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@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
>
>
>
>
>
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>             r@gmail.com>                                               An
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>                                         [R-SIG-Finance] [R-sig-finance]
>             Fax-Deckblatt:              VaR
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>             03.03.2009 12:24
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>
> 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
> --
> View this message in context:
> http://www.nabble.com/VaR-tp22306743p22306743.html
> Sent from the Rmetrics mailing list archive at Nabble.com.
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