# How Does Reinsurance Create Value to an Insurer? A Cost-Benefit Analysis Incorporating Default Risk

## Abstract

**:**

## 1. Introduction

## 2. Three-Party Optimal Insurance–Reinsurance Model with Default Risk

- Translation invariance: If Y is a random variable and c is a real constant, then$${\rho}_{g}[Y+c]={\rho}_{g}[Y]+c.$$
- Positive homogeneity: If Y is a random variable and a is a non-negative constant, then$${\rho}_{g}[aY]=a{\rho}_{g}[Y].$$
- Comonotonic additivity: If ${Y}_{1},{Y}_{2},\dots ,{Y}_{n}$ are comonotonic random variables in the sense that they are all non-decreasing functions of a common random variable, then$${\rho}_{g}[{Y}_{1}+{Y}_{2}+\cdots +{Y}_{n}]={\rho}_{g}[{Y}_{1}]+{\rho}_{g}[{Y}_{2}]+\cdots +{\rho}_{g}[{Y}_{n}].$$

**Definition**

**1.**

## 3. Optimal Insurance–Reinsurance Strategies in the Presence of Default Risk

#### 3.1. Analytic Solutions

**Lemma**

**1.**

**Theorem**

**1.**

**Proof.**

#### 3.2. Cost-Benefit Interpretation of Optimal Insurance–Reinsurance Strategies

- The losses that the insurer has decided to bear from the policyholder in the absence of reinsurance (which means ${\psi}_{I}\le 0$, or ${G}_{I}\le {H}_{I}$) may now be ceded to the reinsurer to contribute to further reduction in the insurer’s risk exposure, provided that the marginal benefit of reinsurance captured by the marginal increase in risk exposure, which is ${G}_{I}$ on $[0,{\mathrm{VaR}}_{\beta}(X))$ and $\delta {G}_{I}$ on $[{\mathrm{VaR}}_{\beta}(X),\infty )$) outweighs the marginal cost of reinsurance represented by the marginal reinsurance premium ${H}_{R}$. Mathematically, reinsuring the originally underwritten losses is profitable to the insurer if the cost-benefit difference of reinsurance ${\psi}_{R}$ defined in Formula (5) is non-positive. This is embodied by Case 1 in Figure 2.
- The access to reinsurance opens up new business opportunities to the insurer in that losses that are previously injudicious to assume from the policyholder (because ${\psi}_{I}>0$) may now be borne (because ${\psi}_{R}\le 0$) to the insurer’s advantage. While underwriting these losses from the policyholder undesirably increases the insurer’s risk exposure, transferring these losses to the reinsurer will generate a decrease that outweighs the preceding increase, leading to an overall drop in the insurer’s risk exposure. This corresponds to Case 2 in Figure 2.

**Corollary**

**1.**

**Proof.**

#### 3.3. Numerical Illustrations

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DRM | Distortion risk measure |

VaR | Value-at-Risk |

TVaR | Tail Value-at-Risk |

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**Figure 1.**The marginal costs and benefits of insurance (

**top**) and reinsurance (

**middle**and

**bottom**) to the insurer.

**Figure 2.**Optimal insurance–reinsurance decisions corresponding to different orders of ${G}_{I}(x),{H}_{I}(x)$ and ${H}_{R}(x)$ when $x<{\mathrm{VaR}}_{\beta}(X)$. The decisions for $x\ge {\mathrm{VaR}}_{\beta}(X)$ are obtained by replacing ${H}_{R}(x)$ by $(1-\delta ){G}_{I}(x)+{H}_{R}(x)$ in each case.

**Figure 3.**The plots of ${g}_{I}(x),{h}_{I}(x),(1-\delta ){g}_{I}(x)+{h}_{R}(x)$ for $0\le x<0.05$ (

**left**, magnified) and ${g}_{I}(x),{h}_{I}(x),{h}_{R}(x)$ for $0.05\le x\le 1$ (

**right**). The circled numbers refer to the cases identified in Figure 2.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lo, A. How Does Reinsurance Create Value to an Insurer? A Cost-Benefit Analysis Incorporating Default Risk. *Risks* **2016**, *4*, 48.
https://doi.org/10.3390/risks4040048

**AMA Style**

Lo A. How Does Reinsurance Create Value to an Insurer? A Cost-Benefit Analysis Incorporating Default Risk. *Risks*. 2016; 4(4):48.
https://doi.org/10.3390/risks4040048

**Chicago/Turabian Style**

Lo, Ambrose. 2016. "How Does Reinsurance Create Value to an Insurer? A Cost-Benefit Analysis Incorporating Default Risk" *Risks* 4, no. 4: 48.
https://doi.org/10.3390/risks4040048