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A Simple Geometric Proof that Comonotonic Risks Have the Convex-Largest Sum

Published online by Cambridge University Press:  29 August 2014

R. Kaas ,
J. Dhaene ,
D. Vyncke ,
M.J. Goovaerts  and
M. Denuit
R. Kaas
Affiliation:
Faculty of Economics and Econometrics, Dept KE, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands, E-mail: robkaas@fee.uva.nl
J. Dhaene
Affiliation:
Center for Risk and Insurance Studies, Catholic University of Leuven, Naamsestraat 69, 3000 Leuven, Belgium, E-mail: Jan.Dhaene@econ.kuleuven.ac.be
D. Vyncke
Affiliation:
Center for Risk and Insurance Studies, Catholic University of Leuven, Naamsestraat 69, 3000 Leuven, Belgium, E-mail: David.Vyncke@econ.kuleuven.ac.be
M.J. Goovaerts
Affiliation:
Center for Risk and Insurance Studies, Catholic University of Leuven, Naamsestraat 69, 3000 Leuven, Belgium, E-mail: Marc.Goovaerts@econ.kuleuven.ac.b
M. Denuit
Affiliation:
Institut de Statistique, Université Catholique de Louvain, Voie du Roman Pays 20, 1348 Louvain-la-Neuve, Belgium, E-mail: denuit@stat.ucl.ac.b
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Abstract

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In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X1X2, …, Xn) with given marginals has a comonotonic joint distribution, the sum X1 + X2 + … + Xn is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.

Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 32 , Issue 1 , May 2002 , pp. 71 - 80
Copyright
Copyright © International Actuarial Association 2002

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