Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T08:25:23.132Z Has data issue: false hasContentIssue false
  • English
  • Français

OPTIMAL REINSURANCE WITH LIMITED CEDED RISK: A STOCHASTIC DOMINANCE APPROACH

Published online by Cambridge University Press:  19 November 2013

Yichun Chi  and
X. Sheldon Lin
Yichun Chi*
Affiliation:
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, China, 100081
X. Sheldon Lin
Affiliation:
Department of Statistical Sciences, University of Toronto, Toronto, Ontario, CanadaM5S 3G3 E-Mail: [email protected]
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

An optimal reinsurance problem from the perspective of an insurer is studied in this paper, where an upper limit is imposed on a reinsurer's expected loss over a prescribed level. In order to reduce the moral hazard, we assume that both the insurer and the reinsurer are obligated to pay more as the amount of loss increases in a typical reinsurance treaty. We further assume that the optimization criterion preserves the convex order. Such a criterion is very general as most of the criteria for optimal reinsurance problems in the literature preserve the convex order. When the reinsurance premium is calculated as a function of the actuarial value of coverage, we show via a stochastic dominance approach that any admissible reinsurance policy is dominated by a stop-loss reinsurance or a two-layer reinsurance, depending upon the amount of the reinsurance premium. Moreover, we obtain a similar result to Mossin's Theorem and find that it is optimal for the insurer to cede a loss as much as possible under the net premium principle. To further examine the reinsurance premium for the optimal piecewise linear reinsurance policy, we assume the expected value premium principle and derive the optimal reinsurance explicitly under (1) the criterion of minimizing the variance of the insurer's risk exposure, and (2) the criterion of minimizing the risk-adjusted value of the insurer'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 ordercost of capitalMossin's Theoremstop-loss reinsurancetwo-layer reinsurance
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 44 , Issue 1 , January 2014 , pp. 103 - 126
Copyright
Copyright © ASTIN Bulletin 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 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. Transactions of the 16th International Congress of Actuaries, I, 597–610.Google 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 Tan, K.S. (2011) Optimal reinsurance under VaR and CVaR risk measures: A simplified approach. ASTIN Bulletin, 41 (2), 487–509.Google Scholar
Chi, Y. and Tan, K.S. (2013) Optimal reinsurance with general premium principles. Insurance: Mathematics and Economics, 52 (2), 180–189.Google Scholar
Cummins, J.D. and Mahul, O. (2004) The demand for insurance with an upper limit on coverage. Journal of Risk and Insurance, 71 (2), 253–264.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
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
Guerra, M. and Centeno, M.L. (2012) Are quantile risk measures suitable for risk-transfer decisions? Insurance: Mathematics and Economics, 50 (3), 446–461.Google Scholar
Huberman, G., Mayers, D. and Smith, C.W. (1983) Optimal insurance policy indemnity schedules. Bell Journal of Economics, 14 (2), 415–426.CrossRefGoogle Scholar
Kaluszka, M. (2001) Optimal reinsurance under mean-variance premium principles. Insurance: Mathematics and Economics, 28 (1), 61–67.Google Scholar
Lu, Z.Y., Liu, L.P. and Meng, S.W. (2013) Optimal reinsurance with concave ceded loss functions under VaR and CTE risk measures. Insurance: Mathematics and Economics, 52 (1), 46–51.Google Scholar
Mossin, J. (1968) Aspects of rational insurance purchasing. Journal of Political Economy, 76 (4), 553–568.CrossRefGoogle 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
Zhou, C. and Wu, C. (2008) Optimal insurance under the insurer's risk constraint. Insurance: Mathematics and Economics, 42 (3), 992–999.Google Scholar
Zhou, C., Wu, W. and Wu, C. (2010) Optimal insurance in the presence of insurer's loss limit. Insurance: Mathematics and Economics, 46 (2), 300–307.Google Scholar