# Retrospective Reserves and Bonus with Policyholder Behavior

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Life Insurance Setup

**Definition**

**1.**

#### The Savings Account, the Surplus and the Dividends

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 3. State-Wise Projections without Policyholder Behavior

**Definition**

**2.**

**Lemma**

**1.**

**Theorem**

**1.**

**Proof.**

## 4. Life Insurance Setup Including Policyholder Behavior

**Proposition**

**3.**

**Proposition**

**4.**

**Remark**

**1.**

## 5. State-Wise Projections Including Policyholder Behavior

#### 5.1. Policyholder Behavior Including Bonus

#### 5.2. The Case with All Benefits Regulated by Bonus

**Remark**

**2.**

#### 5.3. Approximation of the Free-Policy Factor

**Corollary**

**1.**

**Proof.**

#### 5.4. Projections with the Approximated Free-Policy Factor

**Lemma**

**2.**

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

## 6. Numerical Simulation Example

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Additions to Lemma 1

## Appendix B. Addition to the Case ${B}_{1}^{+}=0$

## Appendix C. Additions to Lemma 2

## Appendix D. Proof of Theorem 2

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1 | The figure is elaborated by the authors. |

2 | The figure is elaborated by the authors. |

**Figure 4.**

**Left**: State-wise projections of the savings account in the three simulated scenarios of the interest rate.

**Right**: The mean and confidence intervals of the projected savings account.

**Figure 5.**

**Left**: State-wise projections of the surplus in the three simulated scenarios of the interest rate.

**Right**: The mean and confidence intervals of the projected surplus.

**Figure 6.**

**Left**: The expected life annuity in the three simulated scenarios of the interest conditional in the insured being alive and non-free-policy.

**Right**: The expected life annuity and confidence intervals.

Component | Value |
---|---|

Age of policyholder, ${a}_{0}$ | 30 |

Age of retirement, n | 65 |

Termination | 80 |

Premium, $\pi \left(t\right)$ | $0.3021694\xb7{\U0001d7d9}_{\{{a}_{0}+t<n\}}$ |

Annuity, ${b}_{2}^{0}\left(t\right)$ | $1\xb7{\U0001d7d9}_{\{{a}_{0}+t\ge n\}}$ |

Term insurance, ${b}_{1}^{01}\left(t\right)$ | $5\xb7{\U0001d7d9}_{\{{a}_{0}+t<n\}}$ |

$Z\left(0\right)$ | 0 |

${\mu}_{01}^{*}\left(t\right)$ | $0.0005+{10}^{5.88+0.038(t+{a}_{0})-10}$ |

${\mu}_{02}\left(t\right)$ | $0.015\xb7{\U0001d7d9}_{\{{a}_{0}+t<n\}}$ |

${r}^{*}\left(t\right)$ | 0.01 |

$r\left(0\right)$ | 0.05 |

$\varphi $ | 0.008127 |

$\psi $ | −0.162953 |

$\theta $ | 0.000237 |

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Falden, D.K.; Nyegaard, A.K.
Retrospective Reserves and Bonus with Policyholder Behavior. *Risks* **2021**, *9*, 15.
https://doi.org/10.3390/risks9010015

**AMA Style**

Falden DK, Nyegaard AK.
Retrospective Reserves and Bonus with Policyholder Behavior. *Risks*. 2021; 9(1):15.
https://doi.org/10.3390/risks9010015

**Chicago/Turabian Style**

Falden, Debbie Kusch, and Anna Kamille Nyegaard.
2021. "Retrospective Reserves and Bonus with Policyholder Behavior" *Risks* 9, no. 1: 15.
https://doi.org/10.3390/risks9010015