Aggregation-robustness and model uncertainty of regulatory risk measures

• Published in: Finance and stochastics Vol. 19; no. 4; pp. 763 - 790
• Main Authors: Embrechts, Paul, Wang, Bin, Wang, Ruodu
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2015; Springer Nature B.V
• Subjects: Diversification; Economic theory; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Finance; Inequality; Insurance; Management; Mathematical analysis; Mathematics; Mathematics and Statistics; Portfolio management; Probability Theory and Stochastic Processes; Quantitative Finance; Regulation of financial institutions; Risk; Risk management; Robustness; Solvency; Statistics for Business; Studies; Uncertainty; Aggregation-robustness; 60E15; C10; Dependence uncertainty; Expected shortfall; 62P05; Basel III; Risk aggregation; 62G35; Inhomogeneous portfolio; Value-at-risk
• ISSN: 0949-2984; 1432-1122
• DOI: 10.1007/s00780-015-0273-z

Research related to aggregation, robustness and model uncertainty of regulatory risk measures, for instance, value-at-risk (VaR) and expected shortfall (ES), is of fundamental importance within quantitative risk management. In risk aggregation, marginal risks and their dependence structure are often modelled separately, leading to uncertainty arising at the level of a joint model. In this paper, we introduce a notion of qualitative robustness for risk measures, concerning the sensitivity of a risk measure to the uncertainty of dependence in risk aggregation. It turns out that coherent risk measures, such as ES, are more robust than VaR according to the new notion of robustness. We also give approximations and inequalities for aggregation and diversification of VaR under dependence uncertainty, and derive an asymptotic equivalence for worst-case VaR and ES under general conditions. We obtain that for a portfolio of a large number of risks, VaR generally has a larger uncertainty spread compared to ES. The results warn that unjustified diversification arguments for VaR used in risk management need to be taken with much care, and they potentially support the use of ES in risk aggregation. This in particular reflects on the discussions in the recent consultative documents by the Basel Committee on Banking Supervision.