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#' @title Hotspots for normal VaR
#'
#' @description Estimates the VaR hotspots (or vector of incremental VaRs) for
#' 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
#' @return Hotspots for normal VaR
#' @references Dowd, K. Measuring Market Risk, Wiley, 2007.
#'
#' @author Dinesh Acharya
#'
#' @examples
#'
#' # Hotspots for ES for randomly generated portfolio
#' vc.matrix <- matrix(rnorm(16),4,4)
#' mu <- rnorm(4,.08,.04)
#' positions <- c(5,2,6,10)
#' cl <- .95
#' hp <- 280
#' NormalVaRHotspots(vc.matrix, mu, positions, cl, hp)
#'
#' @export
NormalVaRHotspots <- function(vc.matrix, mu, positions, cl, 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")
}
vc.matrix <- as.matrix(vc.matrix)
if (max(dim(vc.matrix)) != max(dim(positions))){
stop("Positions vector and variance-covariance matrix must have compatible dimensions")
}
# 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 <- - mu %*% t(positions) * hp - qnorm(1 - cl, 0, 1) * (positions %*% vc.matrix %*% t(positions)) * sqrt(hp) # VaR
iVaR <- double(length(positions))
for (i in 1:length(positions)){
x <- positions
x[i] <- 0
iVaR[i] <- VaR + mu %*% t(x) %*% hp + qnorm(1 - cl, 0, 1) * (x %*% vc.matrix %*% t(x)) * sqrt(hp)
}
y <- iVaR
return(y)
}
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