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#' @title Hotspots for ES adjusted by Cornish-Fisher correction
#'
#' @description Estimates the ES hotspots (or vector of incremental ESs) 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
#' AdjustedNormalESHotspots(vc.matrix, mu, skew, kurtosis, positions, cl, hp)
#'
#' @export
AdjustedNormalESHotspots <- 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.")
}
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
# Begin with portfolio ES
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) # Initial VaR
n <- 1000 # Number of slives into which tail is divided
cl0 <- cl # Initial confidence level
term <- VaR
delta.cl <- (1 - cl) / n # Increment to confidence level
for (k in 1:(n - 1)) {
cl <- cl0 + k * delta.cl # Revised cl
z <- qnorm(1 - cl, 0, 1)
adjustment=(1 / 6) * (z ^ 2 - 1) * skew + (1 / 24) * (z ^ 3 - 3 * z) *
(kurtosis - 3) - (1 / 36) * (2 * z ^ 3 - 5 * z) * skew ^ 2
term <- term - mu %*% t(positions) * hp - (z + adjustment) * sigma *
(sum(positions)^2) * sqrt(hp)
}
portfolio.ES <- term/n
# Portfolio ES
es <- double(length(positions))
ies <- double(length(positions))
for (j in 1:length(positions)) {
x <- positions
x[j] <- 0
sigma <- x %*% vc.matrix %*% t(x) / (sum(x)^2)
term[j] <- - mu %*% t(x) * hp - qnorm(1-cl, 0, 1) * x %*%
vc.matrix %*% t(x) * sqrt(hp)
for (k in 1:(n - 1)){
cl <- cl0 + k * delta.cl # Revised cl
z <- qnorm(1-cl, 0, 1)
adjustment=(1 / 6) * (z ^ 2 - 1) * skew + (1 / 24) * (z ^ 3 - 3 * z) *
(kurtosis - 3) - (1 / 36) * (2 * z ^ 3 - 5 * z) * skew ^ 2
term[j] <- term[j] - mu %*% t(positions) * hp - (z + adjustment) *
sigma * (sum(positions)^2) * sqrt(hp)
}
es[j] <- term[j]/n # ES on portfolio minus position j
ies [j] <- portfolio.ES - es[j] # Incremental ES
}
y <- ies
return(ies)
}
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