Finance and Stochastics

, Volume 17, Issue 2, pp 395–417 | Cite as

Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities

  • Ruodu Wang
  • Liang Peng
  • Jingping YangEmail author


In quantitative risk management, it is important and challenging to find sharp bounds for the distribution of the sum of dependent risks with given marginal distributions, but an unspecified dependence structure. These bounds are directly related to the problem of obtaining the worst Value-at-Risk of the total risk. Using the idea of complete mixability, we provide a new lower bound for any given marginal distributions and give a necessary and sufficient condition for the sharpness of this new bound. For the sum of dependent risks with an identical distribution, which has either a monotone density or a tail-monotone density, the explicit values of the worst Value-at-Risk and bounds on the distribution of the total risk are obtained. Some examples are given to illustrate the new results.


Complete mixability Monotone density Sum of dependent risks Value-at-Risk 

Mathematics Subject Classification (2000)

60E05 60E15 

JEL Classification




We thank the co-editor Kerry Back, an associate editor and two reviewers for their helpful comments which significantly improved this paper. Wang’s research was partly supported by the Bob Price Fellowship at the Georgia Institute of Technology. Peng’s research was supported by NSF Grant DMS-1005336. Yang’s research was supported by the Key Program of National Natural Science Foundation of China (Grants No. 11131002) and the National Natural Science Foundation of China (Grants No. 11271033).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada
  3. 3.LMEQF and LMAM, Department of Financial Mathematics, Center for Statistical SciencePeking UniversityBeijingChina

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