  We consider the problem of risk diversification of $\alpha$-stable heavy
tailed risks. We study the behaviour of the aggregated Value-at-Risk, with
particular reference to the impact of different tail dependence structures on
the limits to diversification. We confirm the large evidence of sub-additivity
violations, particularly for risks with low tail index values and positive
dependence. So, reinsurance strategies are not allowed to exploit
diversification gains, or only a very limited amount of them. Concerning the
impact of tail dependence, we find the peculiar results that for high tail
dependence levels the limits to diversification are uniformly lower for all the
levels of dependence, and for all levels of $\alpha<2$. The result is confirmed
as we move towards extreme points in the tail: in this case, we show that at
some point in the tail the aggregated VaR becomes additive above some level of
dependence, but this critical dependence level is lower for copulas with lower
tail dependence.
