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Annals of Operations Research

, Volume 176, Issue 1, pp 259–269 | Cite as

Some consequences of correlation aversion in decision science

  • Michel DenuitEmail author
  • Louis Eeckhoudt
  • Béatrice Rey
Article

Abstract

Very often in decision problems with uni- or multivariate objective, many results depend upon the signs of successive direct or cross derivatives of the utility function at least up to the 4th order. The purpose of the present paper is to provide a new and unified interpretation of these signs. It is based on the observation that decision-makers like to combine assets the return of which are negatively correlated (i.e., they have a preference for hedging). More specifically, this attitude is modelled through the concept of an “elementary correlation increasing transformation” defined by Epstein and Tanny (Can. J. Econ. 13:16–34, 1980). Decision-makers are said to be correlation averse if they dislike such a transformation. It will be shown that correlation aversion underlies many aspects of a decision-maker’s behavior under risk, including risk aversion, prudence, and temperance. Hence, correlation aversion provides a unifying, elegant and powerful framework to analyze risky decisions in the bivariate case. In this framework, also the concave version of the bivariate stochastic orderings introduced in Denuit, Lefèvre and Mesfioui (Insur. Math. Econ. 24:31–50, 1999a) turns out to be appropriate for comparing correlated outcomes and for comparing bivariate distributions with ordered marginals. The main result of this paper states that a decision-maker who is averse to correlation would rank bivariate outcomes as if using such higher order concave stochastic orderings. In particular, some features of decision-making under bidimensional risk, such as cross-prudence and cross-temperance, can also be linked to correlation aversion.

Keywords

Risk aversion Prudence Temperance Bivariate (s1,s2)-increasing concave orders Stochastic dominance 

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References

  1. Atkinson, A. B., & Bourguignon, F. (1982). The comparison of multi-dimensional distributions of economic status. Review of Economic Studies, 49, 183–201. CrossRefGoogle Scholar
  2. Chiu, W. H. (2005). Skewness preference, risk aversion and the precedence relations on stochastic changes. Management Science, 51, 1816–1828. CrossRefGoogle Scholar
  3. Denuit, M., & Eeckhoudt, L. (2008). Bivariate stochastic dominance and common preferences of decision-makers with risk independent utilities (Working Paper 08-03). Institut des Sciences Actuarielles, Université Catholique de Louvain, Louvain-la-Neuve, Belgium. Google Scholar
  4. Denuit, M., Lefèvre, Cl., & Shaked, M. (1998). The s-convex orders among real random variables, with applications. Mathematical Inequalities and Their Applications, 1, 585–613. Google Scholar
  5. Denuit, M., Lefèvre, Cl., & Mesfioui, M. (1999a). A class of bivariate stochastic orderings with applications in actuarial sciences. Insurance: Mathematics and Economics, 24, 31–50. CrossRefGoogle Scholar
  6. Denuit, M., De Vijlder, F. E., & Lefèvre, Cl. (1999b). Extremal generators and extremal distributions for the continuous s-convex stochastic orderings. Insurance: Mathematics and Economics, 24, 201–217. CrossRefGoogle Scholar
  7. Doherty, N., & Schlesinger, H. (1983). Optimal insurance in incomplete markets. Journal of Political Economy, 91, 1045–1054. CrossRefGoogle Scholar
  8. Drèze, J., & Modigliani, F. (1972). Consumption decision under uncertainty. Journal of Economic Theory, 5, 308–335. CrossRefGoogle Scholar
  9. Eeckhoudt, L., & Schlesinger, H. (2006). Putting risk in proper place. American Economic Review, 96, 280–289. CrossRefGoogle Scholar
  10. Eeckhoudt, L., Rey, B., & Schlesinger, H. (2007). A good sign for multivariate risk taking. Management Science, 53, 117–124. CrossRefGoogle Scholar
  11. Ekern, S. (1980). Increasing nth degree risk. Economics Letters, 6, 329–333. CrossRefGoogle Scholar
  12. Engelbrecht, R. (1977). A note on multivariate risk and separable utility functions. Management Science, 23, 1143–1144. CrossRefGoogle Scholar
  13. Epstein, L. G., & Tanny, S. M. (1980). Increasing generalized correlation: A definition and some economic consequences. Canadian Journal of Economics, 13, 16–34. CrossRefGoogle Scholar
  14. Fagart, M. C., & Sinclair-Desgagné, B. (2007). Ranking contingent monitoring systems. Management Science, 53, 1501–1509. CrossRefGoogle Scholar
  15. Fishburn, P. C. (1976). Continua of stochastic dominance relations for bounded probability distributions. Journal of Mathematical Economics, 3, 295–311. CrossRefGoogle Scholar
  16. Fishburn, P. C. (1980). Stochastic dominance and moments of distributions. Mathematics of Operations Research, 5, 94–100. CrossRefGoogle Scholar
  17. Gollier, C., & Pratt, J. (1996). Risk vulnerability and the tempering effect of background risk. Econometrica, 64, 1109–1123. CrossRefGoogle Scholar
  18. Kimball, M. S. (1990). Precautionary savings in the small and in the large. Econometrica, 58, 53–73. CrossRefGoogle Scholar
  19. Kimball, M. S. (1992). Precautionary motives for holding assets. In P. Newman, M. Milgate, & J. Falwell (Eds.), New palgrave dictionary of money and finance (Vol. 3, pp. 158–161). London: MacMillan. Google Scholar
  20. Kimball, M. S. (1993). Standard risk aversion. Econometrica, 61, 589–611. CrossRefGoogle Scholar
  21. Menezes, C., & Wang, X. (2005). Increasing outer risk. Journal of Mathematical Economics, 41, 875–886. CrossRefGoogle Scholar
  22. Menezes, C., Geiss, C., & Tressler, J. (1980). Increasing downside risk. American Economic Review, 70, 921–932. Google Scholar
  23. O’Brien, G. (1984). Stochastic dominance and moment inequalities. Mathematics of Operations Research, 9, 475–477. CrossRefGoogle Scholar
  24. Popescu, I. (2007). Robust mean covariance solution for stochastic optimization and applications. Operations Research, 55, 98–112. CrossRefGoogle Scholar
  25. Pratt, J., & Zeckhauser, R. (1987). Proper risk aversion. Econometrica, 55, 143–154. CrossRefGoogle Scholar
  26. Richard, S. (1975). Multivariate risk aversion utility independence and separable utility functions. Management Science, 22, 12–21. CrossRefGoogle Scholar
  27. Rolski, T. (1976). Order relations in the set of probability distribution functions and their applications in queueing theory. Dissertationes Mathematicae, 132, 5–47. Google Scholar
  28. Tchen, A. H. (1980). Inequalities for distributions with given marginals. Annals of Probability, 8, 814–827. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Michel Denuit
    • 1
    Email author
  • Louis Eeckhoudt
    • 2
    • 3
  • Béatrice Rey
    • 4
  1. 1.Institut de Sciences Actuarielles & Institut de StatistiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.IESEG School of ManagementLEMLilleFrance
  3. 3.COREUniversité Catholique de LouvainLouvain-la-NeuveBelgium
  4. 4.Institut de Science Financière et d’AssurancesUniversité de LyonLyonFrance

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