# Diversification preferences in the theory of choice

## Abstract

Diversification represents the idea of choosing variety over uniformity. Within the theory of choice, desirability of diversification is axiomatized as preference for a convex combination of choices that are equivalently ranked. This corresponds to the notion of risk aversion when one assumes the von Neumann–Morgenstern expected utility model, but the equivalence fails to hold in other models. This paper analyzes axiomatizations of the concept of diversification and their relationship to the related notions of risk aversion and convex preferences within different choice theoretic models. Implications of these notions on portfolio choice are discussed. We cover model-independent diversification preferences, preferences within models of choice under risk, including expected utility theory and the more general rank-dependent expected utility theory, as well as models of choice under uncertainty axiomatized via Choquet expected utility theory. Remarks on interpretations of diversification preferences within models of behavioral choice are given in the conclusion.

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1. 1.

To see this, suppose u is a concave utility function representing a convex preference relation $$\succsim$$. Then, if a function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ is strictly increasing, the composite function $$f\circ u$$ is another utility representation of $$\succsim$$. However, for a given concave utility function u, one can relatively easily construct a strictly increasing function f such that $$f\circ u$$ is not concave.

2. 2.

“Diversification is both observed and sensible; a rule of behavior which does not imply the superiority of diversification must be rejected.”

3. 3.

Drapeau and Kupper (2013) refer to the convexity property as quasiconvexity, which we believe is a mathematically more appropriate nomenclature. However, we stick to the more widely used convexity terminology for consistency.

4. 4.

This definition of tail mean holds only under the assumption of continuous distributions, that is for integrable x.

5. 5.

A risk measure $$\rho :\mathcal {X}\rightarrow \mathbb {R}$$ is convex if for all $$x,y\in \mathcal {X}$$ and $$\lambda \in [0,1], \rho (\lambda x +(1-\lambda )y) \le \lambda \rho (x) +(1-\lambda )\rho (y)$$.

6. 6.

A risk measure $$\rho :\mathcal {X}\rightarrow \mathbb {R}$$ is subadditive if for all $$x,y\in \mathcal {X}, \rho (x+y) \le \rho (x)+\rho (y)$$.

7. 7.

A risk measure $$\rho :\mathcal {X}\rightarrow \mathbb {R}$$ is translation invariant (or cash-additive) if for all $$x\in \mathcal {X}$$ and $$m\in \mathbb {R}, \rho (x+m) = \rho (x)-m$$.

8. 8.

A preference relation $$\succsim$$ is compact continuous if $$x\succsim y$$ whenever a bounded sequence $$(x_n)_{n\in \mathbb {N}}$$ converges in distribution to x and $$x_n\succsim y$$ for each n. As noted by Chew and Mao (1995), many widely used examples of expected utility preferences are in fact compact continuous and not continuous when the corresponding utility function is discontinuous or unbounded.

9. 9.

The theoretical setup of Dekel (1989) used to derive the results reviewed in this section is a very particular one, and we encourage the reader to read his article for the details.

10. 10.

This is despite the evidence supporting alternative descriptive models, showing that people’s actual behavior deviates significantly from this normative model. See Stanovich (2009) and Hastie and Dawes (2009) for a discussion.

11. 11.

For a more complete review of the notions of risk aversion within the theory of choice under risk, we refer the reader to Cohen (1995).

12. 12.

A number of papers have studied portfolio theory, risk sharing, and insurance contracting in the RDEU framework; see Bernard et al. (2013) for a detailed review.

13. 13.

Objective risk is typically available in games of chance, such as a series of coin flips where the probabilities are objectively known. In practice, the notion of risk also encompasses situations, in which reliable statistical information is available, and from which objective probabilities are inferred. Uncertainty, on the other hand, can arise in practice from situations of complete ignorance or when insufficient statistical data are available, for example.

14. 14.

Ellsberg (1961) proposed experiments where choices violate the postulates of subjective expected utility, more specifically the Sure–Thing Principle. The basic idea is that a decision maker will always choose a known probability of winning over an unknown probability of winning even if the known probability is low and the unknown probability could be a guarantee of winning. His paradox holds independent of the utility function and risk aversion characteristics of the decision maker and implies a notion of uncertainty aversion, which is an attitude of preference for known risks over unknown risks.

15. 15.

Recall that two acts $$f,g\in \mathbb {L}$$ are said to be comonotonic if for no $$s,t\in S, f(s) > f(t)$$ and $$g(s) < g(t)$$.

16. 16.

See Example 1 in Chateauneuf and Tallon (2002).

17. 17.

Uncertainty averse preferences are a general class of preferences. Special cases that can be obtained by suitably specifying the uncertainty aversion index G defined below include, among others, variational preferences and smooth ambiguity preferences. See Cerreia-Vioglio et al. (2011b) for more details.

18. 18.

“The nature of one’s information concerning the relative likelihood of events...a quality depending on the amount, type, reliability and ‘unanimity’ of information, and giving rise to one’s degree of ‘confidence’ in an estimation of relative likelihoods.” Ellsberg (1961).

19. 19.

They are referred to as the Two-Urn Paradox and the Three-Color Paradox—see Ellsberg (1961).

20. 20.

“As time goes on I get more and more convinced that the right method in investment is to put fairly large sums into enterprises which one thinks one knows something about and in the management of which one thoroughly believes. It is a mistake to think that one limits one’s risk by spreading too much between enterprises about which one knows little and has no reason for special confidence. [...] One’s knowledge and experience are definitely limited and there are seldom more than two or three enterprises at any given time in which I personally feel myself entitled to put full confidence.” See Keynes (1983).

21. 21.

See De Giorgi et al. (2016) for a review of empirical evidence suggesting underdiversification.

22. 22.

A number of other research efforts empirically studying the effect of ambiguity aversion on portfolio choice reach the conclusion of under-diversification in some form, including the works of Uppal and Wang (2003), Maenhout (2004), Maenhout (2006), Garlappi et al. (2007), Liu (2010), Campanale (2011), and Chen et al. (2014).

## References

1. Acerbi, C., Tasche, D.: Expected shortfall: a natural coherent alternative to value at risk. Econ. Notes 31(2), 379–388 (2002a)

2. Acerbi, C., Tasche, D.: On the coherence of expected shortfall. J. Bank. Finance 26(7), 1487–1503 (2002b)

3. Allais, M.: The General Theory of Random Choices in Relation to the Invariant Cardinal Utility Function and the Specific Probability Function: The $$(U, q)$$ Model—A General Overview. CNRS, Paris (1987)

4. Anscombe, F., Aumann, R.: A definition of subjective probability. Ann. Math. Stat. 34, 199–205 (1963)

5. Arrow, K.J.: The theory of risk aversion. In: Saatio, Y.J. (ed.) Aspects of the Theory of Risk Bearing. Markham Publ. Co., Chicago (1965)

6. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)

7. Bernard, C., He, X., Yan, J.-A., Zhou, X.Y.: Optimal insurance design under rank-dependent expected utility. Math. Finance 25(1), 154–186 (2013)

8. Bernoulli, D.: Exposition of a new theory on the measurement of risk. Econometrica 22, 23–26 (1738)

9. Boyle, P., Garlappi, L., Uppal, R., Wang, T.: Keynes meets Markowitz: the trade-off between familiarity and diversification. Manag. Sci. 58(2), 253–272 (2012)

10. Campanale, C.: Learning, ambiguity and life-cycle portfolio allocation. Rev. Econ. Dyn. 14, 339–367 (2011)

11. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Risk measures: rationality and diversification. Math. Finance 21(4), 743–774 (2011a)

12. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Uncertainty averse preferences. J. Econ. Theory 146, 1275–1330 (2011b)

13. Chateauneuf, A., Cohen, M.: Risk seeking with diminishing marginal utility in a non-expected utility model. J. Risk Uncertain. 9, 77–91 (1994)

14. Chateauneuf, A., Lakhnati, G.: From sure to strong diversification. Econ. Theory 32, 511–522 (2007)

15. Chateauneuf, A., Tallon, J.-M.: Diversification, convex preferences and non-empty core in the choquet expected utility model. Econ. Theory 19, 509–523 (2002)

16. Chen, H., Ju, N., Miao, J.: Dynamic asset allocation with ambiguous return predictability. Rev. Econ. Stud. 17(4), 799–823 (2014)

17. Chew, S.H., Karni, E., Safra, Z.: Risk aversion in the theory of expected utility with rank dependent probabilities. J. Econ. Theory 42, 370–381 (1987)

18. Chew, S.H., Mao, M.H.: A Schur concave characterization of risk aversion for non-expected utility preferences. J. Econ. Theory 67, 402–435 (1995)

19. Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)

20. Cohen, M.D.: Risk-aversion concepts in expected- and non-expected-utility models. Geneva Pap. Risk Insur. Theory 20, 73–91 (1995)

21. De Giorgi, E., Epper, T., Mahmoud, O.: A behavioral analysis of diversification. Working Paper, University of St. Gallen (2016)

22. Debreu, G.: Continuity properties of Paretian utility. Int. Econ. Rev. 5, 285–293 (1964)

23. Dekel, E.: Asset demands without the independence axiom. Econometrica 57, 163–169 (1989)

24. Dimmock, S. G., Kouwenberg, R., Mitchell, O.S., Peijnenburg, K.: Ambiguity aversion and household portfolio choice. NBER Working Paper No. 18743 (2014)

25. Dow, J., da Costa Werlang, S.R.: Uncertainty aversion, risk aversion, and the optimal choice of portfolio. Econometrica 60(1), 197–204 (1992)

26. Drapeau, S., Kupper, M.: Risk preferences and their robust representation. Math. Oper. Res. 38(1), 28–62 (2013)

27. Ellsberg, D.: Risk, ambiguity and savage axioms. Q. J. Econ. 75, 643–679 (1961)

28. Epstein, L.G.: A definition of uncertainty aversion. Rev. Econ. Stud. 66(3), 579–608 (1999)

29. Föllmer, H., Schied, A.: Coherent and convex risk measures. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance, pp. 355–363. Wiley, New York (2010)

30. Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter, Berlin (2011)

31. Garlappi, L., Uppal, R., Wang, T.: Portfolio selection with parameter and model uncertainty: a multi-prior approach. Rev. Financ. Stud. 20, 41–81 (2007)

32. Ghirardato, P., Marinacci, M.: Ambiguity made precise: a comparative foundation. J. Econ. Theory 102, 251–289 (2002)

33. Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econ. 16, 65–88 (1987)

34. Hadar, J., Russell, W.R.: Rules for ordering uncertain prospects. Am. Econ. Rev. 59, 25–34 (1969)

35. Hastie, R., Dawes, R.M.: Rational Choice in an Uncertain World: The Psychology of Judgement and Decision Making. SAGE Publishing Inc., Beverly Hills (2009)

36. Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291 (1979)

37. Kahneman, D., Tversky, A.: Advances in prospect theory: cumulative representation of uncertainty. J. Risk Uncertain. 5, 297–323 (1992)

38. Keynes, J.M.: A Treatise on Probability. Macmillan & Co, London (1921)

39. Keynes, J.M.: The Collected Writings of John Maynard Keynes. Cambridge University Press, Cambridge (1983)

40. Knight, F.: Risk, Uncertainty and Profit. Houghton Miffin, Boston (1921)

41. Landsberger, M., Meilijson, I.: Mean-preserving portfolio dominance. Rev. Econ. Stud. 60, 479–485 (1993)

42. Lintner, J.: The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econ. Stat. 47(1), 13–37 (1965)

43. Liu, H.: Robust consumption and portfolio choice for time varying investment opportunities. Ann. Finance 6, 435–454 (2010)

44. Machina, M.J.: A stronger characterization of declining risk aversion. Econometrica 50, 1069–1079 (1982)

45. Machina, M.J.: The Economic Theory of Individual Behavior Toward Risk. Cambridge University Press, First published in 1983 as Paper No. 433, Center for Research on Organizational Efficiency, Stanford University (2008)

46. Machina, M.J., Siniscalchi, M.: Ambiguity and ambiguity aversion. In: Machina, M.J., Viscusi, K. (eds.) Handbook of the Economics of Risk and Uncertainty, Chap. 13, vol. 1, pp. 729–807. Elsevier, Amsterdam (2014)

47. Maenhout, P.: Robust portfolio rules and asset pricing. Rev. Financ. Stud. 17, 951–983 (2004)

48. Machina, M.J., Siniscalchi, M.: Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium. J. Econ. Theory 128, 136–163 (2006)

49. Markowitz, H.M.: Portfolio selection. J. Finance 7, 77–91 (1952)

50. Mossin, J.: Equilibrium in a capital asset market. Econometrica 34(4), 768–783 (1966)

51. Pratt, J.W.: Risk aversion in the small and in the large. Econometrica 32, 122–136 (1964)

52. Quiggin, J.: A theory of anticipated utility. J. Econ. Behav. Organ. 3, 323–343 (1982)

53. Quiggin, J.: Comparative statics for rank-dependent expected utility theory. J. Risk Uncertain. 4, 339–350 (1991)

54. Quiggin, J.: Increasing risk: another definition. In: Chikan, A. (ed.) Progress in Decision, Utility and Risk Theory. Kluwer, Dordrecht (1992)

55. Quiggin, J.: Generalized expected utility theory: the rank-dependent model. Kluwer Academic Publishers, Dordrecht (1993)

56. Ramsey, F.J.: Truth and probability. In: Braithwaite, R.B. (ed.) The Foundations of Mathematics and Other Logical Essays. Kegan Paul, London (1926)

57. Röell, A.: Risk aversion in Quiggin and Yaari’s rank-order model of choice under uncertainty. Econ. J. 97, 143–159 (1987)

58. Ross, S.A.: Some stronger measures of risk aversion in the small and in the large. Econometrica 49(3), 621–638 (1981)

59. Rothchild, M., Stiglitz, J.E.: Increasing risk: I. A definition. J. Econ. Theory 2, 225–243 (1970)

60. Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954)

61. Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 5, 571–587 (1989)

62. Segal, U.: Anticipated utility: a measure representation approach. Ann. Oper. Res. 19, 359–374 (1989)

63. Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Finance 19(3), 425–442 (1964)

64. Stanovich, K.E.: Decision Making and Rationality in the Modern World. Oxford University Press, Oxford (2009)

65. Uppal, R., Wang, T.: Model misspecification and underdiversification. J. Finance 58, 2465–2486 (2003)

66. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)

67. Wakker, P.: Characterizing optimism and pessimism directly through comonotonicity. J. Econ. Theory 52, 453–463 (1990)

68. Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)

69. Zhang, J.: Subjective ambiguity, expected utility and choquet expected utility. Econ. Theory 20(1), 159–181 (2002)

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Correspondence to Enrico G. De Giorgi.

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