R/tES.R

In Dowd: Functions Ported from 'MMR2' Toolbox Offered in Kevin Dowd's Book Measuring Market Risk

#' ES for t distributed P/L
#' 
#' Estimates the ES of a portfolio assuming that P/L are 
#' t-distributed, for specified confidence level and holding period.
#' 
#' @param ... The input arguments contain either return data or else mean and 
#'  standard deviation data. Accordingly, number of input arguments is either 4 
#'  or 5. In case there 4 input arguments, the mean and standard deviation of 
#'  data is computed from return data. See examples for details.
#'  
#'  returns Vector of daily P/L data
#' 
#'  mu Mean of daily geometric return data
#' 
#'  sigma Standard deviation of daily geometric return data
#' 
#'  df Number of degrees of freedom in the t-distribution
#' 
#'  cl ES confidence level
#' 
#'  hp ES holding period in days
#'  
#' @return Matrix of ES whose dimension depends on dimension of hp and cl. If 
#' cl and hp are both scalars, the matrix is 1 by 1. If cl is a vector and hp is
#'  a scalar, the matrix is row matrix, if cl is a scalar and hp is a vector, 
#'  the matrix is column matrix and if both cl and hp are vectors, the matrix 
#'  has dimension length of cl * length of hp.
#'  
#' @references Dowd, K. Measuring Market Risk, Wiley, 2007.
#' 
#' Evans, M., Hastings, M. and Peacock, B. Statistical Distributions, 3rd 
#' edition, New York: John Wiley, ch. 38,39.
#'
#' @author Dinesh Acharya
#' @examples
#' 
#'    # Computes ES given P/L data
#'    data <- runif(5, min = 0, max = .2)
#'    tES(returns = data, df = 6, cl = .95, hp = 90)
#'    
#'    # Computes ES given mean and standard deviation of P/L data
#'    tES(mu = .012, sigma = .03, df = 6, cl = .95, hp = 90)
#'
#'
#' @export
tES <- function(...){
  if (nargs() < 4) {
    stop("Too few arguments")
  }
  if (nargs() > 5) {
    stop("Too many arguments")
  }
  args <- list(...)
  if (nargs() == 5) {
    mu <- args$mu
    df <- args$df
    cl <- args$cl
    sigma <- args$sigma
    hp <- args$hp
  }
  if (nargs() == 4) {
    mu <- mean(args$returns)
    df <- args$df
    cl <- args$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")
  }
  cl <- as.matrix(cl)
  cl.row <- dim(cl)[1]
  cl.col <- dim(cl)[2]
  if (min(cl.row, cl.col) > 1) {
    stop("Confidence level must be a scalar or a vector")
  }
  hp <- as.matrix(hp)
  hp.row <- dim(hp)[1]
  hp.col <- dim(hp)[2]
  if (min(hp.row, hp.col) > 1) {
    stop("Holding period must be a scalar or a vector")
  }
  df <- as.matrix(df)
  df.row <- dim(df)[1]
  df.col <- dim(df)[2]
  if (max(df.row, df.col) > 1) {
    stop("Number of degrees of freedom must be a scalar")
  }
  
  # Check that cl and hp are read as row and column vectors respectively
  if (cl.row > cl.col) {
    cl <- t(cl)
  }
  if (hp.row > hp.col) {
    hp <- t(hp)
  }
  
  # Check that inputs obey sign and value restrictions
  if (sigma < 0) {
    stop("Standard deviation must be non-negative")
  }
  if (df < 3) {
    stop("Number of degrees of freedom must be at least 3 for first two moments of distribution to be defined")
  }
  if (max(cl) >= 1){
    stop("Confidence level(s) must be less than 1")
  }
  if (min(cl) <= 0){
    stop("Confidence level(s) must be greater than 0")
  }
  if (min(hp) <= 0){
    stop("Holding Period(s) must be greater than 0")
  }
  
  # ES estimation
  ES <- matrix(0, length(hp), length(cl))
  for (i in 1:length(cl)) {
    for (j in 1:length(hp)) {
      ES[j, i] <- Univariate.tES(mu, sigma, df, cl[i], hp[j])
    }
  }
  return (ES)
}

# Accessory function
Univariate.tES <- function (mu, sigma, df, cl, hp) {
  # This function estimates univariate t-ES using average tail quantile algorithm
  number.slices <- 1000
  delta.p <- (1 - cl)/number.slices
  p <- seq(cl + delta.p, 1 - delta.p, delta.p) # Tail confidence levels or cumulative probs
  tail.VaRs <- -sigma * sqrt(hp) * sqrt((df - 2)/df) * qt((1 - p), df) - mu * hp # Tail VaRs
  y <- mean(tail.VaRs)
  return(y)
}