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The determination of the dependence structure giving rise to the minimal convex sum in a general Fréchet space is a practical, yet challenging problem in quantitative risk management. In this article, we consider the closely related problem of finding lower bounds on three kinds of convex functionals, namely, convex expectations, Tail Value-at-Risk and the Haezendonck–Goovaerts risk measure, of a sum of random variables with arbitrary distributions. The sharpness of the lower bounds on the first two types of convex functionals is characterized via the extreme negative dependence structure of mutual exclusivity. Compared to existing results in the literature, our new lower bounds enjoy the advantages of generality and analytic tractability.