[R-SIG-Finance] Antwort: [R-sig-finance] VaR

[Brian G. Peterson]

Subadditivity is critically important in a portfolio context if you wish 
to dis-aggregate or slice the portfolio in different ways, as I 
mentioned in my previous email.  You need to understand risk 
contribution in a coherent fashion if you want to be able to rearrange a 
large portfolio into sub-portfolios and have a rational and fungible 
understanding of what each of those slices contributes to the total risk 
of your portfolio.

That said, ES/CVaR is often just an excuse for not getting the tail 
distribution correct.  By providing the "mean loss beyond the VaR" 
CVaR/ES try to smooth out the tail risk into one number.  This has its 
own risks, but is at least honest about the fact that you don't truly 
know the distribution of the returns under all circumstances.

As for the "dependence structure between asset[s]", my personal 
preference is to use more than two moments, by extending things to 
include skewness and kurtosis.  In a portfolio context, you then use all 
four moments of each asset, and all the co-moments (covariance, 
coskewness, cokurtosis).  I find this to be a more intellectually 
satisfying approach than just finding the best-fitting distribution out 
of some arbitrary list of distributions (or copulae, etc.) because the 
first four moments of the observed returns are easily understood, have 
real economic meaning, and can be communicated to most other investment 
professionals.  Contrast that with the greek-alphabet-soup of parameters 
to the arbitrary distribution of your choice that have mathematical 
meaning but not direct financial meaning.

I agree completely that subadditivity is not always important, but 
neither is it unimportant.  The trick is knowing when you need a 
subadditive measure.

Regards,

  - Brian

-- 
Brian G. Peterson
http://braverock.com/brian/
Ph: 773-459-4973
IM: bgpbraverock


Adams, Zeno wrote:
> I don't know what you think about the topic but I feel that this matter of subadditivity is strongly overemphasized. Many authors argue in their papers that they will use the CVaR instead of the VaR because of the subbaditivity property (which goes back to Artzner, 1999). From my point of view the matter of getting the return distribution right, especially its variation over time, as well as the dependence structure between asset returns if the distribution is not elliptic is far more important for modeling the VaR adequately.
>
> -----Ursprüngliche Nachricht-----
> Von: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch] Im Auftrag von Micha Keijzers
> Gesendet: Dienstag, 3. März 2009 13:22
> An: Matthias.Koberstein at hsbctrinkaus.de
> Cc: r-sig-finance at stat.math.ethz.ch; Bogaso
> Betreff: Re: [R-SIG-Finance] Antwort: [R-sig-finance] VaR
>
> Matthias and others,
>
> Indeed, correlation possibly has something to do with it. But it's not the
> whole story. VaR is a quantile of a distribution and you can draw up
> examples that go wrong specifically there, regardless of correlation. I
> constructed or adapted one, which must have been about three years ago I
> think, based on an example which came from IIRC Föllmer's book "Stochastic
> Finance" or "Quantitative Risk Management" by McNeil, Frey and Embrechts. I
> would have to do some serious digging to be sure... The example was based on
> a very simple example of defaults in a loan portfolio. Explicitly showing
> the quantiles in the loss distribution you could show that subadditivity did
> not hold when VaR is used as a risk measure.
>
> Kind regards,
> Micha Keijzers
>
> 2009/3/3 <Matthias.Koberstein at hsbctrinkaus.de>
>
>   
>> Hi Christofer,
>>
>> I think the analogy is allowed if you assume normal distributions for the
>> assets.
>> Since then the VaR is dependent on the volatility.
>> The variance of two random variables (combined assets in this case) is
>> given by
>>
>> Var(x+y)= E((x+y)^2) - E(x+y)^2
>>
>> which transforms to
>> Var( x+y) = Var(x) + Var(y) + 2  * Covariance(x, y)
>>
>> So it all depends on the covariance of x to y.
>> To give it a better feel this can be expressed in Correlation
>>
>> Var(x+y)= Var(x) + Var(y) + 2 * Vol(x) * Vol(y) * Correlation
>>
>> To better see  the effect throw some weights in w1, and w2 which combine to
>> one.
>> Then
>>
>> Var( w1 x + w2 y)= Var(x) w1^2 + Var(y) w2^2 + 2 * w1 * w2 * Vol(x) * Vol
>> (y) * Correlation
>>
>> the volatility used to estimate VaR is the square root of the variance.
>> So you see that if correlation is 1 VaR is not sub-additive.
>>
>> Another point is if the distributions you use for the assets are not the
>> same,
>> the VaR can not even been combined easily but you have to find the combined
>> distributions of the assets in the portfolio (which can be quite painful).
>>
>> I hope that helps. All the best
>>
>> Matthias
>>
>> ---------------------------------------------------------------------
>> HSBC Trinkaus & Burkhardt AG
>> Sitz: Düsseldorf, Königsallee 21/23, 40212 Düsseldorf, Handelsregister:
>> Amtsgericht Düsseldorf HRB 54447
>> Mitglieder des Vorstands: Andreas Schmitz (Sprecher), Paul Hagen, Dr. Olaf
>> Huth, Carola Gräfin v. Schmettow
>> Vorsitzender des Aufsichtsrats: Dr. Sieghardt Rometsch
>>
>>
>>
>>             Bogaso
>>             <bogaso.christofe
>>             r at gmail.com>                                               An
>>             Gesendet von:               r-sig-finance at stat.math.ethz.ch
>>             r-sig-finance-bou                                       Kopie
>>             nces at stat.math.et
>>             hz.ch                                                   Thema
>>                                         [R-SIG-Finance] [R-sig-finance]
>>             Fax-Deckblatt:              VaR
>>             HSBCTuB
>>             03.03.2009 12:24
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> I frequently hear Value at risk i.e. VaR is not a coherent risk measure
>> because, sum of VaR for two individual assets may be LOWER than VaR of
>> portfolio consists of that two aseets i.e. VaR may not be sub-additive.
>> However when I calculate VaR for general assets like Equity, commodity etc,
>> I see that VaR is actually sub-addtive i.e. portfolio VaR is always less
>> than sum of individuals, which is reported as "diversification benefit".
>> Can
>> anyone give me a particular example why VaR is not sub-additive?
>>
>> Thanks
>>