Dear R community,

in trying to set up a little simulation study I adapt the ideas found in 
"Carol Alexander: Market Risk IV - Value at Risk Models" on page 111 ff. 
and implement them in R.
This project is about student t mixture distributions and Value at Risk 
/ Expected Shortfall.

The following code is my setup so far, and the syntax is calibrated to 
resemble the Example 2.23 on page 118 in the mentioned book of 
Alexander. There is a  EXCEL-file coming with the book and I noticed 
that my results don`t match the results of the EXCEL implementation.

e.g. my result for theta=0.001 is

0.0841052 (method 1) and

0.0842109 (method 2)
... but the EXCEL-file coming with the book says that it was 0.1152


setting theta=0.01 gives

0.04493586 (method 1) and

0.04490717 (method 2)

... but the EXCEL-file coming with the book says that it was 0.0616


Maybe some of you guys have this book at hand and are able to verify and 
hopefully find a solution for my worries.
Or even if you don`t have the book you might still be able to assess the 
correctness of my approach and implementation?




##################################################################################################
# SET UP MIXTURE INGREDIENTS (calibrate to C.Alexander Market Risk 
Analysis IV Exercise 2.23)

p_quiet <- 0.75
mu_quiet <- 0.0
mu_stress <- -0.0004
df_quiet <- 10
df_stress <- 5
variance_quiet <- 0.0126^2
variance_stress <- 0.0253^2
theta <- 0.001


# METHOD_1)   Backing out mixture VaR from implicit analytic formula:
find_quant <- function(quant) {
      (p_quiet*pdist(distribution = "sstd", 
(quant-mu_quiet)/sqrt(variance_quiet)  , mu = 0, sigma = 1, shape = 
df_quiet)
      + (1-p_quiet)*pdist(distribution = "sstd", 
(quant-mu_stress)/sqrt(variance_stress)  , mu = 0, sigma = 1, shape = 
df_stress) - theta)
}
bestquant <- uniroot(f = find_quant, interval = c(-5, 1))
t_mix_VaR1 <- -bestquant$root


# METHOD_2)   Estimating mixture VaR by simulation:
nsim <- 10000000
u_mix <- x <- 1*(runif(nsim) < p_quiet)
t_quiet <- rdist(distribution = "sstd", nsim  , mu = mu_quiet, sigma = 
sqrt(variance_quiet), shape = df_quiet)
t_stress <- rdist(distribution = "sstd", nsim  , mu = mu_stress, sigma = 
sqrt(variance_stress), shape = df_stress)
t_mixture <- u_mix*t_quiet + (1-u_mix)*t_stress
t_mix_VaR2 <- as.numeric(-quantile( t_mixture , probs=theta  ))

# Compare results
t_mix_VaR1
t_mix_VaR2
##################################################################################################




The EXCEL spreadsheet "EX_IV.2.23" in the workbook "Examples_IV.2.xls" 
has been used and modified as follows:
1) change the risk horizon to 1 day (so that there would be no scaling 
and the autocorrelation doesn`t matter)
2) if necessary change the significance level (1% or 0,1% in my example)
3) press F11 to recalcualte the mixture parameters over the risk horizon
4) apply EXCEL SOLVER to line C24 while allowing for changing cell C25 
to get the t Mixture VaR

Thanks a lot for any ideas or suggestions!
Johannes













	[[alternative HTML version deleted]]

