[R-SIG-Finance] student t mixture VaR in R // imitate example 2.23 in C.Alexander: Market Risk IV

Johannes Moser jzmoser at gmail.com
Fri Apr 25 15:17:07 CEST 2014


Thanks for your help so far, Alexios!

I didn`t want to write down all the datails of the exercise because my 
text alredy was lengthly and also because I didn`t want to violate any 
copyrights.
Rather, I was hoping that someone has the book and the CD at hand 
because I think that the book is pretty popular.

To answer your question: The EXCEL example from the book was adapted by 
me in the following way:

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)

this has already been stated in my first mail. So the numbers for for 
mu_stress, variance_quiet and variance_stress are the (annualized) 
numbers given in the example scaled to one day.
I just did copy and paste with the down-scaled numbers from the EXCEL 
spreadsheet to check if my own R syntax leads to the same results as the 
EXCEL setup given by the CD in the book of C.Alexander.

A general description of my approach might be given by the following sketch:
Calculate 1-day VaR with confidence level 99% (and 99,9%) of a bivariate 
student t mixture distribution by means of backing out the quantile from 
the formula
     L * P(X < (q-m)/r) + (1-L) * P(Y < (q-n)/s) = theta
(see C.Alexander p.113)
(the VaR is just minus the 1% or 0.1% quantile "q")
"L" is the mixing law giving the probability for the mixture random 
variable to follow the same student-t distribution as the random 
variable "X" with mean "m", standard deviation "r" and degrees of 
freedom "df1".
The  mixture random variable follows a student t distribution with mean 
"n", standard deviation "s" and degrees of freedom "df2" with 
probability "(1-L)". This is the distribution for the random variable "Y".
"q" ist the required "theta"-quantile (as pointed out, "theta" is set to 
0.01 and 0.001 in my example)

The notation in my R code is:

p_quiet = L
mu_quiet = m
mu_stress = n
df_quiet = df1
df_stress = df2
variance_quiet = r^2
variance_stress = s^2
theta = theta

I hope that with these details everyone can evaluate if my syntax is set 
up correctly.
The values for these variables as to the modified example from 
Alexanders book can be  read in the R syntax given in my first mail.

My aim (later!) is to simulate a fictive return series in a 
regime-switching manner using student t mixture models and calculate the 
analytical VaR and Expected Shortfall in a dynamical way.
Then I want to apply several other estimation techniques to the 
simulated mixture series and examine their performance in relation to 
the analytic results.
But first I have to make sure that my analytic VaR for the student t 
mixture model is correct.
However, verification of the Example given by C.Alexander indeed 
requires her book and the enclosed CD with the EXCEL workbooks.
I just can`t post all this stuff here.




Am 25.04.2014 14:15, schrieb alexios ghalanos:
> I suggest you take a break and consider that in order to help you, it is
> required that you state ALL the assumptions and provide a complete
> example. Since the book is not generally available, and you have told us
> very little about the problem and its assumptions (other than a page and
> example number), then you shouldn't expect much help.
>
> Alexios
>
> PS How did you come up with the numbers for mu_stress, variance_quiet
> and variance_stress? From what I am told, the book example provides the
> annualized values and you are required to calculate the 10-day VaR (so
> you are required to rescale the numbers to their 10-day equivalents).
>
> On 25/04/2014 11:23, Johannes Moser wrote:
>> Thanks a lot, Alexios!
>> I have corrected this issue. The result is the same, though.
>>
>> I forgot to mention that the RUGARCH-package is required to run the code.
>>
>> At the moment I try to find the error in either
>> - my own theoretical thoughts (e.g. confusing the scale parameter with
>> the standard deviation)
>> - the implementation made by C.Alexander (on page 117 she writes in a
>> footnote: "In general, if X has distribution F(x) and Y=aX, a being a
>> constant, then y has distribution function a^(−1) * F(x)". Either I am
>> completely burnout right now, or this must be "F(a^(−1)*x)" in the end.
>> So maybe this is not just a typo, but also incorrectly implemented in
>> the quite complicated EXCEL formula.)
>> - some EXCEL or R issue. I think her EXCEL syntax has been programmed in
>> version 2003, but I`m running 2010.
>>
>> Any help is appreciated a lot!
>>
>>
>>
>> Am 25.04.2014 11:46, schrieb Alexios Ghalanos:
>>> A quick look at your code suggests that you should use "std" (student)
>>> not "sstd" (skew student) for distribution.
>>>
>>> Alexios
>>>
>>>> On 25 Apr 2014, at 09:49, Johannes Moser <jzmoser at gmail.com> wrote:
>>>>
>>>> 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]]
>>>>
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>>>>

-- 
_________________________
Johannes Moser
Sophie-Charlotten-Str. 35
14059 Berlin
Tel: 0176 2171 2196



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