In the actuarial field, it is important to characterize the distribution of the sum \[ S = \sum_{i=1}^n X_i \] with fixed marginals \(X_i\sim F_i\) which is maximal with respect to the convex order. If the \(\operatorname{Var} (X_i)\)’s are finite, then maximality clearly implies that \(S\) has the largest variance.
It is known that \(S\) is maximal with respect to the convex order if and only if the vector \((X_1, X_2, \dots , X_n)\) is comonotonic, see R. Kaas et al. [Astin Bull. 32, No. 1, 71–80 (2002; Zbl 1061.62511)] and K. C. Cheung [Insur. Math. Econ. 43, No. 3, 403–406 (2008; Zbl 1152.91570)]. The authors of this paper continue on this topic of investigation and show that a vector \((X_1, X_2, \dots ,X_n)\) for which \(\operatorname{Var}(S)\) is largest is a comonotonic vector. This completes the proof of the equivalence of the three properties: largest sum, comonotonicity, and largest variance.
The authors further consider the above problem under the additional assumption that the \(\operatorname{Var}(S)\) is fixed but not maximal. In this case, they show that there does not exist a maximal vector with respect to the convex order, and find conditions under which an upper comonotonic vector exists [K. C. Cheung, Insur. Math. Econ. 45, No. 1, 35–40 (2009; Zbl 1231.91158)]. Equivalence of convex order and variance order is also investigated.