Value-at-Risk Bounds with Two-Sided Dependence Information

Abstract

Value-at-Risk bounds for aggregated risks have been derived in the literature in settings where besides the marginal distributions of the individual risk factors one-sided bounds for the joint distribution respectively the copula of the risks are available. In applications it turns out that these improved standard bounds on Value-at-Risk tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute Value-at-Risk bounds when besides the marginal distributions of the risk factors, two-sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds which we derive by means of the Monge–Kantorovich transportation duality. In several applications we illustrate that two-sided dependence information typically leads to strongly improved bounds on the Value-at-Risk of aggregations.

Keywords: model uncertainty, copulas, duality theory, value-at-risk

JEL Classification: C02, C63, D80, G31