Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T11:46:05.299Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Reinsurance under VaR and CVaR Risk Measures: a Simplified Approach

Published online by Cambridge University Press:  09 August 2013

Yichun Chi  and
Ken Seng Tan
Ken Seng Tan
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, E-mail: kstan@uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study two classes of optimal reinsurance models by minimizing the total risk exposure of an insurer under the criteria of value at risk (VaR) and conditional value at risk (CVaR). We assume that the reinsurance premium is calculated according to the expected value principle. Explicit solutions for the optimal reinsurance policies are derived over ceded loss functions with increasing degrees of generality. More precisely, we establish formally that under the VaR minimization model, (i) the stop-loss reinsurance is optimal among the class of increasing convex ceded loss functions; (ii) when the constraints on both ceded and retained loss functions are relaxed to increasing functions, the stop-loss reinsurance with an upper limit is shown to be optimal; (iii) and finally under the set of general increasing and left-continuous retained loss functions, the truncated stop-loss reinsurance is shown to be optimal. In contrast, under CVaR risk measure, the stop-loss reinsurance is shown to be always optimal. These results suggest that the VaR-based reinsurance models are sensitive with respect to the constraints imposed on both ceded and retained loss functions while the corresponding CVaR-based reinsurance models are quite robust.

Keywords

Conditional value at riskValue at riskStop-loss reinsuranceLimited stop-loss designTruncated stop-loss reinsuranceOptimal reinsurance
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 2 , November 2011 , pp. 487 - 509
Copyright
Copyright © International Actuarial Association 2011

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
Asmussen, S. (2000) Ruin Probabilities. In: Advanced Series on Statistical Science & Applied Probability, vol. 2. World Scientific.CrossRefGoogle 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, 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
Cummins, J.D. and Mahul, O. (2004) The demand for insurance with an upper limit on coverage. The 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, second revised and extended edition. Walter de Gruyter.CrossRefGoogle Scholar
Froot, K.A. (2001) The market for catastrophe risk: a clinical examination. Journal of Financial Economics 60(2–3), 529–571.CrossRefGoogle Scholar
Gajek, L. and Zagrodny, D. (2004a) Optimal reinsurance under general risk measures. Insurance: Mathematics and Economics 34(2), 227–240.Google Scholar
Gajek, L. and Zagrodny, D. (2004b) Reinsurance arrangements maximizing insurer's survival probability. The Journal of Risk and Insurance 71(3), 421–435.CrossRefGoogle Scholar
Kaluszka, M. (2001) Optimal reinsurance under mean-variance premium principles. Insurance: Mathematics and Economics 28(1), 61–67.Google Scholar
Kaluszka, M. (2005) Truncated stop loss as optimal reinsurance agreement in one-period models. Astin Bulletin 35(2), 337–349.CrossRefGoogle 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
Rockafellar, R.T. (1970) Convex Analysis. Princeton University Press.CrossRefGoogle Scholar
Tan, K.S., Weng, C. and Zhang, Y. (2011) Optimality of general reinsurance contracts under CTE risk measure. Insurance: Mathematics and Economics 49(2), 175–187.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