# Optimal Investment and Liability Ratio Policies in a Multidimensional Regime Switching Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Economic Motivations

#### 1.2. Review of Consumption-Investment Models and Reinsurance Models

#### 1.3. Review of the AIG Case

#### 1.4. Contributions

## 2. The Model

**Remark**

**1.**

**Remark**

**2.**

**Problem**

**1.**

## 3. The Verification Theorem

**Theorem**

**1.**

**Proof.**

## 4. Construction of Explicit Solutions

- $U(x)=\mathrm{ln}(x),\phantom{\rule{0.166667em}{0ex}}x>0$;
- $U(x)=\frac{1}{\alpha}{x}^{\alpha},\phantom{\rule{0.166667em}{0ex}}x>0$, where $\alpha <1$ and $\alpha \ne 0$.

#### 4.1. $U(x)=\mathrm{ln}(x),\phantom{\rule{0.166667em}{0ex}}x>0$

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

#### 4.2. $U(x)=\frac{1}{\alpha}{x}^{\alpha},\phantom{\rule{0.166667em}{0ex}}x>0,\phantom{\rule{4.pt}{0ex}}where\phantom{\rule{4.pt}{0ex}}\alpha <1$ and $\alpha \ne 0$

**Lemma**

**2.**

**Proof.**

**Proof.**

**Theorem**

**3.**

## 5. Economic Analysis

**Remark**

**3.**

- Denote the loading factor of the insurer by χ. We apply the expected value principle to calculate the premium rate ${p}_{i}$ by:$${p}_{i}=(1+\chi )\xb7E[{R}_{i}]=(1+\chi )\xb7({a}_{i}+\lambda \phantom{\rule{0.166667em}{0ex}}{\gamma}_{i}),\phantom{\rule{1.em}{0ex}}where\phantom{\rule{4.pt}{0ex}}i=1,2.$$

#### 5.1. Analysis of the Impact of the Risk Aversion Parameter α on the Optimal Policy

#### 5.2. Analysis on the Impact of the Correlation Coefficient ρ on the Optimal Policy

#### 5.3. Analysis of the Impact of the Regimes on the Optimal Policy

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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^{1}We use subscript i to denote the dependence on the regime ${\u03f5}_{t}$ when ${\u03f5}_{t}=i$.

Regime | ${\mathit{\mu}}_{\mathit{i}}$ | ${\mathit{r}}_{\mathit{i}}$ | ${\mathit{\sigma}}_{\mathit{i}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{\gamma}}_{\mathit{i}}$ | ${\mathit{p}}_{\mathit{i}}$ | λ |
---|---|---|---|---|---|---|---|---|

1 (bull market) | 0.1 | 0.03 | 0.15 | 0.04 | 0.05 | 0.2 | 0.1 | 0.1 |

2 (bear market) | 0.05 | 0.01 | 0.25 | 0.08 | 0.1 | 0.4 | 0.2 |

ρ | Regime | ${\mathit{\pi}}^{*}$ (Investment) | ${\mathit{\kappa}}^{*}$ (Liability Ratio) |
---|---|---|---|

−0.9 | 1 | 2.4031 | 2.3600 |

2 | 0.0905 | 1.5263 | |

−0.5 | 1 | 2.6628 | 2.6900 |

2 | 0.3386 | 1.5069 | |

−0.1 | 1 | 3.0108 | 3.0088 |

2 | 0.5787 | 1.5314 |

α | Case of ${\mathit{\rho}}_{1}$ and ${\mathit{\rho}}_{2}$ | ${\mathit{\pi}}_{1}^{*}$ | ${\mathit{\kappa}}_{1}^{*}$ | ${\mathit{\pi}}_{2}^{*}$ | ${\mathit{\kappa}}_{2}^{*}$ |
---|---|---|---|---|---|

$\alpha =-5$ | Case 1 | 0.3932 | 0.5372 | −0.0227 | 0.3594 |

Case 2 | 0.4221 | 0.5785 | 0.0094 | 0.3474 | |

Case 3 | 0.4561 | 0.6237 | 0.0383 | 0.3417 | |

Case 4 | 0.4961 | 0.6737 | 0.0657 | 0.3415 | |

$\alpha =-2$ | Case 1 | 0.7983 | 1.0233 | −0.0274 | 0.6688 |

Case 2 | 0.8537 | 1.1000 | 0.0311 | 0.6508 | |

Case 3 | 0.9188 | 1.1823 | 0.0848 | 0.6426 | |

Case 4 | 0.9947 | 1.2714 | 0.1361 | 0.6433 | |

$\alpha =0$ | Case 1 | 2.5214 | 2.5275 | 0.0905 | 1.5263 |

Case 2 | 2.6628 | 2.6900 | 0.2170 | 1.5108 | |

Case 3 | 2.8261 | 2.8502 | 0.3386 | 1.5069 | |

Case 4 | 3.0108 | 3.0088 | 0.4583 | 1.5140 | |

$\alpha =0.2$ | Case 1 | 3.2027 | 2.9408 | 0.1760 | 1.7333 |

Case 2 | 3.3696 | 3.1157 | 0.3178 | 1.7220 | |

Case 3 | 3.5606 | 3.2827 | 0.4558 | 1.7211 | |

Case 4 | 3.7741 | 3.4426 | 0.5924 | 1.7299 | |

$\alpha =0.5$ | Case 1 | 5.3301 | 3.8234 | 0.5142 | 2.1272 |

Case 2 | 5.5565 | 3.9941 | 0.6848 | 2.1257 | |

Case 3 | 5.8080 | 4.1423 | 0.8539 | 2.1303 | |

Case 4 | 6.0799 | 4.2711 | 1.0232 | 2.1404 |

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Zou, B.; Cadenillas, A. Optimal Investment and Liability Ratio Policies in a Multidimensional Regime Switching Model. *Risks* **2017**, *5*, 6.
https://doi.org/10.3390/risks5010006

**AMA Style**

Zou B, Cadenillas A. Optimal Investment and Liability Ratio Policies in a Multidimensional Regime Switching Model. *Risks*. 2017; 5(1):6.
https://doi.org/10.3390/risks5010006

**Chicago/Turabian Style**

Zou, Bin, and Abel Cadenillas. 2017. "Optimal Investment and Liability Ratio Policies in a Multidimensional Regime Switching Model" *Risks* 5, no. 1: 6.
https://doi.org/10.3390/risks5010006