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Average Value-at-Risk Minimizing Reinsurance Under Wang's Premium Principle with Constraints

Published online by Cambridge University Press:  09 August 2013

K.C. Cheung ,
F. Liu  and
S.C.P. Yam
F. Liu
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, N2L 3G1, Canada
S.C.P. Yam
Affiliation:
Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
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Abstract

In the present work, we study the optimal reinsurance decision problem in which the Average Value-at-Risk of the retained loss is minimized under Wang's premium principle and is also subject to either (1) a budget constraint on reinsurance premium, or (2) a reinsurer's probabilistic benchmark constraint of his potential loss. We show that the optimal reinsurance is a single-insurance layer under Constraint (1), and a cap insurance or a double-insurance layer under Constraint (2); moreover, under Constraint (2), we further establish that under most common circumstances (see Remark after Theorem 3), a cap insurance will suffice to be optimal. Finally, some numerical illustrations will be provided.

Keywords

Optimal reinsuranceAverage Value-at-RiskValue-at-RiskWang's premium principleSingle and double insurance layers
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 2 , November 2012 , pp. 575 - 600
Copyright
Copyright © International Actuarial Association 2012

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References

[1] Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking and Finance 26(7), 1487–1503.Google Scholar
[2] Arrow, K.J. (1963) Uncertainty and the Welfare Economics of Medical Care. American Economic Review 53, 941–973.Google Scholar
[3] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance 9(3), 203–228.Google Scholar
[4] Balbás, A., Balbás, B. and Heras, A. (2009) Optimal reinsurance with general risk measures. Insurance: Mathematics and Economics 44(3), 374–384.Google Scholar
[5] Basak, S. and Shapiro, A. (2001) Value-at-Risk based risk management: Optimal policies and asset prices. The Review of Financial Studies 14, 371–405.CrossRefGoogle Scholar
[6] Blazenko, G. (1985) The design of an optimal insurance policy: Note. The American Economic Review 75, 253–255.Google Scholar
[7] Borch, K. (1960) An attempt to determine the optimum amount of stop loss reinsurance. Transactions of the 16th International Congress of Actuaries, 597–610.Google Scholar
[8] Borch, K. (1975) Optimal insurance arrangements. ASTIN Bulletin 8, 284–290.Google Scholar
[9] Buhlmann, H. and Jewell, W.S. (1979) Optimal risk exchange. ASTIN Bulletin 10, 243–262.Google Scholar
[10] 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.Google Scholar
[11] 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
[12] Cheung, K.C. (2010) Optimal reinsurance revisited – geometric approach. ASTIN Bulletin 40(1), 221–239.Google Scholar
[13] Cheung, K.C., Sung, K.C.J., Yam, S.C.P. and Yung, S.P. (2010) Optimal reinsurance under general law-invariant risk measures. To appear in Scandinavian Actuarial Journal.Google Scholar
[14] Delbaen, F. (2000) Coherent risk measures. Scuola Normale Superiore di Pisa, Cattedra.Google Scholar
[15] Delbaen, F. (2002) Coherent risk measures on general probability spaces. Advances in Finance and Stochastics, Springer-Verlag, Berlin, 1–37.Google Scholar
[16] Denuit, M. and Vermandele, C. (1998) Optimal reinsurance and stop-loss order. Insurance: Mathematics and Economics 22, 229–233.Google Scholar
[17] Dhaene, J., Vanduffel, S., Tang, Q., Goovaerts, M., Kaas, R. and Vyncke, D. (2006) Risk measures and comonotonicity: a review. Stochastic Models 22, 573–606.Google Scholar
[18] Embrechts, P. (2000) Extreme value theory: Potential and limitations as an integrated risk management tool. Derivatives Use, Trading and Regulation 6, 449–456.Google Scholar
[19] Föllmer, H. and Schied, A. (2002) Convex measures of risk and trading constraints. Finance and Stochastics 6(4), 429–447.Google Scholar
[20] Föllmer, H. and Schied, A. (2004) Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter, Berlin.CrossRefGoogle Scholar
[21] Frittelli, M. and Rossaza Gianin, E. (2002) Putting order in risk measures. Journal of Banking and Finance 26, 1473–1486.Google Scholar
[22] Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania, Philadelphia.Google Scholar
[23] Guerra, M. and Centeno, M.L. (2008) Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria. Insurance: Mathematics and Economics 42(2), 529–539.Google Scholar
[24] Gollier, C. (1987) The design of optimal insurance contracts without the nonnegativity constraint on claims. The Journal of Risk and Insurance 54, 314–324.Google Scholar
[25] Heath, D. and Ku, H. (2004) Pareto equilibria with coherent measures of risk. Mathematical Finance 14(2), 163–172.CrossRefGoogle Scholar
[26] Hull, J.C. (2010) Risk Management and Financial Institutions. Pearson, U.S.A. Google Scholar
[27] Kaluszka, M. (2004a) Mean-variance optimal reinsurance arrangements. Scandinarian Actuarial Journal 1, 28–41.Google Scholar
[28] Kaluszka, M. (2004b) An extension of Arrow's result on optimality of a stop loss contract. Insurance: Mathematics and Economics 35(3), 527–536.Google Scholar
[29] Kaluszka, M. (2005) Optimal reinsurance under convex principles of premium calculation. Insurance: Mathematics and Economics 36(3), 375–398.Google Scholar
[30] Moore, K.S. and Young, V.R. (2006) Optimal insurance in a continuous-time model. Insurance Mathematics and Economics 39, 47–68.CrossRefGoogle Scholar
[31] von Neumann, J. and Morgenstern, O. (1944) Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ.Google Scholar
[32] Raviv, A. (1979) The design of an optimal insurance policy. The American Economic Review 69, 84–96.Google Scholar
[33] Rockafellar, R.T. and Uryasev, S. (2001) Conditional Value-at-Risk for general loss distributions. ISE Dept., University of Florida.Google Scholar
[34] Sung, K.C.J., Yam, S.C.P., Yung, S.P. and Zhou, J.H. (2011) Behavioral Optimal Insurance. Insurance: Mathematics and Economics 49(3), 418–428.Google Scholar
[35] 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
[36] Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin 26, 71–92.Google Scholar
[37] 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
[38] Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica 55, 95–115.Google Scholar
[39] Yaari, S., Young V.R. and Panjer, H.H. (1997) Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics 21(2), 173–183.Google Scholar
[40] Young, V. R. (1999) Optimal insurance under Wang's premium principle. Insurance: Mathematics and Economics 25, 109–122.Google Scholar