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THE DESIGN OF AN OPTIMAL RETROSPECTIVE RATING PLAN

Published online by Cambridge University Press:  12 October 2015

Xinxiang Chen ,
Yichun Chi  and
Ken Seng Tan
Xinxiang Chen
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China E-Mail: [email protected]
Yichun Chi*
Affiliation:
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China
Ken Seng Tan
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

A retrospective rating plan, whose insurance premium depends upon an insured's actual loss during the policy period, is a special insurance agreement widely used in liability insurance. In this paper, the design of an optimal retrospective rating plan is analyzed from the perspective of the insured who seeks to minimize its risk exposure in the sense of convex order. In order to reduce the moral hazard, we assume that both the insured and the insurer are obligated to pay more for a larger realization of the loss. Under the further assumptions that the minimum premium is zero, the maximum premium is proportional to the expected indemnity, and the basic premium is the only free parameter in the formula for retrospective premium given by Meyers (2004) and that the basic premium is determined in such a way that the expected retrospective premium equates to the expected indemnity with a positive safety loading, we formally establish the relationship that the insured will suffer more risk for a larger loss conversion factor or a higher maximum premium. These findings suggest that the insured prefers an insurance policy with the expected value premium principle, which is a special retrospective premium principle with zero loss conversion factor. In addition, we show that any admissible retrospective rating plan is dominated by a stop-loss insurance policy. Finally, the optimal retention of a stop-loss insurance is derived numerically under the criterion of minimizing the risk-adjusted value of the insured's liability where the liability valuation is carried out using the cost-of-capital approach based on the conditional value at risk.

Keywords

Conditional value at riskconvex ordermoral hazardoptimal retrospective rating planstop-loss insurance
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 46 , Issue 1 , January 2016 , pp. 141 - 163
Copyright
Copyright © Astin Bulletin 2015 

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References

Arrow, K.J. (1963) Uncertainty and the welfare economics of medical care. American Economic Review, 53 (5), 941–973.Google Scholar
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9 (3), 203–228.CrossRefGoogle Scholar
Asimit, A.V., Badescu, A.M. and Verdonck, T. (2013) Optimal risk transfer under quantile-based risk measurers. Insurance: Mathematics and Economics, 53 (1), 252–265.Google Scholar
Balbás, A., Balbás, B. and Heras, A. (2009) Optimal reinsurance with general risk measures. Insurance: Mathematics and Economics, 44 (3), 374–384.Google Scholar
Bernard, C. and Tian, W. (2009) Optimal reinsurance arrangements under tail risk measures. The Journal of Risk and Insurance, 76 (3), 709–725.CrossRefGoogle Scholar
Borch, K. (1960) An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance. In: Transactions of the 16th International Congress of Actuaries, vol. I, pp. 597–610. Georges Thone, Brussels, Belgium.Google Scholar
Cai, J. and Tan, K.S. (2007) Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures. Astin Bulletin, 37 (1), 93–112.CrossRefGoogle Scholar
Cai, J., Tan, K.S., Weng, C. and Zhang, Y. (2008) Optimal reinsurance under VaR and CTE risk measures. Insurance: Mathematics and Economics, 43 (1), 185–196.Google Scholar
Cheung, K.C. (2010) Optimal reinsurance revisited — a geometric approach. Astin Bulletin, 40 (1), 221–239.CrossRefGoogle Scholar
Chi, Y. (2012) Reinsurance arrangements minimizing the risk-adjusted value of an insurer's liability. Astin Bulletin, 42 (2), 529–557.Google Scholar
Chi, Y. and Lin, X.S. (2014) Optimal reinsurance with limited ceded risk: A stochastic dominance approach. Astin Bulletin, 44 (1), 103–126.CrossRefGoogle Scholar
Chi, Y. and Tan, K.S. (2011) Optimal reinsurance under VaR and CVaR risk measures: A simplified approach. Astin Bulletin, 41 (2), 487–509.Google Scholar
Cong, J. and Tan, K.S. (2014) Optimal VaR-based Risk Management with Reinsurance. Annals of Operations Research, in press. doi:10.1007/s10479-014-1584-8.CrossRefGoogle Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002) The concept of comonotonicity in actuarial science and finance: Theory. Insurance: Mathematics and Economics, 31 (1), 3–33.Google Scholar
European Commission 2009. Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 on the Taking-up and Pursuit of the Business of Insurance and Reinsurance (Solvency II). Official Journal of the European Union L335.Google Scholar
Föllmer, H. and Schied, A., 2004. Stochastic Finance: An Introduction in Discrete Time, 2nd revised and extended edition. Berlin: Walter de Gruyter.CrossRefGoogle Scholar
Gajek, L. and Zagrodny, D. (2004) Optimal reinsurance under general risk measures. Insurance: Mathematics and Economics, 34 (2), 227–240.Google Scholar
Gollier, C. and Schlesinger, H. (1996) Arrow's theorem on the optimality of deductibles: A stochastic dominance approach. Economic Theory, 7 (2), 359–363.Google Scholar
Kaluszka, M. (2001) Optimal reinsurance under mean-variance premium principles. Insurance: Mathematics and Economics, 28 (1), 61–67.Google Scholar
Kaluszka, M. and Okolewski, A. (2008) An extension of Arrow's result on optimal reinsurance contract. The Journal of Risk and Insurance, 75 (2), 275–288.CrossRefGoogle Scholar
Meyers, G.G. (2004) Retrospective premium. In Encyclopedia of Actuarial Science, vol 3, pp. 1452–1456 (ed. Teugels, J. and Sundt, B.) John Wiley & Sons, Chichester.Google Scholar
Ohlin, J. (1969) On a class of measures of dispersion with application to optimal reinsurance. Astin Bulletin, 5 (2), 249–266.CrossRefGoogle Scholar
Raviv, A. (1979) The design of an optimal insurance policy. American Economic Review, 69 (1), 84–96.Google Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. New York: Springer.CrossRefGoogle Scholar
Swiss Federal Office of Private Insurance, (2006) Technical document on the Swiss Solvency Test. FINMA.Google Scholar
Van Heerwaarden, A.E., Kaas, R. and Goovaerts, M.J. (1989) Optimal reinsurance in relation to ordering of risks. Insurance: Mathematics and Economics, 8 (1), 11–17.Google Scholar
Young, V.R. (1999) Optimal insurance under Wang's premium principle. Insurance: Mathematics and Economics, 25 (2), 109–122.Google Scholar