Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-04T19:22:39.095Z Has data issue: false hasContentIssue false

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
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email: [email protected]
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: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Articles
Copyright
Copyright © ASTIN Bulletin 2007

References

Arnold, B.C. (1983) Pareto Distributions. International Co-operative Publishing House, Burtonsville, Maryland.Google Scholar
Artzner, P., Delbaen, F., Eber, J. and Heath, D. (1999) Coherence measures of risk, Mathematical Finance 9, 203228.CrossRefGoogle Scholar
Assaf, D., Langberg, N., Savits, T., and Shaked, M. (1984) Multivariate phase-type distributions. Operations Research 32, 688702.CrossRefGoogle Scholar
Bowers, N.J., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1997) Actuarial Mathematics. Second Edition. The Society of Actuaries, Schaumburg.Google Scholar
Cai, J. (2004) Stop-loss premium. Encyclopedia of Actuarial Science. John Wiley & Sons, Chichester, Volume 3, 16151619.Google Scholar
Cai, J. and Li, H. (2005a) Conditional tail expectations for multivariate phase type distributions. Journal of Applied Probability 42, 810825.CrossRefGoogle Scholar
Cai, J. and Li, H. (2005b) Multivariate risk model of phase-type. Insurance: Mathematics and Economics 36, 137152.Google Scholar
Centeno, M.L. (2002) Measuring the effect of reinsurance by the adjustment coefficient in the Sparre Andersen model. Insurance: Mathematics and Economics 30, 3750.Google Scholar
Centeno, M.L. (2004) Retention and reinsurance programmes. Encyclopedia of Actuarial Science, John Wiley & Sons, Chichester.Google Scholar
Cossette, H., Gaillardetz, P., Marceau, E., and Rioux, J. (2002). On two dependent individual risk models. Insurance: Mathematics and Economics 30, 153166 Google Scholar
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994) Practical Risk Theory for Actuaries. Chapman and Hall, London.Google Scholar
Jorion, P. (2000) Value at Risk: The Benchmark for Controlling Market Risk. Second edition, McGraw-Hill.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory. Kluwer Academic Publishers, Boston.Google Scholar
Krokhmal, P., Palmquist, J., and Uryasev, S. (2002) Portfolio Optimization with Conditional Value-At-Risk Objective and Constraints. The Journal of Risk 4(2), 1127.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2004) Loss Models: From Data to Decisions. Second Edition. John and Wiley & Sons, New York.Google Scholar
Marshall, A.W. and Olkin, I. (1967) A multivariate exponential distribution. Journal of the American Statistical Association 2, 8498.Google Scholar
Seal, H. (1969). Stochastic Theory of a Risk Business, John Wiley & Sons, New York.Google Scholar
Wirch, J.L. and Hardy, M.R. (1999) A synthesis of risk measures for capital adequacy. Insurance: Mathematics and Economics 25, 337347.Google Scholar