sandbox/expectedShortFallFunctions.r
In cloudcell/PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis
## expectedShortFallFunctions.r
## author: Eric Zivot
## created: April 29, 2009
## update history:
## December 2, 2009
## Fixed bootstrapIncrementalES to work with one asset portfolio
##
## purpose: functions for computing expected shortfall
##
## Functions:
## normalES
## normalPortfolioES
## normalMarginalES Not done
## normalComponentES Not done
## normalIncrementalES
## normalESreport Not done
## modifiedES Not done
## modifiedPortfolioES Not done
## modifiedMarginalES Not done
## modifiedComponentES Not done
## modifiedIncrementalES Not done
## modifiedESreport Not done
## bootstrapES
## bootstrapPortfolioES
## bootstrapIncrementalES
## bootstrapMarginalES
## bootstrapComponentES
## bootstrapESreport
##
## References:
## 1. Dowd, K. (2002). Measuring Market Risk, John Wiley and Sons
## 2. Boudt, Peterson and Croux (2008), "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
## 3. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
##
## 1. ES functions based on the normal distribution
##
normalES <- function(mu, sigma, tail.prob = 0.01) {
## compute normal ES for collection of assets given mean and sd vector
## inputs:
## mu n x 1 vector of expected returns
## sigma n x 1 vector of standard deviations
## tail.prob scalar tail probability
## output:
## ES n x 1 vector of left tail average returns reported as a positive number
mu = as.matrix(mu)
sigma = as.matrix(sigma)
if ( nrow(mu) != nrow(sigma) )
stop("mu and sigma must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
#' 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) for univariate, component, and marginal cases using a variety of
#' analytical methods.
#'
#'
#' @aliases ES CVaR ETL
#' @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", "kernel", see
#' Details.
#' @param clean method for data cleaning through \code{\link{Return.clean}}.
#' Current options are "none", "boudt", or "geltner".
#' @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 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 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
#' @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)
#' orConditional 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.
#' @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, Lionel, and Volker Ziemann. Improved Forecasts of Higher-Order
#' Comoments and Implications for Portfolio Selection. 2007. EDHEC Risk and
#' Asset Management Research Centre working paper.
#'
#' 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
#'
#' 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")
#'
ES = mu - sigma*dnorm(qnorm(tail.prob))/tail.prob
return(-ES)
}
normalPortfolioES <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal portfolio ES given portfolio weights, mean vector and
## covariance matrix
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## pES scalar left tail average return of normal portfolio distn returned
## as a positive number.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
pES = crossprod(w, mu) + sqrt( t(w) %*% Sigma %*% w )*dnorm(qnorm(tail.prob))/tail.prob
return(as.numeric(pES))
}
normalIncrementalES <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal incremental ES given portfolio weights, mean vector and
## covariance matrix
## Incremental ES is defined as the change in portfolio ES that occurs
## when an asset is removed from the portfolio
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## iES n x 1 vector of incremental ES values
## References:
## Jorian, P. (2007). Value at Risk, pg. 168.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(mu)
## portfolio ES with all assets
pES = normalPortfolioES(mu, Sigma, w, tail.prob)
temp.w = w
iES = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
pES.new = normalPortfolioES(mu, Sigma, temp.w, tail.prob)
iES[i,1] = pES.new - pES
## reset weight
temp.w = w
}
return(iES)
}
##
## 2. ES functions based on simulated (bootstrapped) data
##
bootstrapES <- function(bootData, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute ES given bootstrap data. ES is computed as the sample mean beyond
## VaR where VaR is computed either as the sample quantile or the quantile using
## the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of B bootstrap returns on n assets in portfolio
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## ES n x 1 matrix of ES values for each asset
## References:
require(ff,quietly=TRUE)
method = method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.assets = ncol(bootData)
ES = matrix(0, n.assets, 1)
rownames(ES) = colnames(bootData)
colnames(ES) = "ES"
## loop over all assets and compute ES
for (i in 1:n.assets) {
if (method == "HS") {
## use empirical quantile
VaR = quantile(bootData[,i], probs=tail.prob)
} else {
## use Cornish-Fisher quantile
VaR = -VaR.CornishFisher(bootData[,i], p=(1-tail.prob))
}
idx = which(bootData[,i] <= VaR)
ES[i,1] = mean(bootData[idx, i])
}
return(-ES)
}
bootstrapPortfolioES <- function(bootData, w, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute portfolio ES given bootstrap data and portfolio weights. VaR is computed
## either as the sample quantile or as an estimated quantile using the
## Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on n assets in portfolio
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## pES scalar, portfolio ES value
## References:
## Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(ff, quietly=TRUE)
method = method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
if (is.ff(bootData)) {
## use on disk ff objects
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
r.p = ffrowapply( bootData[i1:i2, ]%*%w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
## use in RAM objects
r.p = bootData %*% w
}
if (method == "HS") {
## use empirical quantile
pVaR = quantile(r.p[], prob=tail.prob)
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
} else {
## use Cornish-Fisher expansion
pVaR = -VaR.CornishFisher(r.p[], p=(1-tail.prob))
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
}
return(pES)
}
bootstrapIncrementalES <- function(bootData, w, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute incremental ES given bootstrap data and portfolio weights.
## Incremental ES is defined as the change in portfolio ES that occurs
## when an asset is removed from the portfolio and allocation is spread equally
## among remaining assets. VaR used in ES computation is computed either as the
## sample quantile or as an estimated quantile using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of B bootstrap returns on n assets in portfolio
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## iES n x 1 matrix of incremental VaR values for each asset
## References:
## Jorian, P. (2007). Value-at-Risk, pg. 168.
##
require(PerformanceAnalytics)
require(ff)
method = method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
if (ncol(bootData) == 1) {
## EZ: Added 12/2/2009
## one asset portfolio so iES = 0 by construction
iES = 0
} else {
n.w = nrow(w)
## portfolio VaR with all assets
if (is.ff(bootData)) {
## use on disk ff object
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
r.p = ffrowapply( bootData[i1:i2, ]%*%w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
## use in RAM object
r.p = bootData %*% w
}
if (method == "HS") {
## use empirical quantile to compute VaR
pVaR = quantile(r.p[], prob=tail.prob)
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
temp.w = w
iES = matrix(0, n.w, 1)
rownames(iES) = colnames(bootData)
colnames(iES) = "i.ES"
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
temp.r.p = ffrowapply( bootData[i1:i2, ]%*%temp.w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
temp.r.p = bootData %*% temp.w
}
pVaR.new = quantile(temp.r.p[], prob=tail.prob)
idx = which(temp.r.p[] <= pVaR.new)
pES.new = -mean(temp.r.p[idx])
iES[i,1] = pES.new - pES
## reset weight
temp.w = w
}
} else {
## use Cornish-Fisher VaR
pVaR = -VaR.CornishFisher(r.p[], p=(1-tail.prob))
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
temp.w = w
iES = matrix(0, n.w, 1)
rownames(iES) = colnames(bootData)
colnames(iES) = "i.ES"
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
temp.r.p = ffrowapply( bootData[i1:i2, ]%*%temp.w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
temp.r.p = bootData %*% temp.w
}
pVaR.new = -VaR.CornishFisher(temp.r.p[], p=(1-tail.prob))
idx = which(temp.r.p[] <= pVaR.new)
pES.new = -mean(temp.r.p[idx])
iES[i,1] = pES.new - pES
## reset weight
temp.w = w
}
}
}
return(iES)
}
bootstrapMarginalES <- function(bootData, w, delta.w = 0.001, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher")) {
## Compute marginal ES given bootstrap data and portfolio weights.
## Marginal ES is computed either as the numerical derivative of ES wrt portfolio weight or
## as the expected fund return given portfolio return is less than or equal to portfolio VaR
## VaR is compute either as the sample quantile or as an estimated quantile
## using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of B bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## tail.prob scalar tail probability
## method character, method for computing marginal ES. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## ES wrt fund portfolio weight; "average" for approximating E[Ri | Rp<=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## mES n x 1 matrix of marginal ES values for each fund
## References:
## 1. Hallerback (2003), "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 2. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(PerformanceAnalytics)
require(ff)
method = method[1]
VaR.method = VaR.method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(w)
## portfolio VaR and ES with all assets
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
r.p = ffrowapply( bootData[i1:i2, ]%*%w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
r.p = bootData %*% w
}
if (VaR.method == "HS") {
pVaR = quantile(r.p[], prob=tail.prob)
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
} else {
pVaR = -VaR.CornishFisher(r.p[], p=(1-tail.prob))
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
}
##
## compute marginal ES
##
if (method=="derivative") {
## compute marginal ES as derivative wrt portfolio weight
temp.w = w
mES = matrix(0, n.w, 1)
for (i in 1:n.w) {
## increment weight for asset i by delta.w
temp.w[i,1] = w[i,1] + delta.w
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
temp.r.p = ffrowapply( bootData[i1:i2, ]%*%temp.w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
temp.r.p = bootData %*% temp.w
}
if (VaR.method == "HS") {
pVaR.new = quantile(temp.r.p[], prob=tail.prob)
idx = which(temp.r.p[] <= pVaR.new)
pES.new = -mean(temp.r.p[idx])
} else {
pVaR.new = -VaR.CornishFisher(temp.r.p[], p=(1-tail.prob))
idx = which(temp.r.p[] <= pVaR.new)
pES.new = -mean(temp.r.p[idx])
}
mES[i,1] = (pES.new - pES)/delta.w
## reset weight
temp.w = w
}
} else {
## compute marginal ES as expected value of fund return given portfolio
## return is less than or equal to portfolio VaR
mES = -as.matrix(colMeans(bootData[idx,,drop=FALSE]))
}
## compute correction factor so that sum of weighted marginal ES adds to portfolio VaR
cf = as.numeric( pES / sum(mES*w) )
return(cf*mES)
}
bootstrapComponentES <- function(bootData, w, delta.w = 0.001, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher")) {
## Compute component ES given bootstrap data and portfolio weights.
## Marginal ES is computed either as the derivative of ES wrt portfolio weight or
## as the expected fund return given portfolio return is less than or equal to
## portfolio VaR. VaR is compute either as the sample quantile or as an estimated
## quantile using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## tail.prob scalar tail probability
## method character, method for computing marginal ES. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## ES wrt fund portfolio weight; "average" for approximating E[Ri | Rp<=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## cES n x 1 matrix of component VaR values for each fund
## References:
## 1. Hallerback (2003), "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 2. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(PerformanceAnalytics)
require(ff)
method = method[1]
VaR.method = VaR.method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
mES = bootstrapMarginalES(bootData, w, delta.w, tail.prob,
method, VaR.method)
cES = mES * w
return(cES)
}
bootstrapESreport <- function(bootData, w, delta.w = 0.001, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher"),
nav, nav.p, fundName, fundStrategy) {
## compute ES report for collection of assets in a portfolio given
## simulated (bootstrapped) return data
## Report format follows that of Excel VaR report
## inputs:
## bootData B x n matrix of B bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative.
## Default value is 0.01.
## tail.prob scalar tail probability
## method character, method for computing marginal ES. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## ES wrt fund portfolio weight; "average" for approximating E[Ri | Rp<=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## nav n x 1 vector of net asset values in each fund
## nav.p scalar, net asset value of portfolio
## fundName n x 1 vector of fund names
## fundStrategy n x 1 vector of fund strategies
## output:
## ESreport.df dataframe with the following columns
## w n x 1 vector of asset weights
## aES n x 1 vector of asset specific ES values
## mES n x 1 vector of asset specific marginal ES values
## iES n x 1 vector of asset specific incremental ES values
## cES n x 1 vector of asset specific component ES values
## pcES n x 1 vector of asset specific percent contribution to ES values
##
## To-do: Add information for cash position.
require(PerformanceAnalytics)
require(ff)
method = method[1]
VaR.method = VaR.method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
nav = as.matrix(nav)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
asset.ES = bootstrapES(bootData, tail.prob, VaR.method)
portfolio.ES = bootstrapPortfolioES(bootData, w, tail.prob, VaR.method)
marginal.ES = bootstrapMarginalES(bootData, w, delta.w, tail.prob,
method, VaR.method)
component.ES = marginal.ES*w
incremental.ES = bootstrapIncrementalES(bootData, w, tail.prob, VaR.method)
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
mean.vals = ffrowapply(colMeans(bootData[i1:i2,,drop=FALSE]),
X=bootData, RETURN=TRUE, CFUN="cmean")
sd.vals = ffrowapply(colMeans(bootData[i1:i2,,drop=FALSE]^2) - colMeans(bootData[i1:i2,,drop=FALSE])^2,
X=bootData, RETURN=TRUE, CFUN="cmean")
sd.vals = sqrt(sd.vals)
} else {
mean.vals = colMeans(bootData)
sd.vals = apply(bootData, 2, sd)
}
ESreport.df = data.frame(Strategy = fundStrategy,
Net.Asset.Value = nav,
Allocation = w,
Mean = mean.vals,
Std.Dev = sd.vals,
Asset.ES = asset.ES,
cES = component.ES,
cES.dollar = component.ES*nav.p,
pcES = component.ES/portfolio.ES,
iES = incremental.ES,
iES.dollar = incremental.ES*nav.p,
mES = marginal.ES,
mES.dollar = marginal.ES*nav.p)
rownames(ESreport.df) = fundName
return(ESreport.df)
}
###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# This file originally written and contributed by Eric Zivot, 2010
# package Copyright (c) 2004-2011 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: expectedShortFallFunctions.r 3523 2014-09-04 10:06:12Z braverock $
#
###############################################################################
cloudcell/PerformanceAnalytics documentation built on May 13, 2019, 8:01 p.m.
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## expectedShortFallFunctions.r
## author: Eric Zivot
## created: April 29, 2009
## update history:
## December 2, 2009
## Fixed bootstrapIncrementalES to work with one asset portfolio
##
## purpose: functions for computing expected shortfall
##
## Functions:
## normalES
## normalPortfolioES
## normalMarginalES Not done
## normalComponentES Not done
## normalIncrementalES
## normalESreport Not done
## modifiedES Not done
## modifiedPortfolioES Not done
## modifiedMarginalES Not done
## modifiedComponentES Not done
## modifiedIncrementalES Not done
## modifiedESreport Not done
## bootstrapES
## bootstrapPortfolioES
## bootstrapIncrementalES
## bootstrapMarginalES
## bootstrapComponentES
## bootstrapESreport
##
## References:
## 1. Dowd, K. (2002). Measuring Market Risk, John Wiley and Sons
## 2. Boudt, Peterson and Croux (2008), "Estimation and Decomposition of Downside
## Risk for Portfolios with Non-Normal Returns," Journal of Risk.
## 3. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
##
## 1. ES functions based on the normal distribution
##
normalES <- function(mu, sigma, tail.prob = 0.01) {
## compute normal ES for collection of assets given mean and sd vector
## inputs:
## mu n x 1 vector of expected returns
## sigma n x 1 vector of standard deviations
## tail.prob scalar tail probability
## output:
## ES n x 1 vector of left tail average returns reported as a positive number
mu = as.matrix(mu)
sigma = as.matrix(sigma)
if ( nrow(mu) != nrow(sigma) )
stop("mu and sigma must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
#' 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) for univariate, component, and marginal cases using a variety of
#' analytical methods.
#'
#'
#' @aliases ES CVaR ETL
#' @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", "kernel", see
#' Details.
#' @param clean method for data cleaning through \code{\link{Return.clean}}.
#' Current options are "none", "boudt", or "geltner".
#' @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 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 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
#' @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)
#' orConditional 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.
#' @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, Lionel, and Volker Ziemann. Improved Forecasts of Higher-Order
#' Comoments and Implications for Portfolio Selection. 2007. EDHEC Risk and
#' Asset Management Research Centre working paper.
#'
#' 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
#'
#' 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")
#'
ES = mu - sigma*dnorm(qnorm(tail.prob))/tail.prob
return(-ES)
}
normalPortfolioES <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal portfolio ES given portfolio weights, mean vector and
## covariance matrix
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## pES scalar left tail average return of normal portfolio distn returned
## as a positive number.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
pES = crossprod(w, mu) + sqrt( t(w) %*% Sigma %*% w )*dnorm(qnorm(tail.prob))/tail.prob
return(as.numeric(pES))
}
normalIncrementalES <- function(mu, Sigma, w, tail.prob = 0.01) {
## compute normal incremental ES given portfolio weights, mean vector and
## covariance matrix
## Incremental ES is defined as the change in portfolio ES that occurs
## when an asset is removed from the portfolio
## inputs:
## mu n x 1 vector of expected returns
## Sigma n x n return covariance matrix
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## output:
## iES n x 1 vector of incremental ES values
## References:
## Jorian, P. (2007). Value at Risk, pg. 168.
mu = as.matrix(mu)
Sigma = as.matrix(Sigma)
w = as.matrix(w)
if ( nrow(mu) != nrow(Sigma) )
stop("mu and Sigma must have same number of rows")
if ( nrow(mu) != nrow(w) )
stop("mu and w must have same number of elements")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(mu)
## portfolio ES with all assets
pES = normalPortfolioES(mu, Sigma, w, tail.prob)
temp.w = w
iES = matrix(0, n.w, 1)
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
pES.new = normalPortfolioES(mu, Sigma, temp.w, tail.prob)
iES[i,1] = pES.new - pES
## reset weight
temp.w = w
}
return(iES)
}
##
## 2. ES functions based on simulated (bootstrapped) data
##
bootstrapES <- function(bootData, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute ES given bootstrap data. ES is computed as the sample mean beyond
## VaR where VaR is computed either as the sample quantile or the quantile using
## the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of B bootstrap returns on n assets in portfolio
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## ES n x 1 matrix of ES values for each asset
## References:
require(ff,quietly=TRUE)
method = method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.assets = ncol(bootData)
ES = matrix(0, n.assets, 1)
rownames(ES) = colnames(bootData)
colnames(ES) = "ES"
## loop over all assets and compute ES
for (i in 1:n.assets) {
if (method == "HS") {
## use empirical quantile
VaR = quantile(bootData[,i], probs=tail.prob)
} else {
## use Cornish-Fisher quantile
VaR = -VaR.CornishFisher(bootData[,i], p=(1-tail.prob))
}
idx = which(bootData[,i] <= VaR)
ES[i,1] = mean(bootData[idx, i])
}
return(-ES)
}
bootstrapPortfolioES <- function(bootData, w, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute portfolio ES given bootstrap data and portfolio weights. VaR is computed
## either as the sample quantile or as an estimated quantile using the
## Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on n assets in portfolio
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## pES scalar, portfolio ES value
## References:
## Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(ff, quietly=TRUE)
method = method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
if (is.ff(bootData)) {
## use on disk ff objects
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
r.p = ffrowapply( bootData[i1:i2, ]%*%w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
## use in RAM objects
r.p = bootData %*% w
}
if (method == "HS") {
## use empirical quantile
pVaR = quantile(r.p[], prob=tail.prob)
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
} else {
## use Cornish-Fisher expansion
pVaR = -VaR.CornishFisher(r.p[], p=(1-tail.prob))
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
}
return(pES)
}
bootstrapIncrementalES <- function(bootData, w, tail.prob = 0.01,
method=c("HS", "CornishFisher")) {
## Compute incremental ES given bootstrap data and portfolio weights.
## Incremental ES is defined as the change in portfolio ES that occurs
## when an asset is removed from the portfolio and allocation is spread equally
## among remaining assets. VaR used in ES computation is computed either as the
## sample quantile or as an estimated quantile using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of B bootstrap returns on n assets in portfolio
## w n x 1 vector of portfolio weights
## tail.prob scalar tail probability
## method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## iES n x 1 matrix of incremental VaR values for each asset
## References:
## Jorian, P. (2007). Value-at-Risk, pg. 168.
##
require(PerformanceAnalytics)
require(ff)
method = method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
if (ncol(bootData) == 1) {
## EZ: Added 12/2/2009
## one asset portfolio so iES = 0 by construction
iES = 0
} else {
n.w = nrow(w)
## portfolio VaR with all assets
if (is.ff(bootData)) {
## use on disk ff object
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
r.p = ffrowapply( bootData[i1:i2, ]%*%w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
## use in RAM object
r.p = bootData %*% w
}
if (method == "HS") {
## use empirical quantile to compute VaR
pVaR = quantile(r.p[], prob=tail.prob)
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
temp.w = w
iES = matrix(0, n.w, 1)
rownames(iES) = colnames(bootData)
colnames(iES) = "i.ES"
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
temp.r.p = ffrowapply( bootData[i1:i2, ]%*%temp.w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
temp.r.p = bootData %*% temp.w
}
pVaR.new = quantile(temp.r.p[], prob=tail.prob)
idx = which(temp.r.p[] <= pVaR.new)
pES.new = -mean(temp.r.p[idx])
iES[i,1] = pES.new - pES
## reset weight
temp.w = w
}
} else {
## use Cornish-Fisher VaR
pVaR = -VaR.CornishFisher(r.p[], p=(1-tail.prob))
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
temp.w = w
iES = matrix(0, n.w, 1)
rownames(iES) = colnames(bootData)
colnames(iES) = "i.ES"
for (i in 1:n.w) {
## set weight for asset i to zero and renormalize
temp.w[i,1] = 0
temp.w = temp.w/sum(temp.w)
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
temp.r.p = ffrowapply( bootData[i1:i2, ]%*%temp.w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
temp.r.p = bootData %*% temp.w
}
pVaR.new = -VaR.CornishFisher(temp.r.p[], p=(1-tail.prob))
idx = which(temp.r.p[] <= pVaR.new)
pES.new = -mean(temp.r.p[idx])
iES[i,1] = pES.new - pES
## reset weight
temp.w = w
}
}
}
return(iES)
}
bootstrapMarginalES <- function(bootData, w, delta.w = 0.001, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher")) {
## Compute marginal ES given bootstrap data and portfolio weights.
## Marginal ES is computed either as the numerical derivative of ES wrt portfolio weight or
## as the expected fund return given portfolio return is less than or equal to portfolio VaR
## VaR is compute either as the sample quantile or as an estimated quantile
## using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of B bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## tail.prob scalar tail probability
## method character, method for computing marginal ES. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## ES wrt fund portfolio weight; "average" for approximating E[Ri | Rp<=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## mES n x 1 matrix of marginal ES values for each fund
## References:
## 1. Hallerback (2003), "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 2. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(PerformanceAnalytics)
require(ff)
method = method[1]
VaR.method = VaR.method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
n.w = nrow(w)
## portfolio VaR and ES with all assets
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
r.p = ffrowapply( bootData[i1:i2, ]%*%w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
r.p = bootData %*% w
}
if (VaR.method == "HS") {
pVaR = quantile(r.p[], prob=tail.prob)
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
} else {
pVaR = -VaR.CornishFisher(r.p[], p=(1-tail.prob))
idx = which(r.p[] <= pVaR)
pES = -mean(r.p[idx])
}
##
## compute marginal ES
##
if (method=="derivative") {
## compute marginal ES as derivative wrt portfolio weight
temp.w = w
mES = matrix(0, n.w, 1)
for (i in 1:n.w) {
## increment weight for asset i by delta.w
temp.w[i,1] = w[i,1] + delta.w
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
temp.r.p = ffrowapply( bootData[i1:i2, ]%*%temp.w, X=bootData, RETURN=TRUE, RETCOL=1)
} else {
temp.r.p = bootData %*% temp.w
}
if (VaR.method == "HS") {
pVaR.new = quantile(temp.r.p[], prob=tail.prob)
idx = which(temp.r.p[] <= pVaR.new)
pES.new = -mean(temp.r.p[idx])
} else {
pVaR.new = -VaR.CornishFisher(temp.r.p[], p=(1-tail.prob))
idx = which(temp.r.p[] <= pVaR.new)
pES.new = -mean(temp.r.p[idx])
}
mES[i,1] = (pES.new - pES)/delta.w
## reset weight
temp.w = w
}
} else {
## compute marginal ES as expected value of fund return given portfolio
## return is less than or equal to portfolio VaR
mES = -as.matrix(colMeans(bootData[idx,,drop=FALSE]))
}
## compute correction factor so that sum of weighted marginal ES adds to portfolio VaR
cf = as.numeric( pES / sum(mES*w) )
return(cf*mES)
}
bootstrapComponentES <- function(bootData, w, delta.w = 0.001, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher")) {
## Compute component ES given bootstrap data and portfolio weights.
## Marginal ES is computed either as the derivative of ES wrt portfolio weight or
## as the expected fund return given portfolio return is less than or equal to
## portfolio VaR. VaR is compute either as the sample quantile or as an estimated
## quantile using the Cornish-Fisher expansion
## inputs:
## bootData B x n matrix of bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative
## tail.prob scalar tail probability
## method character, method for computing marginal ES. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## ES wrt fund portfolio weight; "average" for approximating E[Ri | Rp<=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## output:
## cES n x 1 matrix of component VaR values for each fund
## References:
## 1. Hallerback (2003), "Decomposing Portfolio Value-at-Risk: A General Analysis",
## The Journal of Risk 5/2.
## 2. Yamai and Yoshiba (2002). "Comparative Analyses of Expected Shortfall and
## Value-at-Risk: Their Estimation Error, Decomposition, and Optimization
## Bank of Japan.
require(PerformanceAnalytics)
require(ff)
method = method[1]
VaR.method = VaR.method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
mES = bootstrapMarginalES(bootData, w, delta.w, tail.prob,
method, VaR.method)
cES = mES * w
return(cES)
}
bootstrapESreport <- function(bootData, w, delta.w = 0.001, tail.prob = 0.01,
method=c("derivative", "average"),
VaR.method=c("HS", "CornishFisher"),
nav, nav.p, fundName, fundStrategy) {
## compute ES report for collection of assets in a portfolio given
## simulated (bootstrapped) return data
## Report format follows that of Excel VaR report
## inputs:
## bootData B x n matrix of B bootstrap returns on assets in portfolio.
## w n x 1 vector of portfolio weights
## delta.w scalar, change in portfolio weight for computing numerical derivative.
## Default value is 0.01.
## tail.prob scalar tail probability
## method character, method for computing marginal ES. Valid choices are
## "derivative" for numerical computation of the derivative of portfolio
## ES wrt fund portfolio weight; "average" for approximating E[Ri | Rp<=VaR]
## VaR.method character, method for computing VaR. Valid choices are "HS" for
## historical simulation (empirical quantile); "CornishFisher" for
## modified VaR based on Cornish-Fisher quantile estimate. Cornish-Fisher
## computation is done with the VaR.CornishFisher in the PerformanceAnalytics
## package
## nav n x 1 vector of net asset values in each fund
## nav.p scalar, net asset value of portfolio
## fundName n x 1 vector of fund names
## fundStrategy n x 1 vector of fund strategies
## output:
## ESreport.df dataframe with the following columns
## w n x 1 vector of asset weights
## aES n x 1 vector of asset specific ES values
## mES n x 1 vector of asset specific marginal ES values
## iES n x 1 vector of asset specific incremental ES values
## cES n x 1 vector of asset specific component ES values
## pcES n x 1 vector of asset specific percent contribution to ES values
##
## To-do: Add information for cash position.
require(PerformanceAnalytics)
require(ff)
method = method[1]
VaR.method = VaR.method[1]
if (!is.ff(bootData))
bootData = as.matrix(bootData)
w = as.matrix(w)
nav = as.matrix(nav)
if ( ncol(bootData) != nrow(w) )
stop("Columns of bootData and rows of w do not match")
if ( tail.prob < 0 || tail.prob > 1)
stop("tail.prob must be between 0 and 1")
asset.ES = bootstrapES(bootData, tail.prob, VaR.method)
portfolio.ES = bootstrapPortfolioES(bootData, w, tail.prob, VaR.method)
marginal.ES = bootstrapMarginalES(bootData, w, delta.w, tail.prob,
method, VaR.method)
component.ES = marginal.ES*w
incremental.ES = bootstrapIncrementalES(bootData, w, tail.prob, VaR.method)
if (is.ff(bootData)) {
dummy<-TRUE; if(!dummy){i1=1;i2=2} #fake assign to make R CMD check happy
mean.vals = ffrowapply(colMeans(bootData[i1:i2,,drop=FALSE]),
X=bootData, RETURN=TRUE, CFUN="cmean")
sd.vals = ffrowapply(colMeans(bootData[i1:i2,,drop=FALSE]^2) - colMeans(bootData[i1:i2,,drop=FALSE])^2,
X=bootData, RETURN=TRUE, CFUN="cmean")
sd.vals = sqrt(sd.vals)
} else {
mean.vals = colMeans(bootData)
sd.vals = apply(bootData, 2, sd)
}
ESreport.df = data.frame(Strategy = fundStrategy,
Net.Asset.Value = nav,
Allocation = w,
Mean = mean.vals,
Std.Dev = sd.vals,
Asset.ES = asset.ES,
cES = component.ES,
cES.dollar = component.ES*nav.p,
pcES = component.ES/portfolio.ES,
iES = incremental.ES,
iES.dollar = incremental.ES*nav.p,
mES = marginal.ES,
mES.dollar = marginal.ES*nav.p)
rownames(ESreport.df) = fundName
return(ESreport.df)
}
###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# This file originally written and contributed by Eric Zivot, 2010
# package Copyright (c) 2004-2011 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: expectedShortFallFunctions.r 3523 2014-09-04 10:06:12Z braverock $
#
###############################################################################
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