The safest dependence structure among risks

In this paper we investigate the dependence in Fréchet spaces containing mutually exclusive risks. It is shown that, under some reasonable assumptions, the safest dependence structure, in the sense of the minimal stop-loss premiums for the aggregate claims involved, is obtained with the Fréchet lower bound and precisely corresponds to the mutually exclusive risks of the Fréchet space. In that respect, the present paper complements some previous studies by Heilmann (1986) [On the impact of independence of risks on stop-loss premiums. Insurance: Mathematics and Economics 5, 197–199], Dhaene and Goovaerts (1996) [Dependency of risks and stop-loss order. ASTIN Bulletin 26, 201–212], Dhaene and Goovaerts (1997) [On the dependency of risks in the individual life model, Insurance: Mathematics and Economics 19, 243–253], Müller (1997) [Stop-loss order for portfolios of dependent risks. Insurance: Mathematics and Economics 21, 219–224], Taizhong and Zhiqiang (1999) [On the dependence of risks and the stop-loss premiums. Insurance: Mathematics and Economics 24, 323–332]. A couple of actuarial applications enhance the interest of the results derived here.