Skip to main content

An optimal multi-layer reinsurance policy under conditional tail expectation

  • Amir T. Payandeh Najafabadi (a1) and Ali Panahi Bazaz (a1)

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, M 1) and [M 1, ∞). By shifting the origin of Cartesian coordinate system to (M 1, f(M 1)), 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.

Corresponding author
*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:
Hide All
Assa, H. (2015). On optimal reinsurance policy with distortion risk measures and premiums. Insurance: Mathematics and Economics, 61, 7075.
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, 723.
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.
Cai, J., Tan, K.S., Weng, C. & Zhang, Y. (2008). Optimal reinsurance under VaR and CTE risk measures. Insurance: Mathematics and Economics, 43, 185196.
Cai, J. & Weng, C. (2014). Optimal reinsurance with expectile. Scandinavian Actuarial Journal, 7, 122.
Chi, Y. (2012a). Reinsurance arrangements minimizing the risk-adjusted value of an insurer’s liability. Astin Bulletin, 42(2), 529557.
Chi, Y. (2012b). Optimal reinsurance under variance related premium principles. Insurance: Mathematics and Economics, 51(2), 310321.
Chi, Y. & Tan, K.S. (2011). Optimal reinsurance under VaR and CVaR risk measures: a simplified approach. ASTIN Bulletin, 41, 487509.
Chi, Y. & Tan, K.S. (2013). Optimal reinsurance with general premium principles. Insurance: Mathematics and Economics, 52(2), 180189.
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.
Dedu, S. (2012). Optimization of some risk measures in stop-loss reinsurance with multiple retention levels. Mathematical Reports, 14, 131139.
Denuit, M., Dhaene, J., Goovaerts, M. & Kaas, R. (2006). Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons, New York.
Dickson, D.C. (2005). Insurance Risk and Ruin. Cambridge University Press, New York.
England, P. & Verrall, R. (2002). Stochastic claims reserving in general insurance (with discussion). British Actuarial Journal, 8, 443544.
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, 89103.
Hesselager, O. (1990). Some results on optimal reinsurance in terms of the adjustment coefficient. Scandinavian Actuarial Journal, 1990, 8095.
Hesselager, O. & Witting, T. (1988). A credibility model with random fluctuations in delay probabilities for the prediction of IBNR claims. ASTIN Bulletin, 18, 7990.
Hossack, I.B., Pollard, J.H. & Zenwirth, B. (1999). Introductory Statistics With Applications in General Insurance, 2nd edition. University Press, Cambridge.
Kaluszka, M. (2005). Truncated stop loss as optimal reinsurance agreement in one-period models. Astin Bulletin, 35(2), 337349.
Kaluszka, M. & Okolewski, A. (2008). An extension of arrow’s result on optimal reinsurance contract. Journal of Risk and Insurance, 75(2), 275288.
Makov, U.E. (2001). Principal applications of Bayesian methods in actuarial science: a perspective. North American Actuarial Journal, 5, 5373.
Makov, U.E., Smith, A.F.M. & Liu, Y.H. (1996). Bayesian methods in actuarial science. The Statistician, 45, 503515.
Ouyang, Y.X. & Li, Z.Y. (2010). Adverse selection, systematic risks and sustainable development of policy agricultural insurance. Insurance Studies, 4, 19.
Panahi Bazaz, A. & Payandeh Najafabadi, A.T. (2015). An optimal reinsurance contract from insurer’s and reinsurer’s viewpoints. Applications & Applied Mathematics, 10(2), 970982.
Passalacqua, L. (2007). Measuring effects of excess-of-loss reinsurance on credit insurance risk capital. Giornale delł’Istituto Italiano degli Attuari, LXX, 81102.
Payandeh Najafabadi, A.T. (2010). A new approach to the credibility formula. Insurance: Mathematics and Economics, 46, 334338.
Payandeh Najafabadi, A.T., Hatami, H. & Omidi Najafabadi, M. (2012). A maximum entropy approach to the linear credibility formula. Insurance: Mathematics and Economics, 51, 216221.
Payandeh Najafabadi, A.T. & Panahi Bazaz, A.P. (2016). An optimal co-reinsurance strategy. Insurance: Mathematics and Economics, 69, 149155.
Payandeh Najafabadi, A.T. & Qazvini, M. (2015). A GLM approach to estimating copula models. Communications in Statistics-Simulation and Computation, 44(6), 16411656.
Porth, L., Seng Tan, K. & Weng, C. (2013). Optimal reinsurance analysis from a crop insurer’s perspective. Agricultural Finance Review, 73(2), 310328.
Tan, K.S. & Weng, C. (2012). Enhancing insurer value using reinsurance and value-at-risk criterion. The Geneva Risk and Insurance Review, 37, 109140.
Teugels, J.L. & Sundt, B. (2004). Encyclopedia of Actuarial Science (Vol 1). Wiley, New York.
Weng, C. (2009). Optimal Reinsurance Designs: From an Insurer’s Perspective. PhD thesis. University of Waterloo, Waterloo, Canada.
Whitney, A.W. (1918). The theory of experience rating. Proceedings of the Casualty Actuarial Society, 4, 274292.
Zhuang, S.C., Weng, C., Tan, K.S. & Assa, H. (2016). Marginal indemnification function formulation for optimal reinsurance. Insurance: Mathematics and Economics, 67, 6576.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Annals of Actuarial Science
  • ISSN: 1748-4995
  • EISSN: 1748-5002
  • URL: /core/journals/annals-of-actuarial-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 29 *
Loading metrics...

Abstract views

Total abstract views: 208 *
Loading metrics...

* Views captured on Cambridge Core between 26th July 2017 - 17th March 2018. This data will be updated every 24 hours.