×

zbMATH — the first resource for mathematics

Optimal insurance design under rank-dependent expected utility. (English) Zbl 1314.91134
Summary: We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank-dependent expected utility (RDEU) theory with a concave utility function and an inverse-S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.

MSC:
91B30 Risk theory, insurance (MSC2010)
91B16 Utility theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abdellaoui, Parameter-Free Elicitation of Utility and Probability Weighting Functions, Manage. Sci. 46 (11) pp 1497– (2000) · Zbl 1232.91114 · doi:10.1287/mnsc.46.11.1497.12080
[2] Abdellaoui, Loss Aversion under Prospect Theory: A Parameter-Free Measurement, Manage. Sci. 53 (10) pp 1659– (2007) · doi:10.1287/mnsc.1070.0711
[3] Allais, Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Americaine, Econometrica 21 (4) pp 503– (1953) · Zbl 0050.36801 · doi:10.2307/1907921
[4] Arrow, Uncertainty and the Welfare Economics of Medical Care, Am. Econ. Rev. 53 (5) pp 941– (1963)
[5] Arrow, Essays in the Theory of Risk Bearing (1971)
[6] Barberis, Stocks as Lotteries: The Implications of Probability Weighting for Security Prices, Am. Econ. Rev. 98 (5) pp 2066– (2008) · doi:10.1257/aer.98.5.2066
[7] Bernard, Static Portfolio Choice under Cumulative Prospect Theory, Math. Fin. Econ. 2 (4) pp 277– (2010) · Zbl 1255.91389 · doi:10.1007/s11579-009-0021-2
[8] Bernard, Optimal Insurance Policies When Insurers Implement Risk Management Metrics, Geneva Risk Insurance Rev. 35 pp 47– (2010) · doi:10.1057/grir.2009.2
[9] Berner , R. 2004 The Warranty Windfall Business Week
[10] Bleichrodt, A Parameter-Free Elicitation of the Probability Weighting Function in Medical Decision Analysis, Manage. Sci. 46 (11) pp 1485– (2000) · Zbl 06007244 · doi:10.1287/mnsc.46.11.1485.12086
[11] Booij, A Parametric Analysis of Prospect Theory’s Functionals for the General Population, Theory Decision 68 pp 115– (2010) · Zbl 1197.91089 · doi:10.1007/s11238-009-9144-4
[12] Booij, A Parameter-Free Analysis of the Utility of Money for the General Population under Prospect Theory, J. Econ. Psychol. 30 (4) pp 651– (2009) · doi:10.1016/j.joep.2009.05.004
[13] Camerer, Violations of the Betweenness Axiom and Nonlinearity in Probability, J. Risk Uncertainty 8 (2) pp 167– (1994) · Zbl 0925.90038 · doi:10.1007/BF01065371
[14] Carlier, Core of Convex Distortions of a Probability, J. Econ. Theory 113 pp 199– (2003) · Zbl 1078.28003 · doi:10.1016/S0022-0531(03)00122-4
[15] Carlier, Existence and Monotonicity of Solutions to Moral Hazard Problems, J. Math. Econ. 41 pp 826– (2005a) · Zbl 1117.91378 · doi:10.1016/j.jmateco.2004.08.002
[16] Carlier, Rearrangement Inequalities in Non-Convex Insurance Models, J. Math. Econ. 41 pp 483– (2005b) · Zbl 1106.91038 · doi:10.1016/j.jmateco.2004.12.004
[17] Carlier, Two-Persons Efficient Risk-Sharing and Equilibria for Concave Law Invariant Utilities, Econ. Theory 36 (2) pp 189– (2008) · Zbl 1152.91035 · doi:10.1007/s00199-007-0266-z
[18] Carlier, Optimal Demand for Contingent Claims When Agents Have Law Invariant Utilities, Math. Fin. 21 (2) pp 169– (2011) · Zbl 1230.91173
[19] Chateauneuf, Optimal Risk-Sharing Rules and Equilibria with Choquet-Expected-Utility, J. Math. Econ. 34 (2) pp 191– (2000) · Zbl 1161.91434 · doi:10.1016/S0304-4068(00)00041-0
[20] Cummins, Consumer Attitudes toward Auto and Homeowners Insurance (1974)
[21] Dana, Optimal Risk Sharing with Background Risk, J. Econ. Theory 133 (1) pp 152– (2007) · Zbl 1280.91092 · doi:10.1016/j.jet.2005.10.002
[22] Dana , R.-A. N. Shahidi 2000 Optimal Insurance Contracts under Non Expected Utility
[23] Doherty, Optimal Insurance without Expected Utility: The Dual Theory and the Linearity of Insurance Contracts, J. Risk Uncertainty 10 pp 157– (1995) · Zbl 0847.90036 · doi:10.1007/BF01083558
[24] Gollier, Optimal Insurance of Approximate Losses, J. Risk Insurance 63 pp 369– (1996) · Zbl 0894.62111 · doi:10.2307/253617
[25] Gollier, Arrow’s Theorem on the Optimality of Deductibles: A Stochastic Dominance Approach, Econ. Theory 7 pp 359– (1996) · Zbl 0852.90047
[26] He, Portfolio Choice under Cumulative Prospect Theory: An Analytical Treatment, Manage. Sci. 57 (2) pp 315– (2011a) · Zbl 1214.91099 · doi:10.1287/mnsc.1100.1269
[27] He, Portfolio Choice via Quantiles, Math. Fin. 21 (2) pp 203– (2011b)
[28] He , X. D. X. Y. Zhou 2012 Hope, Fear and Aspirations · Zbl 1403.91313
[29] Huysentruyt, How Do People Value Extended Warranties? Evidence from Two Field Surveys, J. Risk Uncertainty 40 pp 197– (2010) · Zbl 05774228 · doi:10.1007/s11166-010-9094-9
[30] Ingersoll, Non-Monotonicity of the Tversky-Kahneman Probability-Weighting Function: A Cautionary Note, Europ. Fin. Manage. 14 (3) pp 385– (2008) · doi:10.1111/j.1468-036X.2007.00439.x
[31] Jin, Behavioral Portfolio Selection in Continous Time, Math. Fin. 18 (3) pp 385– (2008) · doi:10.1111/j.1467-9965.2008.00339.x
[32] Jin, Erratum to ”Behavioral Portfolio Selection in Continuous Time”, Math. Fin. 20 (3) pp 521– (2010) · Zbl 1192.91173 · doi:10.1111/j.1467-9965.2010.00409.x
[33] Johnson, Framing, Probability Distortions and Insurance Decisions, J. Risk Uncertainty 7 pp 35– (1993) · doi:10.1007/BF01065313
[34] Kliger, Theories of Choice under Risk: Insights from Financial Markets, J. Econ. Behav. Organization 71 (2) pp 330– (2009) · doi:10.1016/j.jebo.2009.01.012
[35] Laury, Insurance Decisions for Low-Probability Losses, J. Risk Uncertainty 39 pp 17– (2009) · Zbl 05633544 · doi:10.1007/s11166-009-9072-2
[36] Lopes, Between Hope and Fear: The Psychology of Risk, Adv. Exp. Social Psychol. 20 pp 255– (1987) · doi:10.1016/S0065-2601(08)60416-5
[37] Mossin, Aspects of Rational Insurance Pricing, J. Pol. Econ. 76 (4) pp 553– (1968) · doi:10.1086/259427
[38] Polkovnichenko, Household Portfolio Diversification: A Case for Rank-Dependent Preferences, Rev. Fin. Stud. 18 (4) pp 1467– (2005) · doi:10.1093/rfs/hhi033
[39] Polkovnichenko, Probability Weighting Functions Implied by Options Prices, J. Financial Econ 107 (3) pp 580– (2012) · doi:10.1016/j.jfineco.2012.09.008
[40] Prelec, The Probability Weighting Function, Econometrica 66 (3) pp 497– (1998) · Zbl 1009.91007 · doi:10.2307/2998573
[41] Quiggin, A Theory of Anticipated Utility, J. Econ. Behav. 3 (4) pp 323– (1982) · doi:10.1016/0167-2681(82)90008-7
[42] Quiggin, Comparative Statics for Rank-Dependent Expected Utility Theory, J. Risk Uncertainty 4 pp 339– (1991) · Zbl 0735.90005 · doi:10.1007/BF00056160
[43] Quiggin, Generalized Expected Utility Theory-The Rank-Dependent Model (1993) · Zbl 0915.90081 · doi:10.1007/978-94-011-2182-8
[44] Raviv, The Design of an Optimal Insurance Policy, Am. Econ. Rev. 69 (1) pp 84– (1979)
[45] Rieger, Cumulative Prospect Theory and the St. Petersburg Paradox, Econ. Theory 28 (3) pp 665– (2006) · Zbl 1145.91328 · doi:10.1007/s00199-005-0641-6
[46] Schlesinger, Insurance Demand without the Expected-Utility Paradigm, J. Risk Insurance 64 pp 19– (1997) · doi:10.2307/253910
[47] Schmeidler, Subjective Probability and Expected Utility without Additivity, Econometrica 57 (3) pp 571– (1989) · Zbl 0672.90011 · doi:10.2307/1911053
[48] Schmidt, Efficient Risk-Sharing and the Dual Theory of Choice under Risk, J. Risk Insurance 66 (4) pp 597– (1999) · doi:10.2307/253865
[49] Starmer, Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk, J. Econ. Literature 38 (2) pp 332– (2000) · doi:10.1257/jel.38.2.332
[50] Sung , K. S. Yam S. Yung J. Zhou 2011 Behavioral Optimal Insurance Insur.: Math. Econ.
[51] Tversky, Weighing Risk and Uncertainty, Psychol. Rev. 102 (2) pp 269– (1995) · doi:10.1037/0033-295X.102.2.269
[52] Tversky, Advances in Prospect Theory: Cumulative Representation of Uncertainty, J. Risk Uncertainty 5 (4) pp 297– (1992) · Zbl 0775.90106 · doi:10.1007/BF00122574
[53] Wu, Curvature of the Probability Weighting Function, Manage. Sci. 42 (12) pp 1676– (1996) · Zbl 0893.90003 · doi:10.1287/mnsc.42.12.1676
[54] Yaari, The Dual Theory of Choice under Risk, Econometrica 55 (1) pp 95– (1987) · Zbl 0616.90005 · doi:10.2307/1911158
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.