' the properties a risk measure should have to be coherent and is a convex
#' function of the portfolio weights (Pflug, 2000). With a sufficiently large
#' data set, you may choose to estimate ES with the sample average of all
#' returns that are below the \eqn{c} empirical quantile. More efficient
#' estimates of VaR are obtained if a (correct) assumption is made on the
#' return distribution, such as the normal distribution. If your return series
#' is skewed and/or has excess kurtosis, Cornish-Fisher estimates of ES can be
#' more appropriate. For the ES of a portfolio, it is also of interest to
#' decompose total portfolio ES into the risk contributions of each of the
#' portfolio components. For the above mentioned ES estimators, such a
#' decomposition is possible in a financially meaningful way.
#'
#' @section Univariate estimation of ES:
#'
#' The ES at a probability level \eqn{p} (e.g. 95\%) is the negative value of
#' the expected value of the return when the return is less than its
#' \eqn{c=1-p} quantile. In a set of returns for which sufficently long history
#' exists, the per-period ES can be estimated by the negative value of the
#' sample average of all returns below the quantile. This method is also
#' sometimes called \dQuote{historical ES}, as it is by definition \emph{ex
#' post} analysis of the return distribution, and may be accessed with
#' \code{method="historical"}.
#'
#' When you don't have a sufficiently long set of returns to use non-parametric
#' or historical ES, or wish to more closely model an ideal distribution, it is
#' common to us a parmetric estimate based on the distribution. Parametric ES
#' does a better job of accounting for the tails of the distribution by more
#' precisely estimating shape of the distribution tails of the risk quantile.
#' The most common estimate is a normal (or Gaussian) distribution \eqn{R\sim
#' N(\mu,\sigma)} for the return series. In this case, estimation of ES requires
#' the mean return \eqn{\bar{R}}, the return distribution and the variance of
#' the returns \eqn{\sigma}. In the most common case, parametric VaR is thus
#' calculated by
#'
#' \deqn{\sigma=variance(R)}{sigma=var(R)}
#'
#' \deqn{ES=-\bar{R} + \sqrt{\sigma} \cdot \frac{1}{c}\phi(z_{c}) }{VaR= -mean(R) + sqrt(sigma)*dnorm(z_c)/c}
#'
#' where \eqn{z_{c}} is the \eqn{c}-quantile of the standard normal
#' distribution. Represented in \R by \code{qnorm(c)}, and may be accessed with
#' \code{method="gaussian"}. The function \eqn{\phi}{dnorm} is the Gaussian
#' density function.
#'
#'
#' The limitations of Gaussian ES are well covered in the literature, since most
#' financial return series are non-normal. Boudt, Peterson and Croux (2008)
#' provide a modified ES calculation that takes the higher moments of non-normal
#' distributions (skewness, kurtosis) into account through the use of a
#' Cornish-Fisher expansion, and collapses to standard (traditional) Gaussian ES
#' if the return stream follows a standard distribution. More precisely, for a
#' loss probability \eqn{c}, modified ES is defined as the negative of the
#' expected value of all returns below the \eqn{c} Cornish-Fisher quantile and
#' where the expectation is computed under the second order Edgeworth expansion
#' of the true distribution function.
#'
#' Modified expected shortfall should always be larger than modified Value at
#' Risk. Due to estimation problems, this might not always be the case. Set
#' Operational = TRUE to replace modified ES with modified VaR in the
#' (exceptional) case where the modified ES is smaller than modified VaR.
#'
#' @section Component ES:
#'
#' By setting \code{portfolio_method="component"} you may calculate the ES
#' contribution of each element of the portfolio. The return from the function in
#' this case will be a list with three components: the univariate portfolio ES,
#' the scalar contribution of each component to the portfolio ES (these will sum
#' to the portfolio ES), and a percentage risk contribution (which will sum to
#' 100\%).
#'
#' Both the numerical and percentage component contributions to ES may contain
#' both positive and negative contributions. A negative contribution to Component
#' ES indicates a portfolio risk diversifier. Increasing the position weight will
#' reduce overall portoflio ES.
#'
#' If a weighting vector is not passed in via \code{weights}, the function will
#' assume an equal weighted (neutral) portfolio.
#'
#' Multiple risk decomposition approaches have been suggested in the literature.
#' A naive approach is to set the risk contribution equal to the stand-alone
#' risk. This approach is overly simplistic and neglects important
#' diversification effects of the units being exposed differently to the
#' underlying risk factors. An alternative approach is to measure the ES
#' contribution as the weight of the position in the portfolio times the partial
#' derivative of the portfolio ES with respect to the component weight. \deqn{C_i
#' \mbox{ES} = w_i \frac{ \partial \mbox{ES} }{\partial w_i}.}{C[i]ES =
#' w[i]*(dES/dw[i]).} Because the portfolio ES is linear in position size, we
#' have that by Euler's theorem the portfolio VaR is the sum of these risk
#' contributions. Scaillet (2002) shows that for ES, this mathematical
#' decomposition of portfolio risk has a financial meaning. It equals the
#' negative value of the asset's expected contribution to the portfolio return
#' when the portfolio return is less or equal to the negative portfolio VaR:
#'
#' \deqn{C_i \mbox{ES} = = -E\left[ w_i r_{i} | r_{p} \leq - \mbox{VaR}\right]}{C[i]ES = -E( w[i]r[i]|rp<=-VaR ) }
#'
#' For the decomposition of Gaussian ES, the estimated mean and covariance
#' matrix are needed. For the decomposition of modified ES, also estimates of
#' the coskewness and cokurtosis matrices are needed. If \eqn{r} denotes the
#' \eqn{Nx1} return vector and \eqn{mu} is the mean vector, then the \eqn{N
#' \times N^2} co-skewness matrix is
#
#' \deqn{ m3 = E\left[ (r - \mu)(r - \mu)' \otimes (r - \mu)'\right]}{m3 = E[ (r - mu)(r - mu)' \%x\% (r - \mu)']}
#'
#' The \eqn{N \times N^3} co-kurtosis matrix is
#'
#' \deqn{ m_{4} =
#' E\left[ (r - \mu)(r - \mu)' \otimes (r - \mu)'\otimes (r - \mu)'
#' \right] }{E[ (r - \mu)(r - \mu)' \%x\% (r - \mu)'\%x\% (r - \mu)']}
#'
#' where \eqn{\otimes}{\%x\%} stands for the Kronecker product. The matrices can
#' be estimated through the functions \code{skewness.MM} and \code{kurtosis.MM}.
#' More efficient estimators were proposed by Martellini and Ziemann (2007) and
#' will be implemented in the future.
#'
#' As discussed among others in Cont, Deguest and Scandolo (2007), it is
#' important that the estimation of the ES measure is robust to single outliers.
#' This is especially the case for modified VaR and its decomposition, since
#' they use higher order moments. By default, the portfolio moments are
#' estimated by their sample counterparts. If \code{clean="boudt"} then the
#' \eqn{1-p} most extreme observations are winsorized if they are detected as
#' being outliers. For more information, see Boudt, Peterson and Croux (2008)
#' and \code{\link{Return.clean}}. If your data consist of returns for highly
#' illiquid assets, then \code{clean="geltner"} may be more appropriate to
#' reduce distortion caused by autocorrelation, see \code{\link{Return.Geltner}}
#' for details.
#'
#' @author Brian G. Peterson and Kris Boudt
#' @seealso \code{\link{VaR}} \cr \code{\link{SharpeRatio.modified}} \cr
#' \code{\link{chart.VaRSensitivity}} \cr \code{\link{Return.clean}}
#'
#' @references Boudt, Kris, Peterson, Brian, and Christophe Croux. 2008.
#' Estimation and decomposition of downside risk for portfolios with non-normal
#' returns. 2008. The Journal of Risk, vol. 11, 79-103.
#'
#' Cont, Rama, Deguest, Romain and Giacomo Scandolo. Robustness and sensitivity
#' analysis of risk measurement procedures. Financial Engineering Report No.
#' 2007-06, Columbia University Center for Financial Engineering.
#'
#' Laurent Favre and Jose-Antonio Galeano. Mean-Modified Value-at-Risk
#' Optimization with Hedge Funds. Journal of Alternative Investment, Fall 2002,
#' v 5.
#'
#' Martellini, L. and Ziemann, V., 2010. Improved estimates of higher-order
#' comoments and implications for portfolio selection. Review of Financial
#' Studies, 23(4):1467-1502.
#'
#' Pflug, G. Ch. Some remarks on the value-at-risk and the conditional
#' value-at-risk. In S. Uryasev, ed., Probabilistic Constrained Optimization:
#' Methodology and Applications, Dordrecht: Kluwer, 2000, 272-281.
#'
#' Scaillet, Olivier. Nonparametric estimation and sensitivity analysis of
#' expected shortfall. Mathematical Finance, 2002, vol. 14, 74-86.
###keywords ts multivariate distribution models
#' @examples
#'
#' if(!( Sys.info()[['sysname']]=="Windows") ){
#' # if on Windows, cut and paste this example
#'
#' data(edhec)
#'
#' # first do normal ES calc
#' ES(edhec, p=.95, method="historical")
#'
#' # now use Gaussian
#' ES(edhec, p=.95, method="gaussian")
#'
#' # now use modified Cornish Fisher calc to take non-normal distribution into account
#' ES(edhec, p=.95, method="modified")
#'
#' # now use p=.99
#' ES(edhec, p=.99)
#' # or the equivalent alpha=.01
#' ES(edhec, p=.01)
#'
#' # now with outliers squished
#' ES(edhec, clean="boudt")
#'
#' # add Component ES for the equal weighted portfolio
#' ES(edhec, clean="boudt", portfolio_method="component")
#'
#' } # end CRAN Windows check
#'
#' @export ETL CVaR ES
ETL <- CVaR <- ES <- function (R=NULL , p=0.95, ...,
method=c("modified","gaussian","historical"),
clean=c("none","boudt", "geltner", "locScaleRob"),
portfolio_method=c("single","component"),
weights=NULL, mu=NULL, sigma=NULL, m3=NULL, m4=NULL,
invert=TRUE, operational=TRUE,
SE=FALSE, SE.control=NULL)
{ # @author Brian G. Peterson
# Descripion:
# wrapper for univariate and multivariate ES functions.
# Fix parameters if SE=TRUE
if(SE){
# Setting the control parameters
if(is.null(SE.control))
SE.control <- RPESE.control(estimator="ES")
# Fix the method
method="historical"
portfolio_method="single"
invert=FALSE
if(SE.control$cleanOutliers=="locScaleRob")
clean="locScaleRob" else
clean="none"
}
# Setup:
#if(exists(modified)({if( modified == TRUE) { method="modified" }}
#if(method == TRUE or is.null(method) ) { method="modified" }
method = method[1]
clean = clean[1]
portfolio_method = portfolio_method[1]
if (is.null(weights) & portfolio_method != "single"){
message("no weights passed in, assuming equal weighted portfolio")
weights=t(rep(1/dim(R)[[2]], dim(R)[[2]]))
}
if(!is.null(R)){
R <- checkData(R, method="xts")
columns=colnames(R)
if (!is.null(weights) & portfolio_method != "single") {
if ( length(weights) != ncol(R)) {
stop("number of items in weights not equal to number of columns in R")
}
}
# weights = checkData(weights, method="matrix", ...) #is this necessary?
# TODO check for date overlap with R and weights
if(clean!="none" & is.null(mu)){ # the assumption here is that if you've passed in any moments, we'll leave R alone
R = as.matrix(Return.clean(R, method=clean))
}
if(portfolio_method != "single"){
# get the moments ready
if (is.null(mu)) { mu = apply(R,2,'mean' ) }
if (is.null(sigma)) { sigma = cov(R) }
if(method=="modified"){
if (is.null(m3)) {m3 = M3.MM(R,as.mat=FALSE,mu=mu)}
if (is.null(m4)) {m4 = M4.MM(R,as.mat=FALSE,mu=mu)}
}
}
} else {
#R is null, check for moments
if(is.null(mu)) stop("Nothing to do! You must pass either R or the moments mu, sigma, etc.")
if ( length(weights) != length(mu)) {
stop("number of items in weights not equal to number of items in the mean vector")
}
}
if(isTRUE(SE)){
if(!requireNamespace("RPESE", quietly = TRUE)){
stop("Package \"pkg\" needed for standard errors computation. Please install it.",
call. = FALSE)
}
# Computation of SE (optional)
ses=list()
# For each of the method specified in se.method, compute the standard error
for(mymethod in SE.control$se.method){
ses[[mymethod]]=RPESE::EstimatorSE(R, estimator.fun = "ES", se.method = mymethod,
cleanOutliers=SE.control$cleanOutliers,
fitting.method=SE.control$fitting.method,
freq.include=SE.control$freq.include,
freq.par=SE.control$freq.par,
a=SE.control$a, b=SE.control$b,
p=p, # Additional parameter
...)
}
ses <- t(data.frame(ses))
}
switch(portfolio_method,
single = {
if(is.null(weights)){
columns=colnames(R)
switch(method,
modified = { if (operational) rES = operES.CornishFisher(R=R,p=p)
else rES = ES.CornishFisher(R=R,p=p)
}, # mu=mu, sigma=sigma, skew=skew, exkurt=exkurt))},)
gaussian = { rES = ES.Gaussian(R=R,p=p) },
historical = { rES = ES.historical(R=R,p=p) }
) # end single method switch calc
# convert from vector to columns
rES=as.matrix(rES)
colnames(rES)=columns
} else { # we have weights, so we should use the .MM calc
weights=as.vector(weights)
switch(method,
modified = { rES=mES.MM(w=weights, mu=mu, sigma=sigma, M3=m3 , M4=m4 , p=p) },
gaussian = { rES=GES.MM(w=weights, mu=mu, sigma=sigma, p=p) },
historical = { rES = ES.historical(R=R,p=p) %*% weights }, # note that this is not tested for weighting the univariate calc by the weights,
) # end multivariate method
}
# check for unreasonable results
columns<-ncol(rES)
for(column in 1:columns) {
tmp=rES[,column]
if (!is.finite(tmp)) {
message(c("ES calculation returned non-finite result for column: ", column, " : ", rES[, column]))
# set ES to NA, since risk is unreasonable
rES[, column] <- NA
} else
if (eval(0 > tmp)) { #eval added previously to get around Sweave bitching
message(c("ES calculation produces unreliable result (inverse risk) for column: ",column," : ",rES[,column]))
# set ES to NA, since inverse risk is unreasonable
rES[,column] <- NA
} else
if (eval(1 < tmp)) { #eval added previously to get around Sweave bitching
message(c("ES calculation produces unreliable result (risk over 100%) for column: ",column," : ",rES[,column]))
# set ES to 1, since greater than 100% is unreasonable
rES[,column] <- 1
}
} # end reasonableness checks
if(invert) rES <- -rES
rownames(rES) <- "ES"
if(SE) # Check if SE computation
return(rbind(rES, ses)) else
return(rES)
}, # end single portfolio switch
component = {
# @todo need to add another loop here for subsetting, I think, when weights is a timeseries
#if (mu=NULL or sigma=NULL) {
# pfolioret = Return.portfolio(R, weights, wealth.index = FALSE, contribution=FALSE, method = c("simple"))
#}
# for now, use as.vector
weights=as.vector(weights)
names(weights)<-colnames(R)
switch(method,
modified = { if (operational) return(operES.CornishFisher.portfolio(p,weights,mu,sigma,m3,m4))
else return(ES.CornishFisher.portfolio(p,weights,mu,sigma,m3,m4))
},
gaussian = { return(ES.Gaussian.portfolio(p,weights,mu,sigma)) },
historical = { return(ES.historical.portfolio(R, p,weights)) },
kernel = { return(ES.kernel.portfolio(R=R,p=p,w=weights)) }
)
} # end component portfolio switch
)
} # end ES wrapper function
###############################################################################
# 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
#
# $Id$
#
###############################################################################
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###############################################################################
# $Id$
###############################################################################
#' calculates Expected Shortfall(ES) (or Conditional Value-at-Risk(CVaR) for
#' univariate and component, using a variety of analytical methods.
#'
#' Calculates Expected Shortfall(ES) (also known as) Conditional Value at
#' Risk(CVaR) or Expected Tail Loss (ETL) for univariate, component,
#' and marginal cases using a variety of analytical methods.
#'
#'
#' @export
#' @aliases ES CVaR ETL
#' @rdname ES
#' @param R a vector, matrix, data frame, timeSeries or zoo object of asset
#' returns
#' @param p confidence level for calculation, default p=.95
#' @param method one of "modified","gaussian","historical", see
#' Details.
#' @param clean method for data cleaning through \code{\link{Return.clean}}.
#' Current options are "none", "boudt", "geltner", or "locScaleRob".
#' @param portfolio_method one of "single","component","marginal" defining
#' whether to do univariate, component, or marginal calc, see Details.
#' @param weights portfolio weighting vector, default NULL, see Details
#' @param mu If univariate, mu is the mean of the series. Otherwise mu is the
#' vector of means of the return series, default NULL, see Details
#' @param sigma If univariate, sigma is the variance of the series. Otherwise
#' sigma is the covariance matrix of the return series, default NULL, see
#' Details
#' @param m3 If univariate, m3 is the skewness of the series. Otherwise m3 is
#' the coskewness matrix (or vector with unique coskewness values) of the
#' returns series, default NULL, see Details
#' @param m4 If univariate, m4 is the excess kurtosis of the series. Otherwise
#' m4 is the cokurtosis matrix (or vector with unique cokurtosis values) of the
#' return series, default NULL, see Details
#' @param invert TRUE/FALSE whether to invert the VaR measure, see Details.
#' @param operational TRUE/FALSE, default TRUE, see Details.
#' @param \dots any other passthru parameters
#' @param SE TRUE/FALSE whether to ouput the standard errors of the estimates of the risk measures, default FALSE.
#' @param SE.control Control parameters for the computation of standard errors. Should be done using the \code{\link{RPESE.control}} function.
#' @note The option to \code{invert} the ES measure should appease both
#' academics and practitioners. The mathematical definition of ES as the
#' negative value of extreme losses will (usually) produce a positive number.
#' Practitioners will argue that ES denotes a loss, and should be internally
#' consistent with the quantile (a negative number). For tables and charts,
#' different preferences may apply for clarity and compactness. As such, we
#' provide the option, and set the default to TRUE to keep the return
#' consistent with prior versions of PerformanceAnalytics, but make no value
#' judgement on which approach is preferable.
#'
#' @section Background:
#'
#' This function provides several estimation methods for
#' the Expected Shortfall (ES) (also called Expected Tail Loss (ETL)
#' or Conditional Value at Risk (CVaR)) of a return series and the Component ES
#' (ETL/CVaR) of a portfolio.
#'
#' At a preset probability level denoted \eqn{c}, which typically is between 1
#' and 5 per cent, the ES of a return series is the negative value of the
#' expected value of the return when the return is less than its
#' \eqn{c}-quantile. Unlike value-at-risk, conditional value-at-risk has all
#' the properties a risk measure should have to be coherent and is a convex
#' function of the portfolio weights (Pflug, 2000). With a sufficiently large
#' data set, you may choose to estimate ES with the sample average of all
#' returns that are below the \eqn{c} empirical quantile. More efficient
#' estimates of VaR are obtained if a (correct) assumption is made on the
#' return distribution, such as the normal distribution. If your return series
#' is skewed and/or has excess kurtosis, Cornish-Fisher estimates of ES can be
#' more appropriate. For the ES of a portfolio, it is also of interest to
#' decompose total portfolio ES into the risk contributions of each of the
#' portfolio components. For the above mentioned ES estimators, such a
#' decomposition is possible in a financially meaningful way.
#'
#' @section Univariate estimation of ES:
#'
#' The ES at a probability level \eqn{p} (e.g. 95\%) is the negative value of
#' the expected value of the return when the return is less than its
#' \eqn{c=1-p} quantile. In a set of returns for which sufficently long history
#' exists, the per-period ES can be estimated by the negative value of the
#' sample average of all returns below the quantile. This method is also
#' sometimes called \dQuote{historical ES}, as it is by definition \emph{ex
#' post} analysis of the return distribution, and may be accessed with
#' \code{method="historical"}.
#'
#' When you don't have a sufficiently long set of returns to use non-parametric
#' or historical ES, or wish to more closely model an ideal distribution, it is
#' common to us a parmetric estimate based on the distribution. Parametric ES
#' does a better job of accounting for the tails of the distribution by more
#' precisely estimating shape of the distribution tails of the risk quantile.
#' The most common estimate is a normal (or Gaussian) distribution \eqn{R\sim
#' N(\mu,\sigma)} for the return series. In this case, estimation of ES requires
#' the mean return \eqn{\bar{R}}, the return distribution and the variance of
#' the returns \eqn{\sigma}. In the most common case, parametric VaR is thus
#' calculated by
#'
#' \deqn{\sigma=variance(R)}{sigma=var(R)}
#'
#' \deqn{ES=-\bar{R} + \sqrt{\sigma} \cdot \frac{1}{c}\phi(z_{c}) }{VaR= -mean(R) + sqrt(sigma)*dnorm(z_c)/c}
#'
#' where \eqn{z_{c}} is the \eqn{c}-quantile of the standard normal
#' distribution. Represented in \R by \code{qnorm(c)}, and may be accessed with
#' \code{method="gaussian"}. The function \eqn{\phi}{dnorm} is the Gaussian
#' density function.
#'
#'
#' The limitations of Gaussian ES are well covered in the literature, since most
#' financial return series are non-normal. Boudt, Peterson and Croux (2008)
#' provide a modified ES calculation that takes the higher moments of non-normal
#' distributions (skewness, kurtosis) into account through the use of a
#' Cornish-Fisher expansion, and collapses to standard (traditional) Gaussian ES
#' if the return stream follows a standard distribution. More precisely, for a
#' loss probability \eqn{c}, modified ES is defined as the negative of the
#' expected value of all returns below the \eqn{c} Cornish-Fisher quantile and
#' where the expectation is computed under the second order Edgeworth expansion
#' of the true distribution function.
#'
#' Modified expected shortfall should always be larger than modified Value at
#' Risk. Due to estimation problems, this might not always be the case. Set
#' Operational = TRUE to replace modified ES with modified VaR in the
#' (exceptional) case where the modified ES is smaller than modified VaR.
#'
#' @section Component ES:
#'
#' By setting \code{portfolio_method="component"} you may calculate the ES
#' contribution of each element of the portfolio. The return from the function in
#' this case will be a list with three components: the univariate portfolio ES,
#' the scalar contribution of each component to the portfolio ES (these will sum
#' to the portfolio ES), and a percentage risk contribution (which will sum to
#' 100\%).
#'
#' Both the numerical and percentage component contributions to ES may contain
#' both positive and negative contributions. A negative contribution to Component
#' ES indicates a portfolio risk diversifier. Increasing the position weight will
#' reduce overall portoflio ES.
#'
#' If a weighting vector is not passed in via \code{weights}, the function will
#' assume an equal weighted (neutral) portfolio.
#'
#' Multiple risk decomposition approaches have been suggested in the literature.
#' A naive approach is to set the risk contribution equal to the stand-alone
#' risk. This approach is overly simplistic and neglects important
#' diversification effects of the units being exposed differently to the
#' underlying risk factors. An alternative approach is to measure the ES
#' contribution as the weight of the position in the portfolio times the partial
#' derivative of the portfolio ES with respect to the component weight. \deqn{C_i
#' \mbox{ES} = w_i \frac{ \partial \mbox{ES} }{\partial w_i}.}{C[i]ES =
#' w[i]*(dES/dw[i]).} Because the portfolio ES is linear in position size, we
#' have that by Euler's theorem the portfolio VaR is the sum of these risk
#' contributions. Scaillet (2002) shows that for ES, this mathematical
#' decomposition of portfolio risk has a financial meaning. It equals the
#' negative value of the asset's expected contribution to the portfolio return
#' when the portfolio return is less or equal to the negative portfolio VaR:
#'
#' \deqn{C_i \mbox{ES} = = -E\left[ w_i r_{i} | r_{p} \leq - \mbox{VaR}\right]}{C[i]ES = -E( w[i]r[i]|rp<=-VaR ) }
#'
#' For the decomposition of Gaussian ES, the estimated mean and covariance
#' matrix are needed. For the decomposition of modified ES, also estimates of
#' the coskewness and cokurtosis matrices are needed. If \eqn{r} denotes the
#' \eqn{Nx1} return vector and \eqn{mu} is the mean vector, then the \eqn{N
#' \times N^2} co-skewness matrix is
#
#' \deqn{ m3 = E\left[ (r - \mu)(r - \mu)' \otimes (r - \mu)'\right]}{m3 = E[ (r - mu)(r - mu)' \%x\% (r - \mu)']}
#'
#' The \eqn{N \times N^3} co-kurtosis matrix is
#'
#' \deqn{ m_{4} =
#' E\left[ (r - \mu)(r - \mu)' \otimes (r - \mu)'\otimes (r - \mu)'
#' \right] }{E[ (r - \mu)(r - \mu)' \%x\% (r - \mu)'\%x\% (r - \mu)']}
#'
#' where \eqn{\otimes}{\%x\%} stands for the Kronecker product. The matrices can
#' be estimated through the functions \code{skewness.MM} and \code{kurtosis.MM}.
#' More efficient estimators were proposed by Martellini and Ziemann (2007) and
#' will be implemented in the future.
#'
#' As discussed among others in Cont, Deguest and Scandolo (2007), it is
#' important that the estimation of the ES measure is robust to single outliers.
#' This is especially the case for modified VaR and its decomposition, since
#' they use higher order moments. By default, the portfolio moments are
#' estimated by their sample counterparts. If \code{clean="boudt"} then the
#' \eqn{1-p} most extreme observations are winsorized if they are detected as
#' being outliers. For more information, see Boudt, Peterson and Croux (2008)
#' and \code{\link{Return.clean}}. If your data consist of returns for highly
#' illiquid assets, then \code{clean="geltner"} may be more appropriate to
#' reduce distortion caused by autocorrelation, see \code{\link{Return.Geltner}}
#' for details.
#'
#' @author Brian G. Peterson and Kris Boudt
#' @seealso \code{\link{VaR}} \cr \code{\link{SharpeRatio.modified}} \cr
#' \code{\link{chart.VaRSensitivity}} \cr \code{\link{Return.clean}}
#'
#' @references Boudt, Kris, Peterson, Brian, and Christophe Croux. 2008.
#' Estimation and decomposition of downside risk for portfolios with non-normal
#' returns. 2008. The Journal of Risk, vol. 11, 79-103.
#'
#' Cont, Rama, Deguest, Romain and Giacomo Scandolo. Robustness and sensitivity
#' analysis of risk measurement procedures. Financial Engineering Report No.
#' 2007-06, Columbia University Center for Financial Engineering.
#'
#' Laurent Favre and Jose-Antonio Galeano. Mean-Modified Value-at-Risk
#' Optimization with Hedge Funds. Journal of Alternative Investment, Fall 2002,
#' v 5.
#'
#' Martellini, L. and Ziemann, V., 2010. Improved estimates of higher-order
#' comoments and implications for portfolio selection. Review of Financial
#' Studies, 23(4):1467-1502.
#'
#' Pflug, G. Ch. Some remarks on the value-at-risk and the conditional
#' value-at-risk. In S. Uryasev, ed., Probabilistic Constrained Optimization:
#' Methodology and Applications, Dordrecht: Kluwer, 2000, 272-281.
#'
#' Scaillet, Olivier. Nonparametric estimation and sensitivity analysis of
#' expected shortfall. Mathematical Finance, 2002, vol. 14, 74-86.
###keywords ts multivariate distribution models
#' @examples
#'
#' if(!( Sys.info()[['sysname']]=="Windows") ){
#' # if on Windows, cut and paste this example
#'
#' data(edhec)
#'
#' # first do normal ES calc
#' ES(edhec, p=.95, method="historical")
#'
#' # now use Gaussian
#' ES(edhec, p=.95, method="gaussian")
#'
#' # now use modified Cornish Fisher calc to take non-normal distribution into account
#' ES(edhec, p=.95, method="modified")
#'
#' # now use p=.99
#' ES(edhec, p=.99)
#' # or the equivalent alpha=.01
#' ES(edhec, p=.01)
#'
#' # now with outliers squished
#' ES(edhec, clean="boudt")
#'
#' # add Component ES for the equal weighted portfolio
#' ES(edhec, clean="boudt", portfolio_method="component")
#'
#' } # end CRAN Windows check
#'
#' @export ETL CVaR ES
ETL <- CVaR <- ES <- function (R=NULL , p=0.95, ...,
method=c("modified","gaussian","historical"),
clean=c("none","boudt", "geltner", "locScaleRob"),
portfolio_method=c("single","component"),
weights=NULL, mu=NULL, sigma=NULL, m3=NULL, m4=NULL,
invert=TRUE, operational=TRUE,
SE=FALSE, SE.control=NULL)
{ # @author Brian G. Peterson
# Descripion:
# wrapper for univariate and multivariate ES functions.
# Fix parameters if SE=TRUE
if(SE){
# Setting the control parameters
if(is.null(SE.control))
SE.control <- RPESE.control(estimator="ES")
# Fix the method
method="historical"
portfolio_method="single"
invert=FALSE
if(SE.control$cleanOutliers=="locScaleRob")
clean="locScaleRob" else
clean="none"
}
# Setup:
#if(exists(modified)({if( modified == TRUE) { method="modified" }}
#if(method == TRUE or is.null(method) ) { method="modified" }
method = method[1]
clean = clean[1]
portfolio_method = portfolio_method[1]
if (is.null(weights) & portfolio_method != "single"){
message("no weights passed in, assuming equal weighted portfolio")
weights=t(rep(1/dim(R)[[2]], dim(R)[[2]]))
}
if(!is.null(R)){
R <- checkData(R, method="xts")
columns=colnames(R)
if (!is.null(weights) & portfolio_method != "single") {
if ( length(weights) != ncol(R)) {
stop("number of items in weights not equal to number of columns in R")
}
}
# weights = checkData(weights, method="matrix", ...) #is this necessary?
# TODO check for date overlap with R and weights
if(clean!="none" & is.null(mu)){ # the assumption here is that if you've passed in any moments, we'll leave R alone
R = as.matrix(Return.clean(R, method=clean))
}
if(portfolio_method != "single"){
# get the moments ready
if (is.null(mu)) { mu = apply(R,2,'mean' ) }
if (is.null(sigma)) { sigma = cov(R) }
if(method=="modified"){
if (is.null(m3)) {m3 = M3.MM(R,as.mat=FALSE,mu=mu)}
if (is.null(m4)) {m4 = M4.MM(R,as.mat=FALSE,mu=mu)}
}
}
} else {
#R is null, check for moments
if(is.null(mu)) stop("Nothing to do! You must pass either R or the moments mu, sigma, etc.")
if ( length(weights) != length(mu)) {
stop("number of items in weights not equal to number of items in the mean vector")
}
}
if(isTRUE(SE)){
if(!requireNamespace("RPESE", quietly = TRUE)){
stop("Package \"pkg\" needed for standard errors computation. Please install it.",
call. = FALSE)
}
# Computation of SE (optional)
ses=list()
# For each of the method specified in se.method, compute the standard error
for(mymethod in SE.control$se.method){
ses[[mymethod]]=RPESE::EstimatorSE(R, estimator.fun = "ES", se.method = mymethod,
cleanOutliers=SE.control$cleanOutliers,
fitting.method=SE.control$fitting.method,
freq.include=SE.control$freq.include,
freq.par=SE.control$freq.par,
a=SE.control$a, b=SE.control$b,
p=p, # Additional parameter
...)
}
ses <- t(data.frame(ses))
}
switch(portfolio_method,
single = {
if(is.null(weights)){
columns=colnames(R)
switch(method,
modified = { if (operational) rES = operES.CornishFisher(R=R,p=p)
else rES = ES.CornishFisher(R=R,p=p)
}, # mu=mu, sigma=sigma, skew=skew, exkurt=exkurt))},)
gaussian = { rES = ES.Gaussian(R=R,p=p) },
historical = { rES = ES.historical(R=R,p=p) }
) # end single method switch calc
# convert from vector to columns
rES=as.matrix(rES)
colnames(rES)=columns
} else { # we have weights, so we should use the .MM calc
weights=as.vector(weights)
switch(method,
modified = { rES=mES.MM(w=weights, mu=mu, sigma=sigma, M3=m3 , M4=m4 , p=p) },
gaussian = { rES=GES.MM(w=weights, mu=mu, sigma=sigma, p=p) },
historical = { rES = ES.historical(R=R,p=p) %*% weights }, # note that this is not tested for weighting the univariate calc by the weights,
) # end multivariate method
}
# check for unreasonable results
columns<-ncol(rES)
for(column in 1:columns) {
tmp=rES[,column]
if (!is.finite(tmp)) {
message(c("ES calculation returned non-finite result for column: ", column, " : ", rES[, column]))
# set ES to NA, since risk is unreasonable
rES[, column] <- NA
} else
if (eval(0 > tmp)) { #eval added previously to get around Sweave bitching
message(c("ES calculation produces unreliable result (inverse risk) for column: ",column," : ",rES[,column]))
# set ES to NA, since inverse risk is unreasonable
rES[,column] <- NA
} else
if (eval(1 < tmp)) { #eval added previously to get around Sweave bitching
message(c("ES calculation produces unreliable result (risk over 100%) for column: ",column," : ",rES[,column]))
# set ES to 1, since greater than 100% is unreasonable
rES[,column] <- 1
}
} # end reasonableness checks
if(invert) rES <- -rES
rownames(rES) <- "ES"
if(SE) # Check if SE computation
return(rbind(rES, ses)) else
return(rES)
}, # end single portfolio switch
component = {
# @todo need to add another loop here for subsetting, I think, when weights is a timeseries
#if (mu=NULL or sigma=NULL) {
# pfolioret = Return.portfolio(R, weights, wealth.index = FALSE, contribution=FALSE, method = c("simple"))
#}
# for now, use as.vector
weights=as.vector(weights)
names(weights)<-colnames(R)
switch(method,
modified = { if (operational) return(operES.CornishFisher.portfolio(p,weights,mu,sigma,m3,m4))
else return(ES.CornishFisher.portfolio(p,weights,mu,sigma,m3,m4))
},
gaussian = { return(ES.Gaussian.portfolio(p,weights,mu,sigma)) },
historical = { return(ES.historical.portfolio(R, p,weights)) },
kernel = { return(ES.kernel.portfolio(R=R,p=p,w=weights)) }
)
} # end component portfolio switch
)
} # end ES wrapper function
###############################################################################
# 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
#
# $Id$
#
###############################################################################
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