Reduction of Value-at-Risk bounds via independence and variance information

Abstract

We derive lower and upper bounds for the Value-at-Risk of a portfolio of losses when the marginal distributions are known and independence among (some) subgroups of the marginal components is assumed. We provide several actuarial examples showing that the newly proposed bounds strongly improve those available in the literature that are based on the sole knowledge of the marginal distributions. When the variance of the joint portfolio loss is small enough, further improvements can be obtained.

Keywords:

• Value-at-Risk
• dependence uncertainty
• model risk
• expected shortfall

AMS Subject Classification:

• 91B30 (primary)
• 60E15 (secondary)

Acknowledgements

The authors would like to thank two anonymous reviewers for constructive comments. They also thank Gerhard Stahl, Chief Risk Officer of Talanx, for various discussions that inspired some of the results treated in this paper. Giovanni Puccetti acknowledges a grant under the call PRIN 2010-2011 from MIUR within the project Robust decision making in markets and organizations. Steven Vanduffel acknowledges support from FWO and from BNP Paribas Fortis (BNP Paribas Fortis Chair in Banking). Ludger Rüschendorf acknowledges support from grant RU 704/11-1 of the German science foundation DFG.

Notes

No potential conflict of interest was reported by the author(s).