Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T17:23:32.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Reinsurance Arrangements Minimizing the Risk-Adjusted Value of an Insurer's Liability

Published online by Cambridge University Press:  09 August 2013

Yichun Chi
Yichun Chi*
Affiliation:
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we investigate the problem of purchasing a reinsurance policy that minimizes the risk-adjusted value of an insurer's liability, where the valuation is carried out using a cost-of-capital approach. In order to exclude the moral hazard, we assume that both the insurer and reinsurer are obligated to pay more for larger loss in a typical reinsurance treaty. Moreover, the reinsurance premium principle is assumed to satisfy three axioms: law invariance, risk loading and preserving convex order. The proposed class of premium principles is quite general in the sense that it contains all the widely used premium principles except Esscher principle listed in Young (2004). When capital at risk is measured by the value at risk (VaR) or conditional value at risk (CVaR), we find it is optimal for the insurer to cede two separate layers over the prescribed premium principles. By imposing an additional weak constraint on the premium principle, we further get that the reinsurance in the form of a layer is optimal. Finally, to illustrate the applicability of our results, we derive explicitly the optimal one-layer reinsurance for expected value principle and Wang's premium principle, and show that two-layer reinsurance may be optimal for Dutch premium principle.

Keywords

Capital at riskCost of capitalConditional value at riskValue at riskOptimal reinsuranceWang's premium principleDutch premium principleLayer reinsurance
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 2 , November 2012 , pp. 529 - 557
Copyright
Copyright © International Actuarial Association 2012

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, V., Badescu, A. and Tsanakas, A. (2012) Optimal risk transfers in insurance groups. Working paper. Available at SSRN: http://ssrn.com/abstract=1987783.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.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) Optimal reinsurance under variance related premium principles. Insurance: Mathematics and Economics 51(2), 310–321.Google Scholar
Chi, Y. and Tan, K.S. (2011a) 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. (2011b) Optimal reinsurance with general premium principles. Working paper. Available at SSRN: http://ssrn.com/abstract=1836142.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, second revised and extended edition. Walter de Gruyter.CrossRefGoogle Scholar
Fu, L. and Khury, C.K. (2010) Optimal layers for catastrophe reinsurance. Variance: Advancing the Science of Risk 4(2), 191–208.Google Scholar
Guerra, M. and Centeno, M.D.L. (2012) Are quantile risk measures suitable for risk-transfer decisions? Insurance: Mathematics and Economics 50(3), 446–461.Google Scholar
Kaluszka, M. (2001) Optimal reinsurance under mean-variance premium principles. Insurance: Mathematics and Economics 28(1), 61–67.Google Scholar
Kaluszka, M. (2005) Optimal reinsurance under convex principles of premium calculation. Insurance: Mathematics and Economics 36(3), 375–398.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
Ohlin, J. (1969) On a class of measures of dispersion with application to optimal reinsurance. Astin Bulletin 5(2), 249–266.CrossRefGoogle Scholar
Pelsser, A. (2010) Time-consistent and market-consistent actuarial valuations. Working paper. Available at SSRN: http://ssrn.com/abstract=1576612.CrossRefGoogle Scholar
Risk Margin Working Group (2009) Measurement of liabilities for insurance contracts: current estimates and risk margins. International Actuarial Association.Google Scholar
Swiss Federal Office of Private Insurance (2006) Technical document on the Swiss Solvency Test. FINMA.Google Scholar
Tan, K.S., Weng, C. and Zhang, Y. (2009) VaR and CTE criteria for optimal quota-share and stop-loss reinsurance. North American Actuarial Journal 13(4), 459–482.CrossRefGoogle Scholar
Van Heerwaarden, A.E. and Kaas, R. (1992) The Dutch premium principle. Insurance: Mathematics and Economics 11(2), 129–133.Google Scholar
Van Heerwaarden, A.E., Kaas, R. and Goovaerts, M.J. (1989) Properties of the Esscher premium calculation principle. Insurance: Mathematics and Economics 8(4), 261–267.Google Scholar
Wang, S. (1995) Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics 17(1), 43–54.Google Scholar
Wang, S., Young, V.R. and Panjer, H.H. (1997) Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics 21(2), 173–183.Google Scholar
Wüthrich, M.V., Bühlmann, H. and Furrer, H. (2010) Market-Consistent Actuarial Valuation, second, revised and enlarged edition. Springer.CrossRefGoogle Scholar
Young, V.R. (1999) Optimal insurance under Wang's premium principle. Insurance: Mathematics and Economics 25(2), 109–122.Google Scholar
Young, V.R. (2004) Premium Principles. In Encyclopedia of Actuarial Science, vol. 3, ed. Teugels, J. and Sundt, B.. John Wiley & Sons.Google Scholar