# Surplus Sharing with Coherent Utility Functions

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## Abstract

**:**

## 1. Notation and Motivation

**Definition**

**1.**

- if $0\le \xi \in {L}^{\infty}$, then $u\left(\xi \right)\ge 0$;
- u is concave, i.e., for all $\xi ,\eta \in {L}^{\infty}$, $0\le \lambda \le 1$, we have $u(\lambda \xi +(1-\lambda \left)\eta \right)\ge \lambda u\left(\xi \right)+(1-\lambda )u\left(\eta \right)$;
- for $a\in \mathbb{R}$ and $\xi \in {L}^{\infty}$, $u(\xi +a)=u\left(\xi \right)+a$;
- if ${\xi}_{n}\downarrow \xi $ (with ${\xi}_{n}\in {L}^{\infty}$), then $u\left({\xi}_{n}\right)\to u\left(\xi \right)$.

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Definition**

**2.**

**Remark**

**4.**

**Definition**

**3.**

**Remark**

**5.**

## 2. Description of the Model

## 3. Model 1

- ${\sum}_{i}{X}_{i}>{\pi}_{0}+{k}_{0}$. In this case, the total claim size exceeds the available capital. The excess is supposed to be covered by, for instance, the government, and this at no cost. The initial capital should then be sufficiently high to make the deal acceptable for the government. The determination of this level is beyond the contents of this paper. We denote by A the set $A=\{{\sum}_{i}{X}_{i}>{\pi}_{0}+{k}_{0}\}$.
- ${\pi}_{0}\le {\sum}_{i}{X}_{i}\le {\pi}_{0}+{k}_{0}$. In this case, there is no surplus, and the insurer will lose part of his/her investment. We denote by B the set $B=\{{\pi}_{0}\le {\sum}_{i}{X}_{i}\le {\pi}_{0}+{k}_{0}\}$.
- ${\pi}_{0}>{\sum}_{i}{X}_{i}$. In this case, there is a surplus. The insurer will keep the surplus entirely. This can be defended since the agents already “gained” from the allocation principle, which is their share when entering the insurance. Furthermore, they do not take any risk. We denote by C the set $C=\{{\pi}_{0}>{\sum}_{i}{X}_{i}\}$.

**Theorem**

**2.**

**Proof.**

**Remark**

**6.**

## 4. Model 2

## 5. Model 3

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Remark**

**7.**

## 6. Model 4

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Remark**

**8.**

**Remark**

**9.**

## 7. Discussion of the Models

## Author Contributions

## Funding

## Conflicts of Interest

## References

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## Share and Cite

**MDPI and ACS Style**

Coculescu, D.; Delbaen, F. Surplus Sharing with Coherent Utility Functions. *Risks* **2019**, *7*, 7.
https://doi.org/10.3390/risks7010007

**AMA Style**

Coculescu D, Delbaen F. Surplus Sharing with Coherent Utility Functions. *Risks*. 2019; 7(1):7.
https://doi.org/10.3390/risks7010007

**Chicago/Turabian Style**

Coculescu, Delia, and Freddy Delbaen. 2019. "Surplus Sharing with Coherent Utility Functions" *Risks* 7, no. 1: 7.
https://doi.org/10.3390/risks7010007