# Generalized Multiplicative Risk Apportionment

## Abstract

**:**

## 1. Introduction

## 2. Model

**Definition**

**1.**

**Lemma**

**1.**

## 3. Main Result

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## 4. Applications

#### 4.1. Relate to Multiplicative Risk Apportionment

#### 4.2. Relate to Preferences over Bivariate Lottery Pairs

**Corollary**

**3.**

**Corollary**

**4.**

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Proposition 1

**Proof.**

#### Appendix A.2. Proof of Corollary 2

**Proof.**

## References

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1 | For more examples, we refer to, e.g., Franke et al. (2006). |

2 | |

3 | Here, ${u}^{\left(n\right)}\left(x\right)=\frac{{d}^{n}u\left(x\right)}{d{x}^{n}}$. |

4 | For more details, we refer to Eeckhoudt and Schlesinger (2008) and Chiu et al. (2012, p. 160), among others. |

5 | |

6 | |

7 | For the case of the second-order stochastic dominance, one can further explore by adopting similar approach and noticing the relationship between stochastic dominance and nth-degree risk increase. |

8 | Eeckhoudt and Schlesinger (2008, p. 1331), among others, argued there is little empirical literature considering the case of nth degree risk increase, $n\ge 4$. |

9 | $Int\left(y\right)$ denotes the greatest-integer function, i.e., the greatest integer not exceeding the real number y. |

10 | Denuit and Rey (2013, p. 341) also presented an application of bivariate cross risk apportionment by adopting this utility function. |

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Wang, H.
Generalized Multiplicative Risk Apportionment. *Risks* **2019**, *7*, 65.
https://doi.org/10.3390/risks7020065

**AMA Style**

Wang H.
Generalized Multiplicative Risk Apportionment. *Risks*. 2019; 7(2):65.
https://doi.org/10.3390/risks7020065

**Chicago/Turabian Style**

Wang, Hongxia.
2019. "Generalized Multiplicative Risk Apportionment" *Risks* 7, no. 2: 65.
https://doi.org/10.3390/risks7020065