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Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

Published online by Cambridge University Press:  17 April 2015

Jun Cai  and
Ken Seng Tan
Jun Cai
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email: jcai@uwaterloo.ca
Ken Seng Tan
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1, and China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, China, 100081. Email: kstan@uwaterloo.ca
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Abstract

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We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

Keywords

Stop-loss reinsuranceexpected value principleretentionloading factorvalue-at-risk (VaR)conditional tail expectation (CTE)multivariate phase type distributionmultivariate Pareto distributionindividual risk modelcollective risk model
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 37 , Issue 1 , May 2007 , pp. 93 - 112
Copyright
Copyright © ASTIN Bulletin 2007

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