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Extremes

, Volume 19, Issue 4, pp 719–733 | Cite as

Bounds for randomly shared risk of heavy-tailed loss factors

  • Oliver KleyEmail author
  • Claudia Klüppelberg
Article

Abstract

For a risk vector V, whose components are shared among agents by some random mechanism, we obtain asymptotic lower and upper bounds for the individual agents’ exposure risk and the aggregated risk in the market. Risk is measured by Value-at-Risk or Conditional Tail Expectation. We assume Pareto tails for the components of V and arbitrary dependence structure in a multivariate regular variation setting. Upper and lower bounds are given by asymptotically independent and fully dependent components of V with respect to the tail index α being smaller or larger than 1. Counterexamples, where for non-linear aggregation functions no bounds are available, complete the picture.

Keywords

Multivariate regular variation Individual and systemic risk Pareto tail Risk measure Bounds for aggregated risk Random risk sharing 

AMS 2000 Subject Classifications

Primary: 90B15, 91B30 Secondary: 60E05, 60G70 

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References

  1. Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Process. Appl. 99(1), 95–115 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. Wiley, Chichester (2006)zbMATHGoogle Scholar
  3. Bernard, C., Rüschendorf, L., Vanduffel, S.: Value-at-risk bounds with variance constraints. Forthcoming in Journal of Risk and Insurance. Available at SSRN: http://ssrn.com/abstract=2342068 (2016)
  4. Böcker, K., Klüppelberg, C.: Multivariate models for Operational Risk. Quant. Finan. 10(8), 855–869 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Breiman, L.: On some limit theorems similar to the arc-sine law. Theory. Probab. Appl. 10, 323–331 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Burgert, C., Rüschendorf, L.: Consistent risk measures for portfolio vectors. Insurance: Mathematics and Economics 38(2), 289–297 (2006)MathSciNetzbMATHGoogle Scholar
  7. Caccioli, F., Shrestha, M., Moore, C., Farmer, J.D.: Stability analysis of financial contagion due to overlapping portfolios. J. Bank. Financ. 46, 233–245 (2014)CrossRefGoogle Scholar
  8. Chen, C., Iyengar, G., Moallemi, C.C.: An axiomatic approach to systemic risk. Manag. Sci. 59(6), 1373–1388 (2013)CrossRefGoogle Scholar
  9. Embrechts, P., Lambrigger, D.D., Wüthrich, M.V.: Multivariate extremes and the aggregation of dependent risks: examples and counter-examples. Extremes 12(2), 107–127 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Embrechts, P., Puccetti, G., Rüschendorf, L.: Model uncertainty and VaR aggregation. Journal of Banking and Finance 37(8), 2750–2764 (2013)CrossRefGoogle Scholar
  11. Fernholz, R., Garvy, R., Hannon, J.: Consistent risk measures for portfolio vectors. J. Portf. Manag. 24(2), 74–82 (1998)CrossRefGoogle Scholar
  12. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge mathematical library. cambridge university press (1952)Google Scholar
  13. Kley, O., Klüppelberg, C., Reinert, G.: Conditional risk measures in a bipartite market structure. Submitted (2015)Google Scholar
  14. Kley, O., Klüppelberg, C., Reinert, G.: Risk in a large claims insurance market with bipartite graph structure. Forthcoming in Operations Research (2016)Google Scholar
  15. Loeve, M.: Probability Theory, vol. 1, 4th edn. springer, New York (1977)Google Scholar
  16. Mainik, G., Rüschendorf, L.: Ordering of multivariate risk models with respect to extreme portfolio losses. Statistics & Risk Modeling 29(1), 73–106 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Resnick, S.I.: Heavy-Tail Phenomena. Springer, New York (2007)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Center for Mathematical SciencesTechnische Universität MünchenGarchingGermany

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