##
##  VaR and Tail VaR
##
##--------------------------------



#--- Normal (mean=0, var=1) ---
X <- rnorm(10000, 0, 1)
qnorm(.95,0,1)

mean(X[X>qnorm(.95,0,1)])
mean(X>qnorm(.95,0,1))

  # For 5%:
  # VaR  = 1.65
  # TVaR = 2.085 (estimate)
         


#--- t-distribution (Standardized t. mean=0, var=1) ---
library(fGarch)

X <- rstd(10000,mean=0,sd=1,nu=5)
qstd(.95,0,1,5)
mean(X[X>qstd(.95,0,1,5)])
mean(X>qstd(.95,0,1,5))

  # For 5%:
  # VaR  = 1.56
  # TVaR = 2.27 (estimate)



#--- Exponential (mean=1, var=1) ---
X <- rexp(10000, 1)
qexp(.95,1)
mean(X[X>qexp(.95,1)])
mean(X>qexp(.95,1))

  # For 5%:
  # VaR  = 2.996
  # TVaR = 3.996



#--- Log-normal (mean=1, var=1)---
X <- rlnorm(10000, -log(2)/2, sqrt(log(2)))
mean(X)
var(X)
VaR.LN <- qlnorm(.95, -log(2)/2, sqrt(log(2)))
VaR.LN
mean(X[X>VaR.LN])
mean(X>VaR.LN)

  # For 5%:
  # VaR  = 2.781
  # TVaR = 4.198 (estimate)



#--- 1-param Pareto (Pareto I. mean=1, var=1) ---
library(VGAM)

a = 2.414
th = (a-1)/a
var <- (th^2 * a) / ( (a-1)^2 * (a-2) )
var
mean <- a*th/(a-1)
mean

X <- rpareto(10000, scale=th, shape=a)
VaR.PA <- qpareto(.95, scale=th, shape=a)
VaR.PA
mean(X[X>VaR.PA])
mean(X>VaR.PA)

  # For 5%:
  # VaR  = 2.026
  # TVaR = 3.3234 (estimate)