# Risk Aversion, Loss Aversion, and the Demand for Insurance

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## Abstract

**:**

## 1. Introduction

## 2. Model and Notation

#### First-Order Risk Aversion, Loss Aversion, and Local Risk Attitude

## 3. Proportional Insurance

**Proposition**

**1.**

**Proof.**

#### Comparative Statics Results

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

## 4. Deductible Insurance

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

## 5. Deductible or Proportional?

**Definition**

**1.**

**(Guo et al. 2016)**Consider the reference-dependent utility specified in Equation (1) and assume $u\in {\mathcal{U}}^{2}$, where

**Proposition**

**7.**

**(Guo et al. 2016)**Let the reference point r be fixed. For any two random variables ${W}_{1}$ and ${W}_{2}$, ${W}_{1}$ $SSD$ ${W}_{2}$ implies ${W}_{1}$ $SS{D}^{r}$ ${W}_{2}$, but the converse is not true.

**Proposition**

**8.**

**Proof.**

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | As pointed out by a referee, Guo’s model contains indeed two deviations from Borch’s assumptions: one is the presence of gain/loss utility, the other one is non-differentiability at the reference point. In an earlier study, Eeckhoudt et al. (2016) have considered a gain/loss argument of utility where differentiability holds everywhere. Other authors, contrarily to Guo et al. (2016), have assumed that at least one component of the utility function exhibits risk loving (see, e.g., Bernard and Ghossoub 2010; Köbberling and Wakker 2005, and the literature therein). |

2 | This assumption is further discussed in the last section, in which we also outline some papers proposing possible generalizations. |

3 | As pointed out by a referee, the concept of utility loss aversion is different from the probabilistic loss aversion that comes in cumulative prospect theory from any differences in decision weights between gains and losses (see Schmidt and Zank 2008). In the present work, the term loss aversion is used in the sense of utility loss aversion. |

4 | Notice the difference with non-EU models of loss aversion where the utility function has convex portions. |

5 | see (Segal and Spivak (1990), p. 118). |

6 | Since it is often the case that over insurance is not allowed, we maintain throughout the paper the assumption that the coinsurance rate $\alpha $ cannot exceed 1. As pointed out by a referee, this is also known as the principle of indemnity in the insurance economics literature (see Peter et al. 2017). |

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**MDPI and ACS Style**

Eeckhoudt, L.; Fiori, A.M.; Rosazza Gianin, E.
Risk Aversion, Loss Aversion, and the Demand for Insurance. *Risks* **2018**, *6*, 60.
https://doi.org/10.3390/risks6020060

**AMA Style**

Eeckhoudt L, Fiori AM, Rosazza Gianin E.
Risk Aversion, Loss Aversion, and the Demand for Insurance. *Risks*. 2018; 6(2):60.
https://doi.org/10.3390/risks6020060

**Chicago/Turabian Style**

Eeckhoudt, Louis, Anna Maria Fiori, and Emanuela Rosazza Gianin.
2018. "Risk Aversion, Loss Aversion, and the Demand for Insurance" *Risks* 6, no. 2: 60.
https://doi.org/10.3390/risks6020060