Can diversified VaR be higher than Undiversified vaR?

[Torsleno]

Hi,

If anybody can shed any light on this thought I was having:
I can see how Diversified VaR can be equal to the Undiversified VaR if the underlying positions are perfectly correlated (- no diversification benefit).
I was then wondering if Diversified VaR could become higher than the Undiversified VaR? I can't see how that can happen in the FRM practice exercises, but in the real world where the returns are not normal (@David Harper CFA FRM I saw you mentioned this point several times in subadditivity) and with more 'complex' VaR measure that bootstrap returns (EWMA/GARCH), could that happen?

Thanks

 

[David Harper CFA FRM]

Hi @Torsleno Interesting question. In my opinion, "diversified VaR" is an FRM term (specifically P. Jorion but also K. Dowd), unlike many terms in the FRM that have universal definitions outside the FRM (is a caveat to acknowledge somebody might disagree with me based on some other source). Within the FRM, diversified VaR cannot become higher than undiversified VaR per its definition. You are correct, of course, that some measures are not subadditive. Subadditivity is a property of a risk measure. But in the FRM diversified VaR--which is always seems to be measured in the mean-variance framework (when it does happen to be subadditive)--is a VaR that incorporates the benefits of diversification. So it cannot be higher. If the distributions implied a VaR that is not subadditive, then I think "diversified VaR" would simply be the wrong term for it. I hope that's helpful,