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An optimal multi-layer reinsurance policy under conditional tail expectation

Published online by Cambridge University Press:  26 July 2017

Amir T. Payandeh Najafabadi  and
Ali Panahi Bazaz
Amir T. Payandeh Najafabadi*
Affiliation:
Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, Tehran 1983963113, Iran
Ali Panahi Bazaz
Affiliation:
Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, Tehran 1983963113, Iran
*
*Correspondence to: Amir T. Payandeh Najafabadi, Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, Tehran 1983963113, Iran. Tel: +98 21 2990 3011. Fax: +98 21 2243 1649. E-mail: [email protected]
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Abstract

An usual reinsurance policy for insurance companies admits one or two layers of the payment deductions. Under optimality criterion of minimising the Conditional Tail Expectation (CTE) risk measure of the insurer’s total risk, this article generalises an optimal stop-loss reinsurance policy to an optimal multi-layer reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy f(⋅). In the first step, it cuts down the interval [0, ∞) into intervals [0, M1) and [M1, ∞). By shifting the origin of Cartesian coordinate system to (M1, f(M1)), and showing that under the CTE criteria $$f\left( x \right)I_{{[0,M_{{\rm 1}} )}} \left( x \right){\plus}\left( {f\left( {M_{{\rm 1}} } \right){\plus}f\left( {x{\minus}M_{{\rm 1}} } \right)} \right)I_{{[M_{{\rm 1}} ,{\rm }\infty)}} \left( x \right)$$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the optimal multi-layer reinsurance policy are estimated using some additional appropriate criteria. Three simulation-based studies have been conducted to demonstrate: (1) the practical applications of our findings and (2) how one may employ other appropriate criteria to estimate unknown parameters of an optimal multi-layer contract. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover, it has some other optimal criteria which the original policy does not have. Under optimality criterion of minimising a general translative and monotone risk measure ρ(⋅) of either the insurer’s total risk or both the insurer’s and the reinsurer’s total risks, this article (in its discussion) also extends a given optimal reinsurance contract f(⋅) to a multi-layer and continuous reinsurance policy.

Keywords

Reinsurance policyStop-loss reinsuranceTranslative and monotone risk measuresOptimisationConditional Tail Expectation (CTE)
Type
Papers
Information
Annals of Actuarial Science , Volume 12 , Issue 1 , March 2018 , pp. 130 - 146
Copyright
© Institute and Faculty of Actuaries 2017 

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References

Assa, H. (2015). On optimal reinsurance policy with distortion risk measures and premiums. Insurance: Mathematics and Economics, 61, 70–75.Google Scholar
Bailey, A.L. (1950). Credibility procedures laplace’s generalization of Bayes’ rule and the combination of collateral knowledge with observed data. Proceedings of the Casualty Actuarial Society, 37, 7–23.Google Scholar
Borch, K. (1960). An attempt to determine the optimum amount of stop-loss reinsurance. Transactions of the 16th International Congress of Actuaries, September 1960, pp. 597–610, Brussels Belgium.Google Scholar
Cai, J., Tan, K.S., Weng, C. & Zhang, Y. (2008). Optimal reinsurance under VaR and CTE risk measures. Insurance: Mathematics and Economics, 43, 185–196.Google Scholar
Cai, J. & Weng, C. (2014). Optimal reinsurance with expectile. Scandinavian Actuarial Journal, 7, 1–22.Google Scholar
Chi, Y. (2012a). Reinsurance arrangements minimizing the risk-adjusted value of an insurer’s liability. Astin Bulletin, 42(2), 529–557.Google Scholar
Chi, Y. (2012b). Optimal reinsurance under variance related premium principles. Insurance: Mathematics and Economics, 51(2), 310–321.Google Scholar
Chi, Y. & Tan, K.S. (2011). Optimal reinsurance under VaR and CVaR risk measures: a simplified approach. ASTIN Bulletin, 41, 487–509.Google Scholar
Chi, Y. & Tan, K.S. (2013). Optimal reinsurance with general premium principles. Insurance: Mathematics and Economics, 52(2), 180–189.Google Scholar
Cortes, O.A.C., Rau-Chaplin, A., Wilson, D., Cook, I. & Gaiser-Porter, J. (2013). Efficient optimization of reinsurance contracts using discretized PBIL. Data Analytics: The Second International Conference on Data Analytics, Porto, Portugal, 29 September–3 October, 2013.Google Scholar
Dedu, S. (2012). Optimization of some risk measures in stop-loss reinsurance with multiple retention levels. Mathematical Reports, 14, 131–139.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. & Kaas, R. (2006). Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons, New York.Google Scholar
Dickson, D.C. (2005). Insurance Risk and Ruin. Cambridge University Press, New York.CrossRefGoogle Scholar
England, P. & Verrall, R. (2002). Stochastic claims reserving in general insurance (with discussion). British Actuarial Journal, 8, 443–544.CrossRefGoogle Scholar
Fang, Y. & Qu, Z. (2014). Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability. IMA Journal of Management Mathematics, 25, 89–103.CrossRefGoogle Scholar
Hesselager, O. (1990). Some results on optimal reinsurance in terms of the adjustment coefficient. Scandinavian Actuarial Journal, 1990, 80–95.CrossRefGoogle Scholar
Hesselager, O. & Witting, T. (1988). A credibility model with random fluctuations in delay probabilities for the prediction of IBNR claims. ASTIN Bulletin, 18, 79–90.CrossRefGoogle Scholar
Hossack, I.B., Pollard, J.H. & Zenwirth, B. (1999). Introductory Statistics With Applications in General Insurance, 2nd edition. University Press, Cambridge.CrossRefGoogle Scholar
Kaluszka, M. (2005). Truncated stop loss as optimal reinsurance agreement in one-period models. Astin Bulletin, 35(2), 337–349.CrossRefGoogle Scholar
Kaluszka, M. & Okolewski, A. (2008). An extension of arrow’s result on optimal reinsurance contract. Journal of Risk and Insurance, 75(2), 275–288.CrossRefGoogle Scholar
Makov, U.E. (2001). Principal applications of Bayesian methods in actuarial science: a perspective. North American Actuarial Journal, 5, 53–73.CrossRefGoogle Scholar
Makov, U.E., Smith, A.F.M. & Liu, Y.H. (1996). Bayesian methods in actuarial science. The Statistician, 45, 503–515.CrossRefGoogle Scholar
Ouyang, Y.X. & Li, Z.Y. (2010). Adverse selection, systematic risks and sustainable development of policy agricultural insurance. Insurance Studies, 4, 1–9.Google Scholar
Panahi Bazaz, A. & Payandeh Najafabadi, A.T. (2015). An optimal reinsurance contract from insurer’s and reinsurer’s viewpoints. Applications & Applied Mathematics, 10(2), 970–982.Google Scholar
Passalacqua, L. (2007). Measuring effects of excess-of-loss reinsurance on credit insurance risk capital. Giornale delł’Istituto Italiano degli Attuari, LXX, 81–102.Google Scholar
Payandeh Najafabadi, A.T. (2010). A new approach to the credibility formula. Insurance: Mathematics and Economics, 46, 334–338.Google Scholar
Payandeh Najafabadi, A.T., Hatami, H. & Omidi Najafabadi, M. (2012). A maximum entropy approach to the linear credibility formula. Insurance: Mathematics and Economics, 51, 216–221.Google Scholar
Payandeh Najafabadi, A.T. & Panahi Bazaz, A.P. (2016). An optimal co-reinsurance strategy. Insurance: Mathematics and Economics, 69, 149–155.Google Scholar
Payandeh Najafabadi, A.T. & Qazvini, M. (2015). A GLM approach to estimating copula models. Communications in Statistics-Simulation and Computation, 44(6), 1641–1656.CrossRefGoogle Scholar
Porth, L., Seng Tan, K. & Weng, C. (2013). Optimal reinsurance analysis from a crop insurer’s perspective. Agricultural Finance Review, 73(2), 310–328.CrossRefGoogle Scholar
Tan, K.S. & Weng, C. (2012). Enhancing insurer value using reinsurance and value-at-risk criterion. The Geneva Risk and Insurance Review, 37, 109–140.CrossRefGoogle Scholar
Teugels, J.L. & Sundt, B. (2004). Encyclopedia of Actuarial Science (Vol 1). Wiley, New York.CrossRefGoogle Scholar
Weng, C. (2009). Optimal Reinsurance Designs: From an Insurer’s Perspective. PhD thesis. University of Waterloo, Waterloo, Canada.Google Scholar
Whitney, A.W. (1918). The theory of experience rating. Proceedings of the Casualty Actuarial Society, 4, 274–292.Google Scholar
Zhuang, S.C., Weng, C., Tan, K.S. & Assa, H. (2016). Marginal indemnification function formulation for optimal reinsurance. Insurance: Mathematics and Economics, 67, 65–76.Google Scholar