Risk Aggregation under Dependence Uncertainty and an Order Constraint

TL;DR

This paper investigates risk aggregation with an order constraint, deriving bounds on tail risk measures like Value-at-Risk using the directional lower coupling, and demonstrates improved bounds over traditional dependence uncertainty models.

Contribution

It introduces the use of the directional lower coupling for risk aggregation under an order constraint and derives analytical formulas for bounds on tail risk measures.

Findings

Bounds on Value-at-Risk are tightened with the order constraint.

The directional lower coupling attains extremal aggregate risks under the order constraint.

Numerical results show significant improvement over models with full dependence uncertainty.

Abstract

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 smaller than or equal to 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.