Comparative risk aversion with two risks

Abstract This paper characterizes aversion to one risk in the presence of another, which is invulnerable to the size of exposure to the former risk and consistent with the common bivariate risk preferences for combining good with bad. We show that all bivariate utility functions that satisfy bivariate risk apportionment exhibit risk aversion with two risks if, and only if, the dependence structure of the two risks is characterized by the notion of expectation dependence. We then propose an intensity measure of risk aversion with two risks that is based on the utility premium normalized by the marginal utility evaluated at an arbitrarily chosen pair. We show that the intensity measure being uniformly larger is equivalent to the concept of greater generalized Ross risk aversion. An application for optimal prevention in a two-period model is presented when the dependence structure of the underlying random variables is governed by the notion of expectation dependence.

[1]  Jingyuan Li,et al.  Risk aversion with two risks: A theoretical extension , 2016 .

[2]  Rachel J. Huang,et al.  Almost expectation and excess dependence notions , 2015 .

[3]  Wen Cheng,et al.  Background risk, bivariate risk attitudes, and optimal prevention , 2013, Math. Soc. Sci..

[4]  W. Henry Chiu.  Financial risk taking in the presence of correlated non-financial background risk , 2020, Journal of Mathematical Economics.

[5]  C. Cuadras.  On the Covariance between Functions , 2002 .

[6]  Jingyuan Li,et al.  The monetary utility premium and interpersonal comparisons , 2014 .

[7]  Louis Eeckhoudt,et al.  Putting risk in its proper place , 2006 .

[8]  L. Eeckhoudt,et al.  Intensity of Preferences for Bivariate Risk Apportionment , 2020 .

[9]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[10]  K. Wong.  Precautionary self-insurance-cum-protection , 2016 .

[11]  Béatrice Rey,et al.  Optimal prevention and other risks in a two-period model , 2012, Math. Soc. Sci..

[12]  Christian Gollier,et al.  A general theory of risk apportionment , 2021, J. Econ. Theory.

[13]  Risk Apportionment Via Bivariate Stochastic Dominance , 2009 .

[14]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

[15]  S. Ross.  SOME STRONGER MEASURES OF RISK AVERSION IN THE SMALL AND THE LARGE WITH APPLICATIONS , 1981 .

[16]  K. Wong.  A note on risky targets and effort , 2017 .

[17]  Liqun Liu,et al.  Greater Arrow–Pratt (Absolute) risk aversion of higher orders , 2019, Journal of Mathematical Economics.

[18]  S. Richard.  Multivariate Risk Aversion, Utility Independence and Separable Utility Functions , 1975 .

[19]  R. Wright.  Expectation dependence of random variables, with an application in portfolio theory , 1987 .

[20]  John W. Pratt,et al.  Aversion to one risk in the presence of others , 1988 .

[21]  Liqun Liu,et al.  Substituting one risk increase for another: A method for measuring risk aversion , 2013, J. Econ. Theory.

[22]  L. Eeckhoudt,et al.  On the intensity of downside risk aversion , 2008 .

[24]  Louis Eeckhoudt,et al.  A Good Sign for Multivariate Risk Taking , 2006, Manag. Sci..

[25]  Jingyuan Li,et al.  Comparative Ross Risk Aversion in the Presence of Mean Dependent Risks , 2013 .

[26]  Jingyuan Li,et al.  The demand for a risky asset in the presence of a background risk , 2011, J. Econ. Theory.

[27]  M. Scarsini,et al.  On risk aversion with two risks , 1999 .

[28]  Mario Menegatti,et al.  Optimal prevention and prudence in a two-period model , 2009, Math. Soc. Sci..