General convex order on risk aggregation

Abstract

Using a general notion of convex order, we derive general lower bounds for risk measures of aggregated positions under dependence uncertainty, and this in arbitrary dimensions and for heterogeneous models. We also prove sharpness of the bounds obtained when each marginal distribution has a decreasing density. The main result answers a long-standing open question and yields an insight in optimal dependence structures. A numerical algorithm provides bounds for quantities of interest in risk management. Furthermore, our numerical results suggest that the bounds obtained in this paper are generally sharp for a broader class of models.

Keywords:

• risk aggregation
• dependence uncertainty
• convex order
• risk measures
• heterogeneous models
• joint mixability

Acknowledgements

The authors thank an anonymous referee and Paul Embrechts for valuable comments and suggestions. E. Jakobsons thanks RiskLab Switzerland and the Swiss Finance Institute for financial support. X. Han thanks the Department of Statistics and Actuarial Science, University of Waterloo for financial support during her visit in Waterloo, Summer 2014. R. Wang acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC No. 435844), and the Forschungsinstitut für Mathematik (FIM) at ETH Zurich during his visits in 2013 and 2014.