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#' @title Variance-covariance ES for normally distributed returns
#'
#' @description Estimates the variance-covariance VaR of a
#' portfolio assuming individual asset returns are normally distributed,
#' for specified confidence level and holding period.
#'
#' @param vc.matrix Variance covariance matrix for returns
#' @param mu Vector of expected position returns
#' @param positions Vector of positions
#' @param cl Confidence level and is scalar
#' @param hp Holding period and is scalar
#'
#' @references Dowd, K. Measuring Market Risk, Wiley, 2007.
#'
#' @author Dinesh Acharya
#'
#' @examples
#'
#' # Variance-covariance ES for randomly generated portfolio
#' vc.matrix <- matrix(rnorm(16), 4, 4)
#' mu <- rnorm(4)
#' positions <- c(5, 2, 6, 10)
#' cl <- .95
#' hp <- 280
#' VarianceCovarianceES(vc.matrix, mu, positions, cl, hp)
#'
#' @export
VarianceCovarianceES <- function(vc.matrix, mu, positions, cl, hp){
# Check that cl is read as a row vector
cl <- as.matrix(cl)
if (dim(cl)[1] > dim(cl)[2]) {
cl <- t(cl)
}
# Check that hp is read as a column vector
hp <- as.matrix(hp)
if (dim(hp)[1] < dim(hp)[2]) {
hp <- t(hp)
}
# Check that positions vector read as a scalar or row vector
positions <- as.matrix(positions)
if (dim(positions)[1] > dim(positions)[2]){
positions <- t(positions)
}
# Check that expected returns vector is read as a scalar or row vector
mu <- as.matrix(mu)
if (dim(mu)[1] > dim(mu)[2]){
mu <- t(mu)
}
# Check that dimensions are correct
if (max(dim(mu)) != max(dim(positions))){
stop("Positions vector and expected returns vector must have same size")
}
if (max(dim(vc.matrix)) != max(dim(positions))){
stop("Positions vector and expected returns vector must have same size")
}
# Check that inputs obey sign and value restrictions
if (cl >= 1){
stop("Confidence level must be less than 1")
}
if (cl <= 0){
stop("Confidence level must be greater than 0");
}
if (hp <= 0){
stop("Holding period must be greater than 0");
}
# VaR and ES estimation
VaR <- matrix(0, length(cl), length(hp))
term <- matrix(0, length(cl), length(hp))
es <- matrix(0, length(cl), length(hp))
cl0 <- double(length(cl))
delta.cl <- double(length(cl))
for (i in 1:length(cl)) {
for (j in 1:length(hp)) {
VaR[i,j] <- - mu %*% t(positions) * hp[j] - qnorm(1-cl[i],0,1) *
(positions%*%vc.matrix%*%t(positions))*sqrt(hp[j]) # VaR
# ES Estimation
n <- 1000 # Number of slives into which tail is divided
cl0[i] <- cl[i] # Initial confidence level
term[i, j] <- VaR[i, j]
delta.cl[i] <- (1 - cl[i]) / n # Increment to confidence level as each
# slice is taken
for (k in 1:(n - 1)) {
cl[i] <- cl0[i] + k * delta.cl[i] # Revised cl
term[i, j] <- term[i, j] - mu %*% t(positions) * hp[j] -
(qnorm(1-cl[i],0,1)) * (positions%*%vc.matrix%*%t(positions))*sqrt(hp[j])
}
es[i, j] <- term[i, j]/n
}
}
y <- t(es)
return(y)
}
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