# Hierarchical Markov Model in Life Insurance and Social Benefit Schemes

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## Abstract

**:**

## 1. Introduction

## 2. Joint Laplace Transform of a Multivariate Jump-Diffusion CIR Model

**Proposition**

**1.**

**Proposition**

**2.**

## 3. Multiple State Model for Life Insurance and Social Benefit Covers

## 4. Numerical Example

#### 4.1. Comparison: CIR vs. CIR with Jump Processes

**highest**unemployed/disabled probability in components 2–5. It can be also noted that ${\kappa}_{G}^{\left(u\right)}$ and ${\kappa}_{G}^{\left(r\right)}$, that is, the unemployment/disabled benefit ($\$9.6481$, component 2) and age pension benefit ($$86.3251$, component 4), were the highest paid by a government social welfare system. We recall that component 2 represents the continuous annuity payment paid to individuals who are alive but permanently unemployed/disabled between times ${t}_{1}$ and ${T}_{1}$, while component 4 is the continuous annuity payment to an individual who had become permanently unemployed/disabled between ${t}_{1}$ and ${T}_{1}$ and who survived from ${T}_{1}$ until the maximum age ${T}_{2}$. The analyses of other combinations of $\{{\rho}^{\left(\right)},{\rho}^{\left(\right)}$ are given by points 1–7:

- For the case of zero jumps (i.e., $\{{\rho}^{\left(\right)},{\rho}^{\left(\right)},{\rho}^{\left(\right)}$), the employment probability in component 1
**increased**(on the other hand, the unemployment/disabled probability for components 2–5**decreased**). Thus, the payments of components 2–4 were lower than for their counterpart (i.e., the combination $\{0,2,0,0,0,0\}$), and the payment of component 1 was also higher. Overall, the total reserves decreased to $\$15.8106$. - For the combination $\{1,0,0,0,0,0\}$, the non-zero average jump frequency in the transition intensity from Employed to Death resulted in
**higher**mortality in component 1 and a**lower**survival probability for components 2–4, resulting in a higher payment of immediate death benefits while still in the Employed state and lower payments for components 2–5 than in the zero-jumps scenario. Overall, the total reserve decreased to $\$13.6640$. - While the combination $\{0,0,0,3,0,0\}$ did not affect component 1, the numerical effect showed a
**lower**survival probability for components 2, 4, and 5, as well as**higher**mortality in component 3. These effects translated into lower payments of components 2, 4, and 5, as well as a higher payment for component 3 than for its counterpart $\{0,0,0,0,0,0\}$. Overall, the total reserves decreased, and we had $\$0.7017$. - The non-zero average jump frequency in the transition intensity from Retired to Death, as shown by the combination $\{0,0,0,0,0,5\}$, resulted in a
**lower**survival probability in component 4 as well as**higher**mortality in component 5. Although the combination did not affect components 1–3, these were then translated into a lower payment for component 4 and a higher payment for component 5, the $\{0,0,0,0,0,0\}$ combination. Overall, the total reserves decreased, and we had $\$3.1158$. - Under the combination $\{0,0,1,0,0,0\}$, the non-retired probability while in the state Employed became
**lower**in all components. Hence, the payments of all components were lower than for the all-zero jump-frequency combination $\{0,0,0,0,0,0\}$, resulting in a slightly lower total-reserves amount of $\$10.1170$. - The combination $\{0,0,0,0,1,0\}$ affected only components 2–5 and had no effect on component 1. The
**lower**non-retired probability of the Unemployed/Disabled state then translated into a lower payment than for the $\{0,0,0,0,0,0\}$ combination for components 2–5. Overall, the total reserves decreased to $\$1.7715$. - For a combination with non-zero jump frequency for all states, such as $\{1,2,1,3,1,5\}$, the payments for components 1 and 3 were
**higher**than for its counterpart (i.e., the all-zero average jump frequency $\{0,0,0,0,0,0\}$); this was attributable to**higher**mortality intensities. Simultaneously, the annuity payments for components 2 and 4 were lower than for their counterpart. Component 5 became lower as the survival probability and the non-retired probability from the Employed/Unemployed state became lower; thus the payment for component 5 was lower than for its counterpart. Overall, the total reserves decreased to $\$0.8842$.

#### 4.2. Sensitivity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Six decrement probabilities under the jump-diffusion process when ${\rho}^{\left(j\right)}=0$ for all states, as a function of time t.

**Figure 5.**Sensitivity analysis with respect to ${\alpha}^{\left(\right)}$ by components and total reserves.

**Figure 6.**Sensitivity analysis with respect to ${\rho}^{\left(\right)}$ by components and total reserves.

**Figure 7.**Sensitivity analysis with respect to ${\sigma}^{\left(\right)}$ by components and total reserves.

**Table 1.**Parameter values for the jump-diffusion process used in the Markov chain of Figure 2.

Parameter | State-Wise Transition | |||||
---|---|---|---|---|---|---|

ed | eu | er | ud | ur | rd | |

$c{a}^{(j)}$ (drift coefficient) | $-1$ | $-0.5$ | $-1$ | $-0.3$ | $-1$ | $-0.2$ |

$c{b}^{(j)}$ (mean reversion level) | 0 | 0 | 0 | 0 | 0 | 0 |

${\sigma}^{(j)}$ (diffusion coefficient) | $0.1$ | $0.3$ | $0.05$ | $0.4$ | $0.05$ | $0.5$ |

${\alpha}^{(j)}$ (average jump size) | 4 | 2 | 1 | 1 | 1 | 1 |

${\lambda}^{\left(j\right)}(0)$ (initial intensity) | $0.03$ | $0.1$ | $0.01$ | $0.2$ | $0.01$ | $0.3$ |

${\rho}^{\left(j\right)}$ (average jump frequency) | 1 | 2 | 1 | 3 | 1 | 5 |

**Table 2.**Values of total reserves with various combinations of average jump frequency (rounded to 4 decimal places). The term Comp. denotes Component.

Combination of Average Jump Frequency | Comp. 1 ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{65}\mathbf{;}$ ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{70}$ | Comp. 2 ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{65}\mathbf{;}$ ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{70}$ | Comp. 3 ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{65}\mathbf{;}$ ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{70}$ | Comp. 4 ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{65}\mathbf{;}$ ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{70}$ | Comp. 5 ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{65}\mathbf{;}$ ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{70}$ | Total Reserves ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{65}\mathbf{;}$ ${\mathit{T}}_{\mathbf{1}}\mathbf{=}\mathbf{70}$ |
---|---|---|---|---|---|---|

${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ | $0.0268;$ $0.0268$ | $1.5123;$ $1.7983$ | $0.0494;$ $0.0494$ | $13.5308;$ $13.0327$ | $0.6912;$ $0.8220$ | $15.8106;$ $15.7292$ |

${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ | $0.7658;$ $0.7713$ | $1.2359;$ $1.4643$ | $0.0398;$ $0.0398$ | $11.0577;$ $10.6122$ | $0.5649;$ $0.6693$ | $13.6640;$ $13.5570$ |

${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ | $0.0268;$ $0.0268$ | $0.0507;$ $0.0507$ | $0.1477;$ $0.1477$ | $0.4534;$ $0.3672$ | $0.0232;$ $0.0232$ | $0.7017;$ $0.6156$ |

${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ | $0.0268;$ $0.0268$ | $1.5123;$ $1.7983$ | $0.0494;$ $0.0494$ | $0.0154;$ $0.0146$ | $1.5119;$ $1.7980$ | $3.1158;$ $3.6871$ |

${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ | $0.0181;$ $0.0181$ | $9.6481;$ $11.4474$ | $0.3124;$ $0.3124$ | $86.3251;$ $82.9605$ | $4.4099;$ $5.2323$ | $100.7137;$ $99.9708$ |

${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ | $0.0200;$ $0.0200$ | $0.9675;$ $1.1430$ | $0.0308;$ $0.0308$ | $8.6565;$ $8.2834$ | $0.4422;$ $0.5224$ | $10.1170;$ $9.9996$ |

${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ | $0.0268;$ $0.0268$ | $0.1656;$ $0.1657$ | $0.0212;$ $0.0212$ | $1.4821;$ $1.2005$ | $0.0757;$ $0.0757$ | $1.7715;$ $1.4899$ |

${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ ${\rho}^{\left(\right)}$ | $0.1276;$ $0.1276$ | $0.1708;$ $0.1708$ | $0.4133;$ $0.4133$ | $0.0017;$ $0.0014$ | $0.1707;$ $0.1708$ | $0.8842;$ $0.8838$ |

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Jang, J.; Mohd Ramli, S.N.
Hierarchical Markov Model in Life Insurance and Social Benefit Schemes. *Risks* **2018**, *6*, 63.
https://doi.org/10.3390/risks6030063

**AMA Style**

Jang J, Mohd Ramli SN.
Hierarchical Markov Model in Life Insurance and Social Benefit Schemes. *Risks*. 2018; 6(3):63.
https://doi.org/10.3390/risks6030063

**Chicago/Turabian Style**

Jang, Jiwook, and Siti Norafidah Mohd Ramli.
2018. "Hierarchical Markov Model in Life Insurance and Social Benefit Schemes" *Risks* 6, no. 3: 63.
https://doi.org/10.3390/risks6030063