[R-SIG-Finance] Framework for VAR allocation among traders

[Brian G. Peterson]

elton wang wrote:
> For example, if underlying is a t distribution with
> DOF=4, then kurtosis does not exsit. Any sample
> kurtosis (with any cleaning tech or not) would be a
> false stat of underlying distribution. 
> How can you rule out this possibility of underlying
> distribution?

I am not in general a believer that real returns are generated by 
mathematically ideal distributions.  Ideal distributions may make good 
estimators (thus the rationale for any parametric estimation method), 
but are rarely if ever the actual generator of returns.

You can't rule out the possibility that the "generating" distribution 
has different moments from the observed distribution.  You might be able 
to cast doubt on the observed moments by fitting various distributions, 
but I think that's all you could do with any confidence.

One of the reasons I prefer the Cornish-Fisher expansion over most other 
fitted distributions is precisely because it directly utilizes the 
observable moments of the distribution.  These moments have intuitive 
financial as well as mathematical meaning.  As I discussed below, if the 
magnitude of the observed higher moments are small, the effect on the 
estimate will also be small.

If you believe that you have a better estimator of the moments for your 
data than the observed moments (possibly via fitting to an ideal 
distribution) you can plug those into the Cornish Fisher expansion and 
still take advantage of the component decomposition offered via the CF 
process.

Again, thank you for your thoughtful commentary.

Regards,

   - Brian

> --- "Brian G. Peterson" <brian at braverock.com> wrote:
> 
>> elton wang wrote:
>>> Brian,
>>> I have a question on your paper:
>>> If you use skewness and kurtosis in the VaR
>>> calculation, you want to make sure:
>>  >
>>> 1. these are exist if the underlying distribution
>> is
>>> non-normal.
>> At least one of skewness!=0 or kurtosis!=3 exist if
>> the underlying 
>> distribution is non-normal.  Perhaps I don't
>> understand your first point?
>>
>> If skewness=0 and kurtosis=3, the Cornish-Fisher
>> expansion does not 
>> change the Gaussian normal distribution.  So it
>> should have no adverse 
>> consequences if utilized even if all portfolio
>> assets were normal (which 
>> seems a highly unlikely circumstance).
>>
>>> 2. your sample skewness and kurtosis is good
>> estimates
>>> of true skewness and hurtosis.
>> While it is possible to fit many different
>> fat-tailed distributions to 
>> the sample, and derive skewness and kurtosis from
>> these, I don't see how 
>> this is a better approach than utilizing the sample
>> skewness and 
>> kurtosis.  We did show in the paper how to test the
>> Cornish Fisher and 
>> Edgeworth expansion against a very skewed and
>> fat-tailed Skew Student-t 
>> distribution.
>>
>> Another problem with utilizing a fitted distribution
>> is that many fitted 
>> distributions would  not carry the same properties
>> of being 
>> differentiable by the weight (properties of the
>> Gaussian normal and 
>> Cornish Fisher distributions) in a portfolio to
>> obtain a good estimator 
>> of Component Risk in a portfolio.
>>
>> In the main, the data cleaning method is most
>> valuable for adding 
>> stability to the effects of the co-moments in
>> decomposing the risk to 
>> avoid undue influence by a small number of extreme
>> events.  The method 
>> was developed to specifically not change
>> observations that were not "in 
>> the tail", and to keep the direction (but not the
>> absolute magnitude) of 
>> the extreme events.  As I discussed in the text of
>> the paper, I do not 
>> believe that you would ever use the cleaning method
>> for measuring VaR or 
>> ES ex post, but only to stabilize the predictions of
>> contribution on a 
>> forward-looking ex ante basis.
>>
>>> In part 5 you discussed the Robust estimation but
>> it
>>> could be stronger argument IMHO. For example, do
>> you
>>> have convergence/sensitivity analysis on estimated
>>> skewness/kurtosis results for your cleaning
>> method? 
>>
>> I agree that a sensitivity analysis would be a good
>> addition.  I will 
>> start thinking about how to add that.
>>
>> Regards,
>>
>>    - Brian
>>
>>
>>  > --- "Brian G. Peterson" <brian at braverock.com>
>> wrote:
>>  >
>>  >> On Thursday 13 March 2008 22:32:59
>>  >> adschai at optonline.net wrote:
>>  >>> Hi,I'm looking for VAR allocation framework
>> among
>>  >> traders. I saw some
>>  >>> papers but none of which (at least that I saw)
>>  >> look practical. I am
>>  >>> wondering if anyone can hint me some idea or
>> some
>>  >> reference? The situation
>>  >>> is if at the desk level you were given a
>> certain
>>  >> amount of VAR limit, how
>>  >>> should one allocate the number among traders?
>>  >> Thank you.adschai
>>  >>
>>  >> Calculate Component VaR.
>>  >>
>>  >> The first definition (as far as I know) is in
>> Garman
>>  >> in Risk Magazine.  The
>>  >> article may be found here:
>>  >>
>>  >> Garman, Mark, "Taking VaR to Pieces (Component
>>  >> VaR)," RISK 10, 10, October
>>  >> 1997.
>>  >> http://www.fea.com/pdf/componentvar.pdf
>>  >>
>>  >> He also has a longer working paper on the topic
>>  >> here:
>>  >>
>>  >>
>>  >
>>
> http://www.gloriamundi.org/detailpopup.asp?ID=453055537
>>  >> We implemented Component VaR for assets with
>>  >> non-normal distribution in our
>>  >> recent paper here:
>>  >>
>>  >> Boudt, Kris, Peterson, Brian G. and Croux,
>>  >> Christophe, "Estimation and
>>  >> Decomposition of Downside Risk for Portfolios
>> With
>>  >> Non-Normal Returns"
>>  >> (October 31, 2007).
>>  >> http://ssrn.com/abstract=1024151
>>  >>
>>  >> All code for our paper was implemented in R, and
>> is
>>  >> available.  We will also
>>  >> be cleaning up and documenting the functions in
>> the
>>  >> next version of
>>  >> PerformanceAnalytics.
>>  >>
>>  >> Regards,
>>  >>
>>  >>     - Brian
>>  >>
>>  >> _______