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OPTIMUM INSURANCE CONTRACTS WITH BACKGROUND RISK AND HIGHER-ORDER RISK ATTITUDES

Published online by Cambridge University Press:  25 April 2018

Yichun Chi  and
Wei Wei
Yichun Chi
Affiliation:
China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China E-mail: [email protected]
Wei Wei*
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, 53211, USA
*
E-mail: [email protected]
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Abstract

In this paper, we study an optimal insurance problem in the presence of background risk from the perspective of an insured with higher-order risk attitudes. We introduce several useful dependence notions to model positive dependence structures between the insurable risk and background risk. Under these dependence structures, we compare insurance contracts of different forms in higher-order risk attitudes and establish the optimality of stop-loss insurance form. We also explicitly derive the optimal retention level. Finally, we carry out a comparative analysis and investigate how the change in the insured's initial wealth or background risk affects the optimal retention level.

Keywords

Background riskhigher-order risk attitudesstochastically increasingright tail increasingstop-loss insurance
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 3 , September 2018 , pp. 1025 - 1047
Copyright
Copyright © Astin Bulletin 2018 

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References

Arrow, K.J. (1963) Uncertainty and the welfare economics of medical care. American Economic Review, 53 (5), 941–973.Google Scholar
Barlow, R.E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. New York: Holt, Rinehart and Winston.Google Scholar
Bernard, C., He, X., Yan, J.-A. and Zhou, X.Y. (2015a) Optimal insurance design under rank-dependent expected utility. Mathematical Finance, 25 (1), 154–186.Google Scholar
Bernard, C., Moraux, F., Rüschendorf, L. and Vanduffel, S. (2015b). Optimal payoffs under state-dependent preferences. Quantitative Finance, 15 (7), 1157–1173.Google Scholar
Bernard, C. and Vanduffel, S. (2014) Financial bounds for insurance claims. The Journal of Risk and Insurance, 81 (1), 27–56.Google Scholar
Cai, J. and Wei, W. (2012) Optimal reinsurance with positively dependent risks. Insurance: Mathematics and Economics, 50 (1), 57–63.Google Scholar
Caballé, J. and Pomansky, A. (1996) Mixed risk aversion. Journal of Economic Theory, 71, 485–513.Google Scholar
Chi, Y. (2017) On the optimality of a straight deductible under belief heterogeneity. Available at SSRN: https://ssrn.com/abstract=3018253.Google Scholar
Chi, Y. and Tan, K.S. (2015) Optimal incentive compatible insurance with background risk. Available at SSRN: https://ssrn.com/abstract=2589438.Google Scholar
Dana, R.A. and Scarsini, M. (2007) Optimal risk sharing with background risk. Journal of Economic Theory, 133, 152–176.Google Scholar
Denuit, M. and Eeckhoudt, L. (2013) Risk attitudes and the value of risk transformations. International Journal of Economic Theory, 9 (3), 245–254.Google Scholar
Doherty, N.A. and Schlesinger, H. (1983) The optimal deductible for an insurance policy when initial wealth is random. Journal of Business, 56 (4), 555–565.Google Scholar
Eeckhoudt, L. and Kimball, M. (1992) Background risk, prudence, and the demand for insurance. In Contributions to Insurance Economics (ed. Dionne, G.), pp. 239–254. New York: Springer.Google Scholar
Eeckhoudt, L. and Schlesinger, H. (2013) Higher-order risk attitudes. In Handbook of Insurance, (ed. Dionne, G.), pp. 41–57. New York: Springer.Google Scholar
Ekern, S. (1980) Increasing Nth degree risk. Economics Letters, 6, 329–333.Google Scholar
Gollier, C. (1996) Optimum insurance of approximate losses. The Journal of Risk and Insurance, 63 (3), 369–380.Google Scholar
Gollier, C. (2001) The Economics of Risk and Time. Cambridge, MA: MIT Press.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
Huang, H.H., Shiu, Y.M. and Wang, C.P. (2013) Optimal insurance contract with stochastic background wealth. Scandinavian Actuarial Journal, 2, 119–139.Google Scholar
Huberman, G., Mayers, D. and Smith, C.W. Jr (1983) Optimal insurance policy indemnity schedules. The Bell Journal of Economics, 14 (2), 415–426.Google Scholar
Kaluszka, M. (2001) Optimal reinsurance under mean-variance premium principles. Insurance: Mathematics and Economics, 28 (1), 61–67.Google Scholar
Kimball, M.S. (1990) Precautionary saving in the small and in the large. Econometrica, 58 (1), 53–73.Google Scholar
Lehmann, E.L. (1966) Some concepts of dependence. The Annals of Mathematical Statistics, 37 (5), 1137–1153.Google Scholar
Lu, Z.Y., Liu, L.P., Zhang, J.Y. and Meng, L.L. (2012) Optimal insurance under multiple sources of risk with positive dependence. Insurance: Mathematics and Economics, 51 (2), 462–471.Google Scholar
Mahul, O. (2000) Optimal insurance design with random initial wealth. Economics Letters, 69, 353–358.Google Scholar
Mayers, D. and Smith, C.W. Jr. (1983) The interdependence of individual portfolio decisions and the demand for insurance. Journal of Political Economy, 91 (2), 304–311.Google Scholar
Müller, A. and Stoyan, D. (2002) Comparison Methods for Stochastic Models and Risks. Chichester: John Wiley & Sons.Google Scholar
Noussair, C.N., Trautmann, S.T. and Van de Kuilen, G. (2014) Higher order risk attitudes, demographics, and financial decisions. The Review of Economic Studies, 81 (1), 325–355.Google Scholar
Ohlin, J. (1969) On a class of measures of dispersion with application to optimal reinsurance. ASTIN Bulletin: The Journal of the IAA, 5 (2), 249–266.Google Scholar
Raviv, A. (1979) The design of an optimal insurance policy. American Economic Review, 69 (1), 84–96.Google Scholar
Rothschild, M. and Stiglitz, J. (1971) Increasing risk. I. A definition. Journal of Economic Theory, 2, 225–243.Google Scholar
Schlesinger, H. (1981) The optimal level of deductibility in insurance contracts. Journal of Risk and Insurance, 48 (3), 465–481.Google Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. New York: Springer.Google Scholar
Shanthikumar, J.G. and Yao, D.D. (1991) Bivariate characterization of some stochastic order relations. Advances in Applied Probability, 23 (3), 642–659.Google 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 Teugels, J. and Sundt, B. (eds.), Encyclopedia of Actuarial Science, vol. 3. New York: John Wiley & Sons.Google Scholar