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#' Percentiles of ES distribution function for t-distributed P/L
#'
#' Estimates percentiles of ES distribution function for t-distributed P/L,
#' using the theory of order statistics
#'
#' @param ... The input arguments contain either return data or else mean and
#' standard deviation data. Accordingly, number of input arguments is either 5
#' or 7. In case there 5 input arguments, the mean, standard deviation and assumed sampel size of
#' data is computed from return data. See examples for details.
#'
#' returns Vector of daily geometric return data
#'
#' mu Mean of daily geometric return data
#'
#' sigma Standard deviation of daily geometric return data
#'
#' n Sample size
#'
#' df Degrees of freedom
#'
#' perc Desired percentile
#'
#' df Number of degrees of freedom in the t distribution
#'
#' cl ES confidence level and must be a scalar
#'
#' hp ES holding period and must be a a scalar
#'
#' @return Percentiles of ES distribution function
#'
#' @references Dowd, K. Measuring Market Risk, Wiley, 2007.
#'
#'
#' @author Dinesh Acharya
#' @examples
#'
#' # Estimates Percentiles of ES distribution given P/L data
#' data <- runif(5, min = 0, max = .2)
#' tESDFPerc(returns = data, perc = .7, df = 6, cl = .95, hp = 60)
#'
#' # Estimates Percentiles of ES distribution given mean, std. deviation and sample size
#' tESDFPerc(mu = .012, sigma = .03, n= 10, perc = .8, df = 6, cl = .99, hp = 40)
#'
#'
#' @export
tESDFPerc <- function(...){
if (nargs() < 5) {
stop("Too few arguments")
}
if (nargs() == 6) {
stop("Incorrect number of arguments")
}
if (nargs() > 7) {
stop("Too many arguments")
}
args <- list(...)
if (nargs() == 7) {
mu <- args$mu
df <- args$df
cl <- args$cl
perc <- args$sigma
n <- args$n
sigma <- args$sigma
hp <- args$hp
}
if (nargs() == 5) {
mu <- mean(args$returns)
df <- args$df
n <- max(dim(as.matrix(args$returns)))
perc <- args$perc
cl <- args$cl
sigma <- sd(args$returns)
hp <- args$hp
}
# Check that inputs have correct dimensions
mu <- as.matrix(mu)
mu.row <- dim(mu)[1]
mu.col <- dim(mu)[2]
if (max(mu.row, mu.col) > 1) {
stop("Mean must be a scalar")
}
sigma <- as.matrix(sigma)
sigma.row <- dim(sigma)[1]
sigma.col <- dim(sigma)[2]
if (max(sigma.row, sigma.col) > 1) {
stop("Standard deviation must be a scalar")
}
n <- as.matrix(n)
n.row <- dim(n)[1]
n.col <- dim(n)[2]
if (max(n.row, n.col) > 1) {
stop("Number of observations in a sample must be a scalar")
}
if (n - round(n) > 0 | n - round(n) < 0) {
stop("Number of observations in a sample must be an integer")
}
perc <- as.matrix(perc)
perc.row <- dim(perc)[1]
perc.col <- dim(perc)[2]
if (max(perc.row, perc.col) > 1) {
stop("Chosen percentile of the distribution must be a scalar")
}
cl <- as.matrix(cl)
cl.row <- dim(cl)[1]
cl.col <- dim(cl)[2]
if (max(cl.row, cl.col) > 1) {
stop("Confidence level must be a scalar")
}
hp <- as.matrix(hp)
hp.row <- dim(hp)[1]
hp.col <- dim(hp)[2]
if (max(hp.row, hp.col) > 1) {
stop("Holding period must be a scalar")
}
# Check that inputs obey sign and value restrictions
if (sigma < 0) {
stop("Standard deviation must be non-negative")
}
if (n < 0) {
stop("Number of observations must be non-negative")
}
if (df < 0) {
stop("Number of degrees of freedom must be greater than zero")
}
if (perc > 1){
stop("Chosen percentile must not exceed 1")
}
if (perc <= 0){
stop("Chosen percentile must be positive")
}
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("Honding period must be greater than 0")
}
# Derive order statistic and ensure it is an integer
w <- n * cl # Derive r-th order statistic
r <- round(w) # Round r to nearest integer
# Bisection routine
a <- 0
fa <- -Inf
b <- 1
fb <- Inf
eps <- .Machine$double.eps
while (b - a > eps * b) {
x <- (a + b) / 2
fx <- 1 - pbinom(r - 1, n, x) - perc
if (sign(fx) == sign(fa)){
a = x
fa = fx
} else {
b = x
fb = fx
}
}
# VaR estimation
VaR <- - mu * hp + sigma * sqrt((df-2)/df)*sqrt(hp)*qt(x, df) # Value of VaR percentile, note t VaR formula
# ES estimation
cl <- x
n <- 1000 # Number of slices into which tail is divided
cl0 <- cl # Initial confidence level
delta.cl <- (1 - cl) / n # Increment to confidence level as each slice is taken
term <- VaR
for (i in 1:(n-1)) {
cl <- cl0 + i * delta.cl # Revised cl
term <- term - mu * hp - sigma * sqrt((df-2)/df)*sqrt(hp)*qt(1 - cl, df)
}
y <- term/n
return(y) # Value of ES percentile
}
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