# Pricing Longevity Bonds under a Credibility Framework with Limited Available Data

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## Abstract

**:**

## 1. Introduction

## 2. Mortality Projection under the Credibility Regression Framework

#### 2.1. Notation and Assumptions

#### 2.2. Credibility Estimation

#### 2.3. Constructing Projected Life Tables

**Remark**

**1.**

## 3. Longevity Bond Pricing

#### 3.1. The Mathematical Structure of a Longevity Bond

#### 3.2. Pricing with the Wang Transform under the Credibility Framework

## 4. Numerical Illustration

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | The explanatory variable $t$ is used to introduce calendar year effects. Generally speaking, a design matrix can reflect mortality characteristics by incorporating more explanatory variables. For example, in medical studies, mortality may depend on the genetic background of an individual, the toxicity of the environment, a possible infectious cause, etc. |

2 | The same approximation was utilized by Lin and Cox (2005). |

3 | In Greece, there is no a secondary annuity market to directly obtain this value. |

4 | HAS2012 is developed according to the directive of the European Commission for the equal treatment of individual male and female customers in insurance premiums and benefits. |

5 | These tables are HAS1990, HAS2005, and HAS2012. |

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**Figure 2.**Observed $logm(t,x)$ values of males in Greece, for ages 60, 70, 80, over the period 1981–2019. The straight lines denote the trend in mortality decline.

**Figure 3.**Observed $logm(t,x)$ values of females in Greece, for ages 60, 70, 80, over the period 1981–2019. The straight lines denote the trend in mortality decline.

**Figure 4.**Observed $logm(t,x)$ values of the total population in Greece, for ages 60, 70, 80, over the period 1981–2019. The straight lines denote the trend in mortality decline.

**Figure 5.**Death probabilities $}_{t}{\widehat{q}}_{x}^{\phantom{\rule{4pt}{0ex}}cred$ and $}_{t}{\tilde{q}}_{x}^{\phantom{\rule{4pt}{0ex}}cred$ under the SPM method for males.

**Figure 6.**Death probabilities $}_{t}{\widehat{q}}_{x}^{\phantom{\rule{4pt}{0ex}}cred$ and $}_{t}{\tilde{q}}_{x}^{\phantom{\rule{4pt}{0ex}}cred$ under the SPM method for females.

**Figure 7.**Death probabilities $}_{t}{\widehat{q}}_{x}^{\phantom{\rule{4pt}{0ex}}cred$ and $}_{t}{\tilde{q}}_{x}^{\phantom{\rule{4pt}{0ex}}cred$ under the SPM method for the total population.

**Table 1.**Market price of risk values under each projection method with $i$ varying from 1 to 3 percent.

Market Price of Risk$(\mathit{\lambda})$ | |||
---|---|---|---|

Projection method: SPM | |||

Gender | Males | Females | Total |

i = 1% | 0.26172 | 0.36171 | 0.31084 |

i = 2% | 0.22323 | 0.30706 | 0.27653 |

i = 3% | 0.18933 | 0.25933 | 0.24518 |

Projection method: MPM | |||

i = 1% | 0.24500 | 0.33869 | 0.29373 |

i = 2% | 0.20723 | 0.28599 | 0.25964 |

i = 3% | 0.17404 | 0.23881 | 0.22922 |

Projection method: EPM | |||

i = 1% | 0.26160 | 0.36168 | 0.31075 |

i = 2% | 0.22310 | 0.30702 | 0.27644 |

i = 3% | 0.18919 | 0.25927 | 0.24508 |

**Table 2.**Longevity bond pricing results under each projection method based on males, females, and annuitants from the total population of Greece.

Pricing Results | |||

Initial cohort of annuitants ($l}_{65$) | 10,000 | ||

Annual payment per annuitant | 1000 | ||

Projection method: SPM | |||

Gender | Males | Females | Total |

Longevity bond price (b) | 9,374,112 | 9,655,962 | 9,196,947 |

Premium value (p) | 625,888 | 344,038 | 803,053 |

Projection method: MPM | |||

Longevity bond price (b) | 9,360,957 | 9,667,377 | 9,356,844 |

Premium value (p) | 639,043 | 332,623 | 643,156 |

Projection method: EPM | |||

Longevity bond price (b) | 9,374,314 | 9,656,575 | 9,197,894 |

Premium value (p) | 625,686 | 343,425 | 802,106 |

**Table 3.**The market price of risk and the longevity bond pricing results under the Lee–Carter method, based on males, females, and annuitants from the total population of Greece.

Pricing Results under the Lee–Carter Method | |||
---|---|---|---|

Gender | Males | Females | Total |

Market price of risk (i = 2%) | 0.2507 | 0.3493 | 0.3056 |

Longevity bond price (b) | 9,008,307 | 9,305,596 | 8,942,833 |

Premium value (p) | 991,693 | 694,404 | 1,057,167 |

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**MDPI and ACS Style**

Bozikas, A.; Badounas, I.; Pitselis, G.
Pricing Longevity Bonds under a Credibility Framework with Limited Available Data. *Risks* **2022**, *10*, 96.
https://doi.org/10.3390/risks10050096

**AMA Style**

Bozikas A, Badounas I, Pitselis G.
Pricing Longevity Bonds under a Credibility Framework with Limited Available Data. *Risks*. 2022; 10(5):96.
https://doi.org/10.3390/risks10050096

**Chicago/Turabian Style**

Bozikas, Apostolos, Ioannis Badounas, and Georgios Pitselis.
2022. "Pricing Longevity Bonds under a Credibility Framework with Limited Available Data" *Risks* 10, no. 5: 96.
https://doi.org/10.3390/risks10050096