# Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits

^{1}

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## Abstract

**:**

## 1. Introduction

**Lemma**

**1.**

## 2. Preliminaries

**Remark**

**1.**

## 3. PH Fits of Human Mortality Data

## 4. Valuation of Benefits

## 5. The Factorization: Implementation for Jump Diffusions

**Corollary**

**1.**

**Proof.**

**Example**

**1.**

**Corollary**

**2.**

**Proof.**

**Remark**

**2.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 6. Numerical Examples

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Example**

**11.**

**Example**

**12.**

## 7. Erlangization and Extrapolation

**Example**

**13.**

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Fits Using Sum-of-Exponentials

## Appendix B. Algorithms for Computing U

**Proposition**

**A1.**

- $\mathit{U}=\mathit{H}{\Delta}_{-\mathit{s}}{\mathit{H}}^{-1}$;
- $\mathit{s}$ is the vector with elements ${s}_{j}=-\mu /{\sigma}^{2}+\sqrt{{\mu}^{2}/{\sigma}^{4}+2{\lambda}_{j}/{\sigma}^{2}}$;
- $\mathit{H}$ is the matrix with columns ${\mathit{h}}_{1},\dots ,{\mathit{h}}_{n}$;
- ${\mathit{h}}_{j}$ is the eigenvector of $\mathit{T}+{\lambda}_{j}\mathit{I}$ corresponding to eigenvalue zero, with elements ${h}_{ij}$ given by ${h}_{ij}=0$for $i>j$, ${h}_{1j}=1$ and:$${h}_{ij}=\prod _{k=1}^{i-1}\frac{{\lambda}_{k}-{\lambda}_{j}}{{\lambda}_{k}{q}_{k}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}1<i\le j.$$In particular, $\mathbb{P}({J}_{\overline{\sigma}}=k)=-\mathit{\alpha}{\mathit{U}}^{-1}{\mathit{e}}_{k}\xb7{u}_{k}$ where ${u}_{k}$ is the ${k}^{\mathrm{th}}$ element of $-\mathit{U}\mathit{e}$.

**Remark**

**A1.**

**Proposition**

**A2.**

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**Figure 3.**Probability density functions and hazard rate functions for the life table and phase-type (PH) fits using the general and the generalized Coxian (GC) structures with $p=20$ phases.

**Figure 4.**Probability density functions and hazard rate functions for the life table and phase-type fits using the generalized Coxian structures with $p=35,\text{}40$ phases.

**Figure 6.**Probability density functions and hazard rate functions for the life table and GC fits with $p=50,\text{}75,\text{}100$ phases.

**Figure 8.**Joint distribution of ${M}_{\tau}$ and ${D}_{\tau}$. The joint density is in the left panel and a contour plot in the right.

**Table 1.**Price estimates for $\delta =r$ and varying p. HWB, high-water benefit; GMDB, guaranteed minimum death benefit.

p | 20 | 35 | 40 | 50 | 75 | 100 | Monte Carlo |
---|---|---|---|---|---|---|---|

HWB, $\delta =r=0$ | 2.703 | 2.703 | 2.704 | 2.704 | 2.704 | 2.704 | 2.665 ± 0.008 |

HWB, $\delta =r=0.03$ | 1.698 | 1.699 | 1.699 | 1.699 | 1.699 | 1.699 | 1.680 ± 0.008 |

GMDB, $\delta =r=0$ | 1.468 | 1.468 | 1.468 | 1.468 | 1.468 | 1.468 | 1.467 ± 0.008 |

GMDB, $\delta =r=0.03$ | 1.080 | 1.080 | 1.079 | 1.079 | 1.079 | 1.079 | 1.079 ± 0.010 |

Seed | 1 | 2 | 3 | 4 | 5 | Monte Carlo |
---|---|---|---|---|---|---|

HWB, $\delta =r=0$ | 2.704 | 2.704 | 2.703 | 2.702 | 2.704 | 2.665 ± 0.008 |

HWB, $\delta =r=0.03$ | 1.699 | 1.699 | 1.699 | 1.698 | 1.699 | 1.680 ± 0.008 |

GMDB, $\delta =r=0$ | 1.468 | 1.468 | 1.468 | 1.467 | 1.468 | 1.467 ± 0.008 |

GMDB, $\delta =r=0.03$ | 1.079 | 1.079 | 1.079 | 1.078 | 1.080 | 1.079 ± 0.010 |

$\mathit{\delta}$ | 0.00 | 0.01 | 0.02 | 0.03 |
---|---|---|---|---|

Price | 6.24 | 3.99 | 2.58 | 1.70 |

$\mathit{r}\mathbf{=}\mathit{\delta}$ | 0.00 | 0.01 | 0.02 | 0.03 | 0.05 |
---|---|---|---|---|---|

Price | 2.70 | 2.23 | 1.92 | 1.70 | 1.44 |

$\mathit{\delta}$ | 0.00 | 0.01 | 0.02 | 0.03 | 0.05 |
---|---|---|---|---|---|

Ratio | 0.97 | 0.88 | 0.78 | 0.68 | 0.51 |

q | HWB Simple | HWB Extrapolated | GMDB Simple | GMDB Extrapolated |
---|---|---|---|---|

1 | 1.523 | 1.523 | 1.092 | 1.092 |

2 | 1.583 | 1.642 | 1.097 | 1.102 |

3 | 1.606 | 1.652 | 1.097 | 1.099 |

4 | 1.618 | 1.655 | 1.097 | 1.096 |

5 | 1.626 | 1.657 | 1.097 | 1.095 |

6 | 1.631 | 1.658 | 1.096 | 1.095 |

7 | 1.635 | 1.658 | 1.096 | 1.094 |

8 | 1.638 | 1.659 | 1.096 | 1.094 |

9 | 1.640 | 1.659 | 1.096 | 1.094 |

10 | 1.642 | 1.659 | 1.095 | 1.094 |

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**MDPI and ACS Style**

Asmussen, S.; Laub, P.J.; Yang, H.
Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits. *Risks* **2019**, *7*, 17.
https://doi.org/10.3390/risks7010017

**AMA Style**

Asmussen S, Laub PJ, Yang H.
Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits. *Risks*. 2019; 7(1):17.
https://doi.org/10.3390/risks7010017

**Chicago/Turabian Style**

Asmussen, Søren, Patrick J. Laub, and Hailiang Yang.
2019. "Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits" *Risks* 7, no. 1: 17.
https://doi.org/10.3390/risks7010017