Abstract
Finding the worst-case value of a preference over a set of plausible models is a well-established approach to address the issue of model uncertainty or ambiguity. In this paper, we study the worst-case evaluation of Yaari dual utility functionals of an aggregate risk under dependence uncertainty along with its decision-theoretic implications. To arrive at our main findings, we introduce a technical notion of conditional joint mixability. Lower and upper bounds on dual utilities with dependence uncertainty are established, and in the presence of conditional joint mixability, they are shown to be exact bounds. Moreover, conditional joint mixability is indeed necessary for attaining these exact bounds when the distortion functions are strictly inverse-S-shaped. A particular economic implication of our results is what we call the pessimism effect. We show that a (generally non-convex/non-concave) dual utility-based decision maker under dependence uncertainty behaves as if she had a risk-averse dual utility which is more pessimistic but free of dependence uncertainty.
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References
Acerbi, C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Finance 26, 1505–1518 (2002)
Bernard, C., Jiang, X., Wang, R.: Risk aggregation with dependence uncertainty. Insur. Math. Econ. 54, 93–108 (2014)
Brighi, B., Chipot, M.: Approximated convex envelope of a function. SIAM J. Numer. Anal. 31, 128–148 (1994)
Cai, J., Liu, H., Wang, R.: Asymptotic equivalence of risk measures under dependence uncertainty. Math. Finance 28, 29–49 (2018)
Chateauneuf, A., Cohen, M., Meilijson, I.: More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model. Econ. Theory 25, 649–667 (2005)
Embrechts, P., Puccetti, G., Rüschendorf, L.: Model uncertainty and VaR aggregation. J. Bank. Finance 37, 2750–2764 (2013)
Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R., Beleraj, A.: An academic response to Basel 3.5. Risks 2, 25–48 (2014)
Embrechts, P., Wang, B., Wang, R.: Aggregation-robustness and model uncertainty of regulatory risk measures. Finance Stoch. 19, 763–790 (2015)
Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 (1989)
Hansen, L.P., Sargent, T.J.: Robust control and model uncertainty. Am. Econ. Rev. 91(2), 60–66 (2001)
He, X., Zhou, X.: Hope, fear, and aspirations. Math. Finance 26, 3–50 (2016)
Hou, D., Xu, Z.Q.: A robust Markowitz mean–variance portfolio selection model with an intractable claim. SIAM J. Financ. Math. 7, 124–151 (2016)
Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 46, 171–185 (1979)
Lopes, L.L.: Between hope and fear: the psychology of risk. Adv. Exp. Soc. Psychol. 20, 255–295 (1987)
McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques and Tools. Revised edn., Princeton University Press, Princeton (2015)
Müller, A., Stoyan, D.: Comparison Methods for Statistical Models and Risks. Wiley, England (2002)
Prelec, D.: The probability weighting function. Econometrica 66, 497–527 (1998)
Puccetti, G., Wang, R.: Extremal dependence concepts. Stat. Sci. 30, 485–517 (2015)
Puccetti, G., Wang, R.: Detecting complete and joint mixability. J. Comput. Appl. Math. 280, 174–187 (2015)
Quiggin, J.: A theory of anticipated utility. J. Econ. Behav. Organ. 3, 323–343 (1982)
Rüschendorf, L.: Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer, Heidelberg (2013)
Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)
Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain. 5, 297–323 (1992)
Wang, B., Wang, R.: The complete mixability and convex minimization problems with monotone marginal densities. J. Multivar. Anal. 102, 1344–1360 (2011)
Wang, B., Wang, R.: Joint mixability. Math. Oper. Res. 41, 808–826 (2016)
Wang, R., Bignozzi, V., Tsanakas, A.: How superadditive can a risk measure be? SIAM J. Financ. Math. 6, 776–803 (2015)
Wang, R., Peng, L., Yang, J.: Bounds for the sum of dependent risks and worst value-at-risk with monotone marginal densities. Finance Stoch. 17, 395–417 (2013)
Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)
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Wang acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823, RGPAS-2018-522590). Xu acknowledges financial supports from NSFC (No. 11471276), Hong Kong GRF (No. 15204216 and No. 15202817) and the Hong Kong Polytechnic University. Zhou acknowledges financial supports through a start-up grant at Columbia University and from Columbia’s Nie Center for Intelligent Asset Management.
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Wang, R., Xu, Z.Q. & Zhou, X.Y. Dual utilities on risk aggregation under dependence uncertainty. Finance Stoch 23, 1025–1048 (2019). https://doi.org/10.1007/s00780-019-00399-y
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DOI: https://doi.org/10.1007/s00780-019-00399-y