  We study the aggregation of two risks when the marginal distributions are
known and the dependence structure is unknown, under the additional constraint
that one risk is no larger than the other. Risk aggregation problems with the
order constraint are closely related to the recently introduced notion of the
directional lower (DL) coupling. The largest aggregate risk in concave order
(thus, the smallest aggregate risk in convex order) is attained by the DL
coupling. These results are further generalized to calculate the best-case and
worst-case values of tail risk measures. In particular, we obtain analytical
formulas for bounds on Value-at-Risk. Our numerical results suggest that the
new bounds on risk measures with the extra order constraint can greatly improve
those with full dependence uncertainty.
