# Measurement and Impact of Longevity Risk in Portfolios of Pension Annuity: The Case in Sub Saharan Africa

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## Abstract

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## 1. Introduction

## 2. Theoretical Perspective

#### 2.1. Notations and Data Transformation

#### 2.2. Age-Period-Cohort (APC) Stochastic Mortality Models

- Random element: This is the r.v. ${D}_{xt}\sim $ Poisson or a Binomial distribution.$${D}_{xt}\sim Poisson\left({E}_{xt}^{c}{\mu}_{xt}\right)\phantom{\rule{-3.0pt}{0ex}}$$$${D}_{xt}\sim Binomial\left({E}_{xt}^{0},{q}_{xt}\right)$$
- Systematic element: The predictor, ${\eta}_{xt}$, ensures the capturing of the age x, period (calendar year) t and cohort (year-of-birth) $c=t-x$ effects.$${\eta}_{xt}={\alpha}_{x}+\sum _{i=1}^{N}{\beta}_{x}^{\left(i\right)}{\kappa}_{t}^{\left(i\right)}+{\beta}_{x}^{\left(0\right)}{\gamma}_{t-x}\phantom{\rule{-3.0pt}{0ex}}$$
- Link function: The function g forms the link between the two components, i.e., the random element and the systematic element, such that:$$g\left(\mathbb{E}\left(\frac{{D}_{xt}}{{E}_{xt}}\right)\right)={\eta}_{xt}\phantom{\rule{-3.0pt}{0ex}}$$The canonical link is used to connect the terms, with the log link function for the Poisson distribution and the logit link for the binomial distribution.
- Parameter constraints: This is very important to solve the identifiability issues associated with most stochastic mortality models. These ensure unique estimates for the parameters in the models. The constraint function $nu$ is used to obtain the parameter $theta$ as follows:$$\theta :=\left({\alpha}_{x},{\beta}_{x}^{\left(1\right)},\dots ,{\beta}_{x}^{\left(N\right)},{\kappa}_{t}^{\left(1\right)},\dots ,{\kappa}_{t}^{\left(N\right)},{\beta}_{x}^{\left(0\right)},{\gamma}_{t-x}\right)\phantom{\rule{-3.0pt}{0ex}}$$$$\nu \left(\theta \right)=\tilde{\theta}=\left({\tilde{\alpha}}_{x},{\tilde{\beta}}_{x}^{\left(1\right)},\dots ,{\tilde{\beta}}_{x}^{\left(N\right)},{\tilde{\kappa}}_{t}^{\left(1\right)},\dots ,{\tilde{\kappa}}_{t}^{\left(N\right)},{\tilde{\beta}}_{x}^{\left(0\right)},{\tilde{\gamma}}_{t-x}\right)\phantom{\rule{-3.0pt}{0ex}}$$

#### 2.3. Lee–Carter Model under GAPC

#### 2.4. Renshaw and Haberman Model under GAPC

#### 2.5. Cairns–Blake–Dowd Model under GAPC

#### 2.6. Quadratic Cairns–Blake–Dowd Model under GAPC

#### 2.7. Parameters Estimation

#### 2.8. Forecast of Mortality Rates

#### 2.8.1. Model Selection and Diagnostics

#### 2.8.2. Mortality Improvement

#### 2.9. Present Annuity Value with Pension Ages

## 3. Results and Discussion

#### 3.1. Data

#### 3.2. Mortality Behavior and Rate Improvement in Ghana

#### 3.3. Stochastic Mortality Models Estimation

#### 3.3.1. Lee-Carter Model

#### 3.3.2. Renshaw–Haberman Model

#### 3.3.3. Cairns-Blake-Dowd Model

#### 3.3.4. Quadratic Cairns–Blake–Dowd Model

#### 3.4. Models Goodness of Fit

#### 3.5. Mortality Models Rate Forecasts

#### 3.6. Mortality Improvement

#### 3.7. Present Annuity Value Factor for Pension Ages

#### 3.8. Discussion

## 4. Conclusions and Policy Recommendation

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mortality Rate Forecasts—Pension Ghana

## References

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**Figure 3.**Lee–Carter Parameters Fitted to the Ghana Pension Population for Ages 40–83 and the Period 2010–2020.

**Figure 4.**Renshaw–Haberman Parameters fitted to the Ghana Pension Population for Ages 40–83 and the Period 2010–2020.

**Figure 5.**CBD Parameters Fitted to the Ghana Pension Population for Ages 40–83 and the Period 2010–2020.

**Figure 6.**Quadratic CBD parameters Fitted to the Ghana Pension Population for Ages 40–83 and the Period 2010–2020.

**Figure 7.**LC Deviance Residuals from the Fitted Model to the Ghana Pension Population for Ages 40–84 and the Period 2010–2020.

**Figure 8.**RH Deviance Residuals from the Fitted Model to the Ghana Pension Population for Ages 40–83 and the Period 2010–2020.

**Figure 9.**CBD Deviance Residuals from the Fitted Model to the Ghana Pension Population for Ages 40–83 and the Period 2010–2020.

**Figure 10.**Q-CBD Deviance Residuals from the Fitted Model to the Ghana Pension Population for Ages 40–83 and the Period 2010–2020.

Female | Male | Total | |
---|---|---|---|

Moment Measure | (%) | (%) | (%) |

Mean | 1.393 | 1.078 | 1.236 |

Median | 1.322 | 0.960 | 1.066 |

Standard deviation | 0.650 | 0.565 | 0.588 |

Half-Decades Rate Improvement | |||

2001–2005 | 1.580 | 1.471 | 1.525 |

2006–2010 | 1.604 | 1.406 | 1.505 |

2011–2015 | 1.322 | 0.998 | 1.160 |

2016–2020 | 1.574 | 1.163 | 1.368 |

Metric | LC | RH | CBD | Q-CBD |
---|---|---|---|---|

MAFE | $4.2\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $1.3\times {10}^{-2}$ | $3.9\times {10}^{-3}$ |

RMSFE | $7.4\times {10}^{-3}$ | $2.0\times {10}^{-2}$ | $2.2\times {10}^{-2}$ | $5.8\times {10}^{-3}$ |

MAPFE | 3.4% | 6.7% | 7.6% | 2.9% |

Rank | (2) | (3) | (4) | (1) |

Model 1 | Model 2 | DM Statistics | p-Value |
---|---|---|---|

Q-CBD | LC | 1.2150 | 0.8878 |

Q-CBD | RH | −2.9951 | 0.0014 ** |

Q-CBD | CBD | −1.9594 | 0.0250 ** |

LC | RH | −2.8088 | 0.0025 ** |

LC | CBD | −1.9007 | 0.0287 ** |

LC | Q-CBD | −1.2150 | 0.1122 |

RH | CBD | −1.4370 | 0.0754 * |

Lee-Carter | 2021–2022 | 2021–2025 | 2021–2030 |
---|---|---|---|

60 | 2.651 | 2.652 | 2.653 |

65 | 3.338 | 3.340 | 3.342 |

70 | 3.261 | 3.264 | 3.267 |

75 | 4.116 | 4.123 | 4.132 |

80 | 3.195 | 3.203 | 3.214 |

Overall: 40–80 | 2.575 | ||

Renshaw-Haberman | |||

60 | 3.223 | 4.705 | 5.253 |

65 | 5.460 | 4.711 | 5.350 |

70 | 4.962 | 4.624 | 5.039 |

75 | 3.700 | 4.212 | 5.529 |

80 | 4.957 | 4.477 | 4.582 |

Overall: 40–80 | 4.256 | ||

CBD | |||

60 | 2.771 | 2.772 | 2.773 |

65 | 2.970 | 2.972 | 2.974 |

70 | 3.158 | 3.161 | 3.165 |

75 | 3.327 | 3.332 | 3.339 |

80 | 3.465 | 3.474 | 3.486 |

Overall: 40–80 | 2.805 | ||

Quadratic CBD | |||

60 | 2.073 | 3.321 | 3.217 |

65 | 4.327 | 3.131 | 3.316 |

70 | 3.723 | 3.093 | 3.261 |

75 | 2.656 | 2.974 | 3.875 |

80 | 4.176 | 3.588 | 3.456 |

Overall: 40–80 | 2.534 |

Age | 2021–2030 | 2021–2030 |
---|---|---|

(Annually) | (Monthly) | |

55 | 5.4191 | 5.8775 |

57 | 5.3854 | 5.8438 |

60 | 5.3237 | 5.7820 |

62 | 5.2748 | 5.7332 |

65 | 5.2009 | 5.6593 |

Age | 2021–2030 | 2021–2030 |
---|---|---|

(Annually) | (Monthly) | |

55 | 5.4314 | 5.8897 |

57 | 5.3988 | 5.8571 |

60 | 5.3314 | 5.7897 |

62 | 5.2797 | 5.7381 |

65 | 5.2075 | 5.6658 |

Age | 2021–2030 | 2021–2030 |
---|---|---|

(Annually) | (Monthly) | |

55 | 5.4247 | 5.8830 |

57 | 5.3859 | 5.8443 |

60 | 5.3220 | 5.7803 |

62 | 5.2676 | 5.7260 |

65 | 5.1882 | 5.6465 |

Age | 2021–2030 | 2021–2030 |
---|---|---|

(Annually) | (Monthly) | |

55 | 5.4255 | 5.8838 |

57 | 5.3885 | 5.8468 |

60 | 5.3254 | 5.7838 |

62 | 5.2721 | 5.7304 |

65 | 5.1948 | 5.6532 |

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## Share and Cite

**MDPI and ACS Style**

Gyamerah, S.A.; Arthur, J.; Akuamoah, S.W.; Sithole, Y.
Measurement and Impact of Longevity Risk in Portfolios of Pension Annuity: The Case in Sub Saharan Africa. *FinTech* **2023**, *2*, 48-67.
https://doi.org/10.3390/fintech2010004

**AMA Style**

Gyamerah SA, Arthur J, Akuamoah SW, Sithole Y.
Measurement and Impact of Longevity Risk in Portfolios of Pension Annuity: The Case in Sub Saharan Africa. *FinTech*. 2023; 2(1):48-67.
https://doi.org/10.3390/fintech2010004

**Chicago/Turabian Style**

Gyamerah, Samuel Asante, Janet Arthur, Saviour Worlanyo Akuamoah, and Yethu Sithole.
2023. "Measurement and Impact of Longevity Risk in Portfolios of Pension Annuity: The Case in Sub Saharan Africa" *FinTech* 2, no. 1: 48-67.
https://doi.org/10.3390/fintech2010004