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

[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





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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
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