Bounds and Approximations for Sums of Dependent Log-Elliptical Random Variables

25 Pages Posted: 14 Jul 2009 Last revised: 27 Feb 2011

Emiliano A. Valdez

University of Connecticut

Jan Dhaene

Katholieke Universiteit Leuven - Faculty of Economics and Business

Mateusz Maj

Vrije Universiteit Brussel (VUB) - Faculty of Economic, Social and Political Sciences

Steven Vanduffel

Vrije Universiteit Brussel (VUB)

Date Written: October 10, 2008

Abstract

Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002a,b) 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.

Suggested Citation

Valdez, Emiliano A. and Dhaene, Jan and Maj, Mateusz and Vanduffel, Steven, Bounds and Approximations for Sums of Dependent Log-Elliptical Random Variables (October 10, 2008). Insurance: Mathematics and Economics, Vol. 44, pp. 385-397 . Available at SSRN: https://ssrn.com/abstract=1356483

Emiliano A. Valdez (Contact Author)

University of Connecticut ( email )

341 Mansfield Road U-1009
Storrs, CT 06269-1009
United States

HOME PAGE: http://www.math.uconn.edu/~valdez

Jan Dhaene

Katholieke Universiteit Leuven - Faculty of Economics and Business ( email )

Naamsestraat 69
Leuven, 3000
Belgium

Mateusz Maj

Vrije Universiteit Brussel (VUB) - Faculty of Economic, Social and Political Sciences ( email )

Ave. Franklin D Roosevelt, 50 - C.P. 191
B-1050 Brussels
Belgium

Steven Vanduffel

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
Brussels, Brabant 1050
Belgium

HOME PAGE: http://www.stevenvanduffel.com

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