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We consider the problem of risk diversification of $\alpha$-stable heavytailed risks. We study the behaviour of the aggregated Value-at-Risk, withparticular reference to the impact of different tail dependence structures onthe limits to diversification. We confirm the large evidence of sub-additivityviolations, particularly for risks with low tail index values and positivedependence. So, reinsurance strategies are not allowed to exploitdiversification gains, or only a very limited amount of them. Concerning theimpact of tail dependence, we find the peculiar results that for high taildependence levels the limits to diversification are uniformly lower for all thelevels of dependence, and for all levels of $\alpha<2$. The result is confirmedas we move towards extreme points in the tail: in this case, we show that atsome point in the tail the aggregated VaR becomes additive above some level ofdependence, but this critical dependence level is lower for copulas with lowertail dependence.

Value-at-Risk Diversification of $\alpha$-stable Risks: The Tail-Dependence Puzzle