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Showing that VaR is not coherent for exponentially distributed loss variables

Please bear in mind that in general a multivariate distribution each of the marginal of which is exponentially distributed is not necessarily a member of the exponential family of (multivariate) distributions.

The simplest case to consider is where two loss variables come from the same exponential distribution with parameter . The probability density function of  where  and  are independent random variables each coming from an exponential distribution with parameter  is .

The VaR at confidence level  for ,  is the same as  and is the value of  for which:

The VaR at the same confidence level for  is  the value of  for which

Integrating by parts we have:

The exponential distribution (with independent variables) is not elliptical so VaR’s for it shouldn’t be coherent. More specifically it should be possible to identify a suitable  for which VaR is not sub-additive, i.e. for which , as sub-additivity is one of the axioms required for coherence. Although there is no certainty that this is the case it is likely that such a situation will arise either where  is close to zero or when  is close to 1. The case where  is close to 1 does not look promising as the behaviour of  will tend to be dominated by the exponential term. So instead let us consider the case where when  is close to 0. We then have for :

However in these circumstances:

So if  is small enough we will have  as desired to prove the conjecture.