## 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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## Notes

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

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

*x*. - 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.
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.
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.
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.
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.
- 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.
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.
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.
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.
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.
See Example 1 in Chateauneuf and Tallon (2002).

- 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.
“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.
They are referred to as the

*Two-Urn Paradox*and the*Three-Color Paradox*—see Ellsberg (1961). - 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.
See De Giorgi et al. (2016) for a review of empirical evidence suggesting underdiversification.

- 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).

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De Giorgi, E.G., Mahmoud, O. Diversification preferences in the theory of choice.
*Decisions Econ Finan* **39, **143–174 (2016). https://doi.org/10.1007/s10203-016-0182-4

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### Keywords

- Diversification
- Risk aversion
- Convex preferences
- Portfolio choice

### JEL Classification

- D81
- G11