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Conditional Tail Expectation and Premium Calculation*

Published online by Cambridge University Press:  09 August 2013

Antonio Heras ,
Beatriz Balbás  and
José Luis Vilar
Beatriz Balbás
Affiliation:
Departamento de Análisis Económico y Finanzas, Universidad de Castilla – La Mancha, Talavera, Toledo (Spain), E-Mail: [email protected]
José Luis Vilar
Affiliation:
Departamento de Economía Financiera y Contabilidad 1, (Economía Financiera y Actuarial), Universidad Complutense de Madrid, Madrid (Spain) and Actuaris Ibérica, E-Mail: [email protected]
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Abstract

In this paper we calculate premiums which are based on the minimization of the Expected Tail Loss or Conditional Tail Expectation (CTE) of absolute loss functions. The methodology generalizes well known premium calculation procedures and gives sensible results in practical applications. The choice of the absolute loss becomes advisable in this context since its CTE is easy to calculate and to understand in intuitive terms. The methodology also can be applied to the calculation of the VaR and CTE of the loss associated with a given premium.

Keywords

Premium PrinciplesLoss FunctionsRisk MeasuresValue at RiskConditional Tail Expectation
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 1 , May 2012 , pp. 325 - 342
Copyright
Copyright © International Actuarial Association 2012

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Footnotes

*

Research supported by Comunidad Autónoma de Madrid (Spain), Grant S2009/ESP-1594, and Spanish Ministry of Science and Technology, Grants ECO2009-14457-C04-02 and ECO2010-22065-C03-01.

References

Acerbi, C. (2002) Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion, Journal of Banking and Finance 26(7), 1505–1518.Google Scholar
Artzner, P. (1999) Application of Coherent Risk Measures to Capital Requirements in Insurance, North American Actuarial Journal 3, 11–25.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent Measures of Risk, Mathematical Finance 9(3), 203–228.CrossRefGoogle Scholar
Balbás, A., Balbás, B. and Heras, A. (2009) Optimal Reinsurance with General Risk Measures, Insurance: Mathematics and Economics 44, 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.Google Scholar
Cai, J. and Tan, K. (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., Weng, C. and Zhang, Y. (2008) Optimal Reinsurance under VaR and CTE Risk Measures, Insurance: Mathematics and Economics 43, 185–196.Google Scholar
Denault, M. (2001) Coherent Allocation of Risk Capital, Journal of Risk 4, 7–21.CrossRefGoogle Scholar
Deprez, O. and Gerber, U. (1985) On Convex Principles of Premium Calculation, Insurance: Mathematics and Economics 4, 179–189.Google Scholar
Dhaene, J., Henrard, L., Landsman, Z., Vandendorpe, A. and Vanduffel, S. (2008) Some results on the CTE-based Capital Allocation Rule, Insurance: Mathematics and Economics 42, 855–863.Google Scholar
Föllmer, H. and Schied, A. (2002) Convex Measures of Risk and Trading Constraints, Finance and Stochastics 6(4), 429–447.Google Scholar
Furman, E. and Landsman, Z. (2006) On Some Risk-Adjusted Tail-Based Premium Calculation Principles, Journal of Actuarial Practice 13, 175–191.Google Scholar
Furman, E. and Zitikis, R. (2008a) Weighted Premium Calculation Principles, Insurance: Mathematics and Economics 42, 459–465.Google Scholar
Furman, E. and Zitikis, R. (2008b) Weighted Risk Capital Allocations, Insurance: Mathematics and Economics 43, 263–269.Google Scholar
Goovaerts, M., Etienne, F., De Vylder, C. and Haezendonk, J. (1984) Insurance Premiums, North-Holland Publishing, Amsterdam.Google Scholar
Goovaerts, M., Kaas, R., Dhaene, J. and Tang, Q. (2003) A Unified Approach to Generate Risk Measures, ASTIN Bulletin 33(2), 173–191.CrossRefGoogle Scholar
Goovaerts, M., Kaas, R., Dhaene, J. and Tang, Q. (2004a) A New Class of Consistent Risk Measures, Insurance: Mathematics and Economics 34, 505–516.Google Scholar
Goovaerts, M., Kaas, R., Laeven, R. and Tang, Q. (2004b) A Comonotonic Image of Independence for Additive Risk Measures, Insurance: Mathematics and Economics 35, 581–594.Google Scholar
Goovaerts, M. and Laeven, R. (2008) Actuarial Risk Measures for Financial Derivative Pricing, Insurance: Mathematics and Economics 42, 540–547.Google Scholar
Goovaerts, M., Kaas, R. and Laeven, R. (2010) Decision Principles Derived from Risk Measures, Insurance: Mathematics and Economics 47, 294–302.Google Scholar
Heilmann, W. (1989) Decision Theoretic Foundations of Credibility Theory, Insurance: Mathematics and Economics 8, 77–95.Google Scholar
Hürlimann, W. (2003) Conditional Value-at-Risk Bounds for Compound Poisson Risks and a Normal Approximation, Journal of Applied Mathematics 3, 141–153.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory, Kluwer Academic Publishers, Dordrecht.Google Scholar
Laeven, R. and Goovaerts, M. (2004) An Optimization Approach to the Dynamic Allocation of Economic Capital, Insurance: Mathematics and Economics 35, 299–319.Google Scholar
Landsman, Z. and Sherris, M. (2001) Risk Measures and Insurance Premium Principles, Insurance: Mathematics and Economics 29, 103–115.Google Scholar
Lemaire, J. and Vandermeulen, E. (1983) Une propieté du principe de l'espérance mathématique, Bulletin Trimestriel de l'Institut des Actuaires Français, 5–14.Google Scholar
Ogryczac, W. and Ruszczynski, A. (2002) Dual Stochastic Dominance and Related Mean Risk Models, SIAM Journal on Optimization 13, 60–78.CrossRefGoogle Scholar
Panjer, H. (2001) Measurement of Risk, Solvency Requirements and Allocation of Capital within Financial Conglomerates, Institute of Insurance and Pension Research Report 01-15, University of Waterloo, Waterloo.Google Scholar
Panjer, H. and Wilmott, G.E. (1992) Insurance Risk Models, Society of Actuaries.Google Scholar
Rockafellar, R. and Uryasev, S. (2000) Optimization of Conditional Value at Risk, Journal of Risk 2, 21–41.Google Scholar
Rockafellar, R. and Uryasev, S. (2002) Conditional Value at Risk for General Loss Distributions, Journal of Banking and Finance 26, 1443–1471.CrossRefGoogle Scholar
Rockafellar, R., Uryasev, S. and Zabarankin, M. (2006) Generalized Deviations in Risk Analysis, Finance & Stochastics 10, 51–74.CrossRefGoogle Scholar
Tsanakas, A. and Desli, E. (2003) Risk Measures and Theories of Choice, British Actuarial Journal 9, 959–991.CrossRefGoogle Scholar
Wang, S. (1995) Insurance Pricing and Increased Limits Ratemaking by Proportional Hazards Transform, Insurance: Mathematics and Economics 17(1), 43–54.Google Scholar
Wang, S. (1996) Premium Calculation by Transforming the Premium Layer Density, ASTIN Bulletin 26(1), 71–92.Google Scholar
Wang, S. (2000) A Class of Distortion Operators for Pricing Financial and Insurance Risks, Journal of Risk and Insurance 67(1), 15–36.Google Scholar
Wang, S. (2002) A Universal Framework for Pricing Financial and Insurance Risks, ASTIN Bulletin 32(2), 213–234.Google Scholar
Wang, S., Young, V. and Panjer, H. (1997) Axiomatic Characterization of Insurance Prices, Insurance: Mathematics and Economics 21(2), 173–183.Google Scholar
Wirch, J. and Hardy, M. (1999) A Synthesis of Risk Measures for Capital Adequacy, Insurance: Mathematics and Economics 25(3), 337–347.Google Scholar
Young, V. (2004) Premium Principles, in Teugels and Sundt (editors), Encyclopedia of Actuarial Science, John Wiley and Sons, England, Vol. 3, 1322–1331.Google Scholar