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Diversification Preferences and Risk Attitudes

Xiangxin He, Fangda Liu, Ruodu Wang

Abstract

Portfolio diversification is a cornerstone of modern finance, while risk aversion is central to decision theory; both concepts are long-standing and foundational. We investigate their connections by studying how different forms of diversification correspond to notions of risk aversion. We focus on the classical distinctions between weak and strong risk aversion, and consider diversification preferences for pairs of risks that are identically distributed, comonotonic, antimonotonic, independent, or exchangeable, as well as their intersections. Under a weak continuity condition and without assuming completeness of preferences, diversification for antimonotonic and identically distributed pairs implies weak risk aversion, and diversification for exchangeable pairs is equivalent to strong risk aversion. The implication from diversification for independent pairs to weak risk aversion requires a stronger continuity. We further provide results and examples that clarify the relationships between various diversification preferences and risk attitudes, in particular justifying the one-directional nature of many implications.

Diversification Preferences and Risk Attitudes

Abstract

Portfolio diversification is a cornerstone of modern finance, while risk aversion is central to decision theory; both concepts are long-standing and foundational. We investigate their connections by studying how different forms of diversification correspond to notions of risk aversion. We focus on the classical distinctions between weak and strong risk aversion, and consider diversification preferences for pairs of risks that are identically distributed, comonotonic, antimonotonic, independent, or exchangeable, as well as their intersections. Under a weak continuity condition and without assuming completeness of preferences, diversification for antimonotonic and identically distributed pairs implies weak risk aversion, and diversification for exchangeable pairs is equivalent to strong risk aversion. The implication from diversification for independent pairs to weak risk aversion requires a stronger continuity. We further provide results and examples that clarify the relationships between various diversification preferences and risk attitudes, in particular justifying the one-directional nature of many implications.
Paper Structure (12 sections, 10 theorems, 44 equations, 1 figure)

This paper contains 12 sections, 10 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preferences and risk aversion
  3. Diversification and dependence
  4. Relations between diversification and risk aversion
  5. Comonotonic pairs
  6. Antimonotonic pairs
  7. Exchangeable pairs
  8. Independent pairs
  9. Strict single-directional implications in Figure \ref{['fig:1']}
  10. Neutrality
  11. Extension to unbounded random variables
  12. Conclusion

Key Result

Proposition 1

For a risk preference, diversification on comonotonic pairs does not imply weak risk aversion. Indeed, the risk preference $\succsim$ represented by $U$ via eq:utility with exhibits diversification on comonotonic pairs and strict strong risk seeking in eq:ssrs.

Figures (1)

  • Figure 1: Summary of results for risk preferences, where "AM" stands for "antimonotonic" (we omit "pairs"), "EX" stands for "exchangeable", "IN" stands for "independent", $\centernot\iff$ means incomparable, and $\Downarrow^*$ requires compact upper semicontinuity. The converse statements of all single-direction implications do not hold for general risk preferences.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Example 1: Weak risk aversion $\centernot\implies$ diversification on AM and ID
  • Remark 1: Diversification on AM $\centernot\implies$ strong risk aversion
  • Example 2: Strong risk aversion $\centernot\implies$ diversification on AM
  • Remark 2
  • ...and 23 more