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Bounds and approximations for sums of dependent log-elliptical random variables

  • Valdez, Emiliano A.
  • Dhaene, Jan
  • Maj, Mateusz
  • Vanduffel, Steven

Dhaene, Denuit, Goovaerts, Kaas and Vyncke [Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002a. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31 (1), 3-33; Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002b. The concept of comonotonicity in actuarial science and finance: Applications. Insurance Math. Econom. 31 (2), 133-161] have studied convex bounds for a sum of dependent random variables and applied these to sums of log-normal random variables. In particular, they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In this paper we investigate to which extent their general results on convex bounds can also be applied to sums of log-elliptical random variables which incorporate sums of log-normals as a special case. Firstly, we show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound does no longer result in closed-form approximations for the different risk measures. Secondly, we demonstrate how instead the weaker stop-loss order can be used to derive such closed-form approximations. We also present numerical examples to show the accuracy of the proposed approximations.

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Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

Volume (Year): 44 (2009)
Issue (Month): 3 (June)
Pages: 385-397

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Handle: RePEc:eee:insuma:v:44:y:2009:i:3:p:385-397
Contact details of provider: Web page: http://www.elsevier.com/locate/inca/505554

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