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#' @title Hotspots for VaR adjusted by Cornish-Fisher correction
#'
#' @description Estimates the VaR hotspots (or vector of incremental VaRs) for a
#' portfolio with portfolio return adjusted for non-normality by Cornish-Fisher
#' corerction, for specified confidence level and holding period.
#'
#' @param vc.matrix Variance covariance matrix for returns
#' @param mu Vector of expected position returns
#' @param skew Return skew
#' @param kurtosis Return kurtosis
#' @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
#'
#' # Hotspots for ES for randomly generated portfolio
#' vc.matrix <- matrix(rnorm(16),4,4)
#' mu <- rnorm(4)
#' skew <- .5
#' kurtosis <- 1.2
#' positions <- c(5,2,6,10)
#' cl <- .95
#' hp <- 280
#' AdjustedNormalVaRHotspots(vc.matrix, mu, skew, kurtosis, positions, cl, hp)
#'
#' @export
AdjustedNormalVaRHotspots <- function(vc.matrix, mu, skew, kurtosis, 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
z <- qnorm(1 - cl, 0 ,1)
sigma <- positions %*% vc.matrix %*% t(positions)/(sum(positions)^2) # Initial standard deviation of portfolio returns
adjustment <- (1 / 6) * (z ^ 2 - 1) * skew + (1 / 24) * (z ^ 3 - 3 * z) * (kurtosis - 3) - (1 / 36) * (2 * z ^ 3 - 5 * z) * skew ^ 2
VaR <- - mu %*% t(positions) * hp - (z + adjustment) * sigma * (sum(positions)^2) * sqrt(hp)
# VaR
x <- double(length(positions))
sigma <- double(length(positions))
iVaR <- double(length(positions))
for (i in 1:length(positions)){
x <- positions
x[i] <- 0
sigma[i] <- x %*% vc.matrix %*% t(x)/sum(x)^2 # standard deviation of portfolio returns
iVaR[i] <- VaR + mu %*% t(x) %*% hp + (z + adjustment) * sigma[i] * (sum(x))^2 * sqrt(hp) # Incremental VaR
}
y <- iVaR
return(y)
}
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