Var subadditivity

I have a question in reference to the footnote on page 62. Footnote 36 states that the 'addition of VAR figures is not sub-additive in general. In particular, super-additivity may occur when the tail risks are bigger than if they were normally distributed'.

If VAR has a heavy-tail distribution (while we assumed it is normally distributed), then would the addition of all the VAR figures not be sub-additive (to calculate the EC)?

HH,

The author is talking about how to aggregate VaRs across different books, desks or business units once each of these has had their VaR calculated. Generally - and I say generally in the sense of the textbook definition of VaR where distribution of future portfolio values is assumed normally distributed - VaR is sub-additive because of the diversification effect created by correlations between the bank's different business units being less than +1. Therefore one might think that adding up the different VaRs ignoring the diversification effect may be the conservative thing to do as it will likely give the maximum possible theoretical value.

Yet in real life, the author argues, even this may not be 'safe' or conservative enough. This is because the distribution of future prices and portfolio values can take any shape other than normal, in a way as to produce an 'anti-diversification' effect where VaR actually goes up as a result of what may on the surface look to be diversifying. During the recent crisis, I am sure there must have been situations where each desk or business unit in a bank may have had perfectly acceptable individual VaRs, but their combined positions, concentrations, counterparty exposures, reputation impact, liquidity and market depth implications and other factors may have been such as to produce way larger losses. Kind of points to the fallibility of VaR.

Hope that helps - let me know if there is anything I can try helping with.

Mukul