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Risk Aversion and Insurance Propensity

Fabio Maccheroni, Massimo Marinacci, Ruodu Wang, Qinyu Wu

TL;DR

The paper establishes a unified, instruction-friendly link between risk aversion and insurance behavior, reframing Arrow–Pratt and Rothschild–Stiglitz notions via propensity to insurance across probabilistically sophisticated preferences. It defines precise insurance contracts (full, proportional, deductible-limit) and broader hedging forms (indemnity- and contingency-schedule) and proves equivalences: weak risk aversion ⇔ propensity to full insurance, and strong risk aversion ⇔ propensity to partial insurance (and hedging). The comparative analysis maps Yaari’s and Ross’s notions to comparative propensities to full and partial insurance, respectively, yielding a coherent framework that extends beyond expected utility and accommodates diverse risk models. Practically, the results explain why common insurance policies (proportional and deductible-limit) are prevalent and why insurers' portfolios typically mix contract types, providing normative and empirical insights applicable to financial risk management and policy design, all without relying on EV representations.

Abstract

We provide a new foundation of risk aversion by showing that this attitude is fully captured by the propensity to seize insurance opportunities. Our foundation, which applies to all probabilistically sophisticated preferences, well accords with the commonly held prudential interpretation of risk aversion that dates back to the seminal works of Arrow (1963) and Pratt (1964). In our main results, we first characterize the Arrow-Pratt risk aversion in terms of propensity to full insurance and the stronger notion of risk aversion of Rothschild and Stiglitz (1970) in terms of propensity to partial insurance. We then extend the analysis to comparative risk aversion by showing that the notion of Yaari (1969) corresponds to comparative propensity to full insurance, while the stronger notion of Ross (1981) corresponds to comparative propensity to partial insurance.

Risk Aversion and Insurance Propensity

TL;DR

The paper establishes a unified, instruction-friendly link between risk aversion and insurance behavior, reframing Arrow–Pratt and Rothschild–Stiglitz notions via propensity to insurance across probabilistically sophisticated preferences. It defines precise insurance contracts (full, proportional, deductible-limit) and broader hedging forms (indemnity- and contingency-schedule) and proves equivalences: weak risk aversion ⇔ propensity to full insurance, and strong risk aversion ⇔ propensity to partial insurance (and hedging). The comparative analysis maps Yaari’s and Ross’s notions to comparative propensities to full and partial insurance, respectively, yielding a coherent framework that extends beyond expected utility and accommodates diverse risk models. Practically, the results explain why common insurance policies (proportional and deductible-limit) are prevalent and why insurers' portfolios typically mix contract types, providing normative and empirical insights applicable to financial risk management and policy design, all without relying on EV representations.

Abstract

We provide a new foundation of risk aversion by showing that this attitude is fully captured by the propensity to seize insurance opportunities. Our foundation, which applies to all probabilistically sophisticated preferences, well accords with the commonly held prudential interpretation of risk aversion that dates back to the seminal works of Arrow (1963) and Pratt (1964). In our main results, we first characterize the Arrow-Pratt risk aversion in terms of propensity to full insurance and the stronger notion of risk aversion of Rothschild and Stiglitz (1970) in terms of propensity to partial insurance. We then extend the analysis to comparative risk aversion by showing that the notion of Yaari (1969) corresponds to comparative propensity to full insurance, while the stronger notion of Ross (1981) corresponds to comparative propensity to partial insurance.
Paper Structure (34 sections, 26 theorems, 142 equations, 3 figures)

This paper contains 34 sections, 26 theorems, 142 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. Risk setting
  3. Risk preferences
  4. Classical risk attitudes
  5. Absolute attitudes
  6. Insurance contracts and attitudes
  7. Weak risk aversion
  8. Strong risk aversion
  9. More insurances
  10. Neutrality
  11. Comparative attitudes
  12. Classical comparative risk attitudes
  13. Comparative insurance propensity
  14. Conclusion
  15. Outlines of the proofs of Theorems \ref{['th:omni']}, \ref{['th:comp-wra']}, and \ref{['th:omni-comp']}
  16. ...and 19 more sections

Key Result

Theorem 1

The following properties are equivalent for a risk preference:

Figures (3)

  • Figure 1: Proportional insurance (in red) and deductible-limit insurance (in blue) for loss $-w$
  • Figure 2: Summary tables, where $\rho^w(g,f )$ stands for $\rho(w+g, w+f)$
  • Figure 3: The functions $H$ (left panel) and $K$ (right panel).

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • ...and 34 more