# Evaluation of the Kou-Modified Lee-Carter Model in Mortality Forecasting: Evidence from French Male Mortality Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model Specification

#### 2.1. The Lee-Carter Model

#### 2.2. The Age-Period-Cohort Model

#### 2.3. The Cairns-Blake-Dowd Model

#### 2.4. The Kou-Modified Lee-Carter Model

#### 2.4.1. Equivalence of the PBJD Model and DEJD Process

#### 2.4.2. Parameter Estimation

## 3. Mortality Data

#### 3.1. Data Analysis

#### 3.2. The Backtesting Procedure

## 4. Application of the Models

#### 4.1. Application of the APC Model

#### 4.2. Application of the CBD Model

#### 4.3. Application of the Lee-Carter Model

#### 4.3.1. Random Walk with Drift

#### 4.3.2. Application of the Kou Model on the Lee-Carter Mortality Index

## 5. Performance Metrics

## 6. Results

## 7. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. SVD Procedure Used in the Lee-Carter Model

## References

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1 | Data downloaded on September 2017. Source: https://www.mortality.org/. |

**Figure 4.**The evolution of the first age-dependent parameter of the APC model during the (

**a**) 1900–1938 period and (

**b**) 1961–1990 period.

**Figure 5.**The evolution of the APC model mortality index during the (

**a**) 1900–1938 period and (

**b**) 1961–1990 period.

**Figure 6.**The evolution of the parameter showing the cohort effects during the (

**a**) 1900–1938 period and (

**b**) 1961–1990 period.

**Figure 7.**The forecast of the mortality index using the APC model for the (

**a**) 1939–1960 period and (

**b**) 1991–2015 period.

**Figure 10.**Forecasts of the two time-dependent parameters of the CBD model for the 1939–1960 period.

**Figure 11.**Forecasts of the two time-dependent parameters of the CBD model for the 1991–2015 period.

**Figure 14.**Lee-Carter residuals plotted against (

**a**) age, (

**b**) calendar year, and (

**c**) year of birth for the 1900–1938 period.

**Figure 15.**Lee-Carter residuals plotted against (

**a**) age, (

**b**) calendar year, and (

**c**) year of birth for the 1961–1990 period.

**Figure 16.**Forecasts of the Lee-Carter mortality index for the (

**a**) 1939–1960 period and (

**b**) 1991–2015 period.

**Figure 17.**Comparison between a normal probability curve and the shape distribution of the change in the Lee-Carter mortality index for (

**a**) 1900–1938 period, and (

**b**) 1961–1990 period.

**Figure 18.**Forecasts of the Lee-Carter mortality index using the Kou model for the (

**a**) 1939–1960 period and (

**b**) 1991–2015 period.

**Table 1.**Comparison of accuracy measures for the four models under consideration for the 1900–1960 period.

1900–1960 Period | ||||
---|---|---|---|---|

Models | Average RMSE | Average MAE | Average MPE | Average MAPE |

APC | 0.013063 | 0.006709 | 12.8334 | 16.30235 |

CBD | 0.014837 | 0.007495 | 15.08032 | 18.38768 |

Lee-Carter | 0.012102 | 0.006798 | 13.65589 | 15.49652 |

Lee-Carter Kou-Modified Mortality Index | 0.003029 | 0.003153 | 10.27675 | 13.58456 |

**Table 2.**Comparison of accuracy measures for the four models under consideration for the 1961–2015 period.

1961–2015 Period | ||||
---|---|---|---|---|

Models | Average RMSE | Average MAE | Average MPE | Average MAPE |

APC | 0.008016 | 0.016184 | 16.03057 | 18.92784 |

CBD | 0.013744 | 0.02708 | 25.13116 | 26.68533 |

Lee-Carter | 0.006764 | 0.00852 | 13.48385 | 15.88327 |

Lee-Carter Kou-Modified Mortality Index | 0.018139 | 0.030101 | 28.11825 | 30.23325 |

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**MDPI and ACS Style**

Alijean, M.A.C.; Narsoo, J. Evaluation of the Kou-Modified Lee-Carter Model in Mortality Forecasting: Evidence from French Male Mortality Data. *Risks* **2018**, *6*, 123.
https://doi.org/10.3390/risks6040123

**AMA Style**

Alijean MAC, Narsoo J. Evaluation of the Kou-Modified Lee-Carter Model in Mortality Forecasting: Evidence from French Male Mortality Data. *Risks*. 2018; 6(4):123.
https://doi.org/10.3390/risks6040123

**Chicago/Turabian Style**

Alijean, Marie Angèle Cathleen, and Jason Narsoo. 2018. "Evaluation of the Kou-Modified Lee-Carter Model in Mortality Forecasting: Evidence from French Male Mortality Data" *Risks* 6, no. 4: 123.
https://doi.org/10.3390/risks6040123