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SIZE-BIASED TRANSFORM AND CONDITIONAL MEAN RISK SHARING, WITH APPLICATION TO P2P INSURANCE AND TONTINES

Published online by Cambridge University Press:  17 July 2019

Michel Denuit
Michel Denuit*
Affiliation:
Institut de Statistique, Biostatistique et Sciences Actuarielles - ISBA UCLouvain B-1348 Louvain-la-Neuve, Belgium E-mail: [email protected]
*
E-mail: [email protected]
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Abstract

Using risk-reducing properties of conditional expectations with respect to convex order, Denuit and Dhaene [Denuit, M. and Dhaene, J. (2012). Insurance: Mathematics and Economics 51, 265–270] proposed the conditional mean risk sharing rule to allocate the total risk among participants to an insurance pool. This paper relates the conditional mean risk sharing rule to the size-biased transform when pooled risks are independent. A representation formula is first derived for the conditional expectation of an individual risk given the aggregate loss. This formula is then exploited to obtain explicit expressions for the contributions to the pool when losses are modeled by compound Poisson sums, compound Negative Binomial sums, and compound Binomial sums, to which Panjer recursion applies. Simple formulas are obtained when claim severities are homogeneous. A couple of applications are considered: first, to a peer-to-peer insurance scheme where participants share the first layer of their respective risks while the higher layer is ceded to a (re)insurer; second, to survivor credits to be shared among surviving participants in tontine schemes.

Keywords

Conditional expectationrisk poolingrisk measurescompound distributionsPanjer family of distributions
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 3 , September 2019 , pp. 591 - 617
Copyright
© Astin Bulletin 2019 

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