# An Empirical Study on Stochastic Mortality Modelling under the Age-Period-Cohort Framework: The Case of Greece with Applications to Insurance Pricing

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## Abstract

**:**

## 1. Introduction

## 2. Mortality Modelling

#### 2.1. The Age-Period-Cohort Framework

#### 2.2. Data and Assumptions

#### 2.3. Reviewing Mortality Models

## 3. Model Fit

#### 3.1. Parameter Estimates

#### Robustness

#### 3.2. Goodness of Fit Diagnostics

#### 3.2.1. Information Criteria

#### 3.2.2. Likelihood-Ratio Test

## 4. Mortality Projection

#### 4.1. Assessing Parameter Risk

#### 4.2. Application in Insurance-Related Products

## 5. Results

#### Comparison with Original Papers

## 6. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Animated Plots

## References

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1 | According to Cairns et al. (2009), the force of mortality can be viewed as the instantaneous death rate at exact time $t$ for a person aged exactly $x$ at time $t$. |

2 | For instance, Hyndman and Shahid Ullah (2007) used functional data analysis and penalized regression splines in their modelling framework. |

3 | Due to the limited availability of Greek data in HMD, years 2011–2013 correspond to a percentage of 10% of the whole fitting year span. |

4 | As Hunt and Blake (2015) point out, in practice, $M}_{7$ has proved the most popular extension of the original Cairns et al. (2006) model, since it gives a better fit to their data than $M}_{6$ and the age function for the cohort parameters in $M}_{8$ may be more complicated to fit data due to the estimation of the additional constant parameter $x}_{d$. |

5 | The sum of the estimated parameters minus those that reflect each model’s constraints. |

6 | The probabilities of death in the last year of the fitting period. |

7 | Inconsistency in male ranking results is expected, since BIC criterion penalizes stronger models with more parameters. |

**Figure 1.**$M}_{1$: $\alpha}_{x$, $\beta}_{x}^{\left(1\right)$ and $\kappa}_{t}^{\left(1\right)$ estimated parameters for males (

**top**panels) and females (

**bottom**panels), aged 60–89, fitted in 1981–2010 (solid lines) and 1981–2000 (dotted lines).

**Figure 2.**$M}_{2$: $\beta}_{x}^{\left(1\right)$, $\kappa}_{t}^{\left(1\right)$ and $\gamma}_{c$ estimated parameters for males (

**top**panels) and females (

**bottom**panels), aged 60–89, fitted in 1981–2010 (solid lines) and 1981–2000 (dotted lines).

**Figure 3.**$M}_{3$: $\kappa}_{t}^{\left(1\right)$ and $\gamma}_{c$ estimated parameters for males (

**top**panels) and females (

**bottom**panels), aged 60–89, fitted in 1981–2010 (solid lines) and 1981–2000 (dotted lines).

**Figure 4.**$M}_{4$: $\beta}_{x}^{\left(1\right)$, $\kappa}_{t}^{\left(1\right)$ and $\gamma}_{c$ estimated parameters for males (

**top**panels) and females (

**bottom**panels), aged 60–89, fitted in 1981–2010 (solid lines) and 1981–2000 (dotted lines).

**Figure 5.**$M}_{5$: $\kappa}_{t}^{\left(1\right)$ and $\kappa}_{t}^{\left(2\right)$ estimated parameters for males (

**top**panels) and females (

**bottom**panels), aged 60–89, fitted in 1981–2010 (solid lines) and 1981–2000 (dotted lines).

**Figure 6.**$M}_{6$: $\kappa}_{t}^{\left(1\right)},{\kappa}_{t}^{\left(2\right)$ and $\gamma}_{c$ estimated parameters for males (

**top**panels) and females (

**bottom**panels), aged 60–89, fitted in 1981–2010 (solid lines) and 1981–2000 (dotted lines).

**Figure 7.**$M}_{7$: $\kappa}_{t}^{\left(1\right)},{\kappa}_{t}^{\left(2\right)},{\kappa}_{t}^{\left(3\right)$ and $\gamma}_{c$ estimated parameters for males (

**left**panels) and females (

**right**panels), aged 60–89, fitted in 1981–2010 (solid lines) and 1981–2000 (dotted lines).

**Figure 8.**Residuals deviance of $M}_{1$ for males (

**top**panels) and females (

**bottom**panels) in Greece. Period 1981–2010, ages 60–89.

**Figure 9.**Residuals deviance of $M}_{2$ for males (

**top**panels) and females (

**bottom**panels) in Greece. Period 1981–2010, ages 60–89.

**Figure 10.**Residuals deviance of $M}_{3$ for males (

**top**panels) and females (

**bottom**panels) in Greece. Period 1981–2010, ages 60–89.

**Figure 11.**Residuals deviance of $M}_{4$ for males (

**top**panels) and females (

**bottom**panels) in Greece. Period 1981–2010, ages 60–89.

**Figure 12.**Residuals deviance of $M}_{5$ for males (

**top**panels) and females (

**bottom**panels) in Greece. Period 1981–2010, ages 60–89.

**Figure 13.**Residuals deviance of $M}_{6$ for males (

**top**panels) and females (

**bottom**panels) in Greece. Period 1981–2010, ages 60–89.

**Figure 14.**Residuals deviance of $M}_{7$ for males (

**top**panels) and females (

**bottom**panels) in Greece. Period 1981–2010, ages 60–89.

**Figure 15.**Long-term mortality projection results at ages x = 65 (bottom lines), x = 75 (middle lines) and x = 85 (top lines) derived from models $M}_{1$–$M}_{7$ fitted to males (

**left**panels) and females (

**right**panels) for ages 60–89 of the period 1981–2010. The shades regions in the projection period 2011–2030 denote the 50%, 80% and 95% prediction intervals.

**Figure 16.**95% prediction intervals for the probabilities of death at ages $x=65$, $x=75$ and $x=85$ for models $M}_{1$–$M}_{7$, fitted to males (

**left**panels) and females (

**right**panels) for ages 60–89 and the period 1981–2010 (thick dots). Solid lines denote the corresponding fitted rates and dot-dashed lines depict the 95% confidence intervals including parameter uncertainty. For the projection period 2011–2030, the central forecast values are given by dashed lines. Dashed lines and dot lines show the 95% prediction intervals with and without parameter uncertainty, respectively.

**Figure 17.**Absolute error and absolute percentage error values of life insurance and annuity products for the top four models in ranking for males (

**left**panels) and females (

**right**panels).

Model | Structure | Original Papers |
---|---|---|

$M}_{1$ | $logit{q}_{x,t}={\alpha}_{x}+{\beta}_{x}^{\left(1\right)}{\kappa}_{t}^{\left(1\right)}$ | Lee and Carter (1992) |

$M}_{2$ | $logit{q}_{x,t}={\alpha}_{x}+{\beta}_{x}^{\left(1\right)}{\kappa}_{t}^{\left(1\right)}+{\gamma}_{t-x}$ | Renshaw and Haberman (2006) |

$M}_{3$ | $logit{q}_{x,t}={\alpha}_{x}+{\kappa}_{t}^{\left(1\right)}+{\gamma}_{t-x}$ | Currie (2006) |

$M}_{4$ | $logit{q}_{x,t}={\alpha}_{x}+{\kappa}_{t}^{\left(1\right)}+(x-\overline{x}){\kappa}_{t}^{\left(2\right)}+{\gamma}_{t-x}$ | Plat (2009) |

$M}_{5$ | $logit{q}_{x,t}={\kappa}_{t}^{\left(1\right)}+(x-\overline{x}){\kappa}_{t}^{\left(2\right)}$ | Cairns et al. (2006) |

$M}_{6$ | $logit{q}_{x,t}={\kappa}_{t}^{\left(1\right)}+(x-\overline{x}){\kappa}_{t}^{\left(2\right)}+{\gamma}_{t-x}$ | Cairns et al. (2009) |

$M}_{7$ | $logit{q}_{x,t}={\kappa}_{t}^{\left(1\right)}+(x-\overline{x}){\kappa}_{t}^{\left(2\right)}+({(x-\overline{x})}^{2}-{\widehat{\sigma}}_{x}^{2}){\kappa}_{t}^{\left(3\right)}+{\gamma}_{t-x}$ | Cairns et al. (2009) |

**Table 2.**The maximum log likelihood and the number of the effective parameters along with AIC(c), AIC and BIC values (ranking order in brackets) of the mortality models for males and females.

Males | |||||

Model | Maximum Log Likelihood | Effective Parameters | AIC | AIC(c) | BIC |

$M}_{1$ | $-$4487.643 | 88 | 9151.287(7) | 9172.483(7) | 9566.560(7) |

$M}_{2$ | $-$4191.779 | 129 | 8641.558(4) | 8689.610(4) | 9250.311(4) |

$M}_{3$ | $-$4218.961 | 100 | 8637.922(3) | 8665.708(3) | 9109.823(2) |

$M}_{4$ | $-$4202.953 | 128 | 8661.907(5) | 8709.151(5) | 9265.940(5) |

$M}_{5$ | $-$4501.146 | 60 | 9122.291(6) | 9131.835(6) | 9405.432(6) |

$M}_{6$ | $-$4209.024 | 101 | 8620.048(2) | 8648.429(2) | 9096.669(1) |

$M}_{7$ | $-$4160.547 | 130 | 8581.094(1) | 8629.960(1) | 9194.565(3) |

Females | |||||

$M}_{1$ | $-$4980.632 | 88 | 10,137.265(6) | 10,158.461(6) | 10,552.538(6) |

$M}_{2$ | $-$4254.321 | 129 | 8766.643(3) | 8814.694(3) | 9375.395(3) |

$M}_{3$ | $-$4367.542 | 100 | 8935.085(4) | 8962.870(4) | 9406.986(4) |

$M}_{4$ | $-$4235.015 | 128 | 8726.030(2) | 8773.275(2) | 9330.064(2) |

$M}_{5$ | $-$5279.019 | 60 | 10,678.038(7) | 10,687.581(7) | 10,961.178(7) |

$M}_{6$ | $-$4474.985 | 101 | 9151.969(5) | 9180.349(5) | 9628.590(5) |

$M}_{7$ | $-$4209.487 | 130 | 8678.975(1) | 8727.841(1) | 9292.447(1) |

**Table 3.**Likelihood ratio test statistics for pairs of nested models ($H}_{0$) within general models ($H}_{1$).

Males | ||||

$\mathit{H}}_{\mathbf{0}$: Nested Model | $\mathit{H}}_{\mathbf{1}$: General Model | Likelihood Ratio Test Statistic | Degrees of Freedom | $\mathit{p}$-Value |

$M}_{1$ | $M}_{2$ | 591.730 | 41 | $<$0.0001 |

$M}_{3$ | $M}_{2$ | 54.364 | 29 | $<$0.0001 |

$M}_{3$ | $M}_{4$ | 32.015 | 28 | $<$0.0001 |

$M}_{5$ | $M}_{6$ | 584.240 | 41 | $<$0.0001 |

$M}_{5$ | $M}_{7$ | 681.200 | 70 | $<$0.0001 |

$M}_{6$ | $M}_{7$ | 96.955 | 29 | $<$0.0001 |

Females | ||||

$M}_{1$ | $M}_{2$ | 1452.600 | 41 | $<$0.0001 |

$M}_{3$ | $M}_{2$ | 226.440 | 29 | $<$0.0001 |

$M}_{3$ | $M}_{4$ | 265.050 | 28 | $<$0.0001 |

$M}_{5$ | $M}_{6$ | 1608.100 | 41 | $<$0.0001 |

$M}_{5$ | $M}_{7$ | 2139.100 | 70 | $<$0.0001 |

$M}_{6$ | $M}_{7$ | 530.990 | 29 | $<$0.0001 |

**Table 4.**Selected ARIMA(p,d,q) models for the period index ${\kappa}_{t}^{\left(i\right)},\phantom{\rule{4pt}{0ex}}i=1,2,3$ of male and female mortality models.

Males | |||

Model | $\mathit{\kappa}}_{\mathit{t}}^{\left(\mathbf{1}\right)$ | $\mathit{\kappa}}_{\mathit{t}}^{\left(\mathbf{2}\right)$ | $\mathit{\kappa}}_{\mathit{t}}^{\left(\mathbf{3}\right)$ |

$M}_{1$ | ARIMA(0,2,2) | —– | —– |

$M}_{2$ | ARIMA(0,1,1) with drift | —– | —– |

$M}_{3$ | ARIMA(1,1,0) with drift | —– | —– |

$M}_{4$ | ARIMA(0,2,2) | ARIMA(2,1,0) with drift | —– |

$M}_{5$ | ARIMA(1,2,1) | ARIMA(2,1,0) with drift | —– |

$M}_{6$ | ARIMA(0,2,2) with drift | ARIMA(0,1,1) with drift | —– |

$M}_{7$ | ARIMA(1,2,1) | ARIMA(2,2,0) | ARIMA(0,1,1) with drift |

Females | |||

Model | $\mathit{\kappa}}_{\mathit{t}}^{\left(\mathbf{1}\right)$ | $\mathit{\kappa}}_{\mathit{t}}^{\left(\mathbf{2}\right)$ | $\mathit{\kappa}}_{\mathit{t}}^{\left(\mathbf{3}\right)$ |

$M}_{1$ | ARIMA(1,1,0) with drift | —– | —– |

$M}_{2$ | ARIMA(3,1,0) with drift | —– | —– |

$M}_{3$ | ARIMA(3,1,0) with drift | —– | —– |

$M}_{4$ | ARIMA(1,1,0) with drift | ARIMA(1,1,0) with drift | —– |

$M}_{5$ | ARIMA(0,2,2) | ARIMA(0,1,0) with drift | —– |

$M}_{6$ | ARIMA(0,1,1) with drift | ARIMA(0,1,1) with drift | —– |

$M}_{7$ | ARIMA(2,1,0) with drift | ARIMA(2,2,0) | ARIMA(0,1,1) with drift |

**Table 5.**Selected ARIMA(p,d,q) models for the cohort index $\gamma}_{c$ of male and female mortality models.

Model | $\mathit{\gamma}}_{\mathit{c}$ for Males | $\mathit{\gamma}}_{\mathit{c}$ for Females |
---|---|---|

$M}_{2$ | ARIMA(2,1,0) | ARIMA(2,1,1) with drift |

$M}_{3$ | ARIMA(0,0,1) | ARIMA(4,1,1) |

$M}_{4$ | ARIMA(0,0,2) | ARIMA(4,1,1) |

$M}_{6$ | ARIMA(0,1,3) | ARIMA(3,0,2) |

$M}_{7$ | ARIMA(0,0,1) | ARIMA(4,0,1) |

**Table 6.**Averaged values (ranking order in brackets) of the mean absolute error (MAE) and mean absolute percentage error (MAPE) measures of the forecasting period 2011–2013 using fitted or actual jump-off rates for males and females.

Fitted Jump-off Rates | |||||||

Males | |||||||

Error | $M}_{1$ | $M}_{2$ | $M}_{3$ | $M}_{4$ | $M}_{5$ | $M}_{6$ | $M}_{7$ |

$MA{E}_{avg}$ | 0.332(6) | 0.251(1) | 0.253(2) | 0.287(3) | 0.327(5) | 0.295(4) | 0.346(7) |

$MAP{E}_{avg}$ | 10.194(4) | 6.496(1) | 6.583(2) | 9.385(3) | 10.935(6) | 10.559(5) | 15.697(7) |

Females | |||||||

$MA{E}_{avg}$ | 0.207(4) | 0.147(1) | 0.165(2) | 0.219(5) | 0.234(6) | 0.198(3) | 0.281(7) |

$MAP{E}_{avg}$ | 10.363(3) | 6.052(1) | 7.981(2) | 12.239(5) | 13.396(6) | 11.216(4) | 22.340(7) |

Actual Jump-off Rates | |||||||

Males | |||||||

Error | $M}_{1$ | $M}_{2$ | $M}_{3$ | $M}_{4$ | $M}_{5$ | $M}_{6$ | $M}_{7$ |

$MA{E}_{avg}$ | 0.273(6) | 0.213(3) | 0.208(2) | 0.192(1) | 0.289(7) | 0.237(4) | 0.247(5) |

$MAP{E}_{avg}$ | 6.780(5) | 5.222(2) | 5.086(1) | 5.371(3) | 6.916(6) | 6.020(4) | 8.545(7) |

Females | |||||||

$MA{E}_{avg}$ | 0.213(6) | 0.180(3) | 0.168(2) | 0.196(4) | 0.200(5) | 0.165(1) | 0.250(7) |

$MAP{E}_{avg}$ | 7.073(5) | 5.570(2) | 5.336(1) | 6.225(4) | 7.283(6) | 5.866(3) | 11.818(7) |

**Table 7.**Averaged values (ranking order in brackets) of MAE and MAPE measures for 10 year forecasted life insurance, pure endowment and life annuity values using actual jump-off rates for males and females, aged 60–79 in 2001–2010.

Life Insurance | |||||||

Males | |||||||

Error | $M}_{1$ | $M}_{2$ | $M}_{3$ | $M}_{4$ | $M}_{5$ | $M}_{6$ | $M}_{7$ |

$MA{E}_{x}$ | 2.222(6) | 1.242(1) | 2.284(7) | 2.199(5) | 2.020(4) | 1.456(2) | 1.799(3) |

$MAP{E}_{x}$ | 7.651(6) | 5.536(1) | 8.895(7) | 7.626(5) | 7.412(4) | 5.557(2) | 6.490(3) |

Females | |||||||

$MA{E}_{x}$ | 1.605(6) | 0.870(1) | 0.885(2) | 1.494(5) | 0.914(3) | 1.016(4) | 2.150(7) |

$MAP{E}_{x}$ | 9.264(5) | 6.404(1) | 6.901(3) | 9.268(6) | 6.426(2) | 6.930(4) | 11.883(7) |

Pure Endowment | |||||||

Males | |||||||

Error | $M}_{1$ | $M}_{2$ | $M}_{3$ | $M}_{4$ | $M}_{5$ | $M}_{6$ | $M}_{7$ |

$MA{E}_{x}$ | 1.605(6) | 0.927(1) | 1.666(7) | 1.590(5) | 1.451(4) | 1.039(2) | 1.293(3) |

$MAP{E}_{x}$ | 4.114(7) | 2.190(1) | 4.094(6) | 4.064(5) | 3.619(4) | 2.531(2) | 3.212(3) |

Females | |||||||

$MA{E}_{x}$ | 1.198(6) | 0.623(1) | 0.651(2) | 1.091(5) | 0.690(3) | 0.738(4) | 1.556(7) |

$MAP{E}_{x}$ | 2.615(6) | 1.282(2) | 1.242(1) | 2.250(5) | 1.408(3) | 1.565(4) | 3.240(7) |

Life Annuity | |||||||

Males | |||||||

Error | $M}_{1$ | $M}_{2$ | $M}_{3$ | $M}_{4$ | $M}_{5$ | $M}_{6$ | $M}_{7$ |

$MA{E}_{x}$ | 7.711(6) | 5.506(2) | 8.132(7) | 7.637(5) | 6.781(4) | 5.225(1) | 5.924(3) |

$MAP{E}_{x}$ | 1.127(6) | 0.785(2) | 1.168(7) | 1.112(5) | 0.980(4) | 0.748(1) | 0.856(3) |

Females | |||||||

$MA{E}_{x}$ | 5.484(6) | 2.465(1) | 2.944(2) | 4.995(5) | 3.254(4) | 3.091(3) | 6.466(7) |

$MAP{E}_{x}$ | 0.754(6) | 0.325(1) | 0.386(2) | 0.673(5) | 0.439(4) | 0.416(3) | 0.873(7) |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bozikas, A.; Pitselis, G.
An Empirical Study on Stochastic Mortality Modelling under the Age-Period-Cohort Framework: The Case of Greece with Applications to Insurance Pricing. *Risks* **2018**, *6*, 44.
https://doi.org/10.3390/risks6020044

**AMA Style**

Bozikas A, Pitselis G.
An Empirical Study on Stochastic Mortality Modelling under the Age-Period-Cohort Framework: The Case of Greece with Applications to Insurance Pricing. *Risks*. 2018; 6(2):44.
https://doi.org/10.3390/risks6020044

**Chicago/Turabian Style**

Bozikas, Apostolos, and Georgios Pitselis.
2018. "An Empirical Study on Stochastic Mortality Modelling under the Age-Period-Cohort Framework: The Case of Greece with Applications to Insurance Pricing" *Risks* 6, no. 2: 44.
https://doi.org/10.3390/risks6020044