# Adding Shocks to a Prospective Mortality Model

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Stochastic Mortality Model

#### 2.1. Specification

#### 2.2. Log-Likelihood Determination

#### 2.3. Parameter Estimation

#### 2.4. Calculating Prospective Residual Life Expectancies

## 3. Numerical Application

#### 3.1. Model Adjustment

#### 3.1.1. Estimation of Gamma Distribution Parameters

#### 3.1.2. Estimation of Model Parameters

#### 3.1.3. Extrapolation of Time Coefficients

#### 3.2. Projected Mortality Forces

#### 3.3. Estimating Prospective Residual Life Expectancies

#### 3.4. Sensitivity to Frailty Parameter

- -
- the severity of the COVID-19 pandemic remains below the Solvency 2 bicentennial event. It is associated with a 10-fold higher probability of occurrence.
- -
- the calibration of frailty with a volatility of 5.5% is consistent with that of the Solvency 2 standard formula for mortality risk.

#### 3.5. Consequences for the Capital Requirement of an Annuity Plan

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Alpha | ||||||||
---|---|---|---|---|---|---|---|---|

Age | Model Studied | LC Reference Model | Age | Model Studied | LC Reference Model | Age | Model Studied | LC Reference Model |

0 | −5.6542 | −5.6553 | 36 | −7.0173 | −7.0186 | 72 | −4.0422 | −4.0456 |

1 | −7.3447 | −7.3457 | 37 | −6.9390 | −6.9396 | 73 | −3.9592 | −3.9625 |

2 | −8.2790 | −8.2804 | 38 | −6.8474 | −6.8480 | 74 | −3.8668 | −3.8713 |

3 | −8.6679 | −8.6695 | 39 | −6.7630 | −6.7633 | 75 | −3.7867 | −3.7882 |

4 | −8.9672 | −8.9721 | 40 | −6.6729 | −6.6733 | 76 | −3.6851 | −3.6862 |

5 | −9.0765 | −9.0803 | 41 | −6.5862 | −6.5867 | 77 | −3.5790 | −3.5803 |

6 | −9.1958 | −9.2028 | 42 | −6.4878 | −6.4880 | 78 | −3.4774 | −3.4791 |

7 | −9.2892 | −9.2924 | 43 | −6.3872 | −6.3877 | 79 | −3.3674 | −3.3679 |

8 | −9.4087 | −9.4124 | 44 | −6.2849 | −6.2854 | 80 | −3.2163 | −3.2216 |

9 | −9.3935 | −9.4009 | 45 | −6.1800 | −6.1803 | 81 | −3.0838 | −3.0881 |

10 | −9.4223 | −9.4329 | 46 | −6.0917 | −6.0922 | 82 | −2.9510 | −2.9556 |

11 | −9.3577 | −9.3662 | 47 | −5.9813 | −5.9817 | 83 | −2.8178 | −2.8236 |

12 | −9.2768 | −9.2809 | 48 | −5.8840 | −5.8845 | 84 | −2.6937 | −2.7014 |

13 | −9.1674 | −9.1742 | 49 | −5.7947 | −5.7952 | 85 | −2.5725 | −2.5850 |

14 | −8.9411 | −8.9443 | 50 | −5.7068 | −5.7077 | 86 | −2.4381 | −2.4498 |

15 | −8.6907 | −8.6949 | 51 | −5.6157 | −5.6166 | 87 | −2.3015 | −2.3126 |

16 | −8.4690 | −8.4721 | 52 | −5.5399 | −5.5411 | 88 | −2.1619 | −2.1733 |

17 | −8.2022 | −8.2047 | 53 | −5.4596 | −5.4605 | 89 | −2.0266 | −2.0390 |

18 | −7.9805 | −7.9814 | 54 | −5.3712 | −5.3722 | 90 | −1.8871 | −1.9001 |

19 | −7.7424 | −7.7441 | 55 | −5.2853 | −5.2866 | 91 | −1.7522 | −1.7651 |

20 | −7.6642 | −7.6654 | 56 | −5.2062 | −5.2078 | 92 | −1.6216 | −1.6351 |

21 | −7.6042 | −7.6051 | 57 | −5.1270 | −5.1289 | 93 | −1.4936 | −1.5072 |

22 | −7.5935 | −7.5952 | 58 | −5.0598 | −5.0611 | 94 | −1.3657 | −1.3777 |

23 | −7.5824 | −7.5837 | 59 | −4.9902 | −4.9925 | 95 | −1.2453 | −1.2594 |

24 | −7.5520 | −7.5540 | 60 | −4.9156 | −4.9178 | 96 | −1.1239 | −1.1367 |

25 | −7.5513 | −7.5531 | 61 | −4.8551 | −4.8570 | 97 | −1.0154 | −1.0262 |

26 | −7.5195 | −7.5211 | 62 | −4.7853 | −4.7877 | 98 | −0.9056 | −0.9177 |

27 | −7.4982 | −7.4995 | 63 | −4.7165 | −4.7182 | 99 | −0.8059 | −0.8166 |

28 | −7.4795 | −7.4816 | 64 | −4.6552 | −4.6571 | 100 | −0.7093 | −0.7208 |

29 | −7.4393 | −7.4409 | 65 | −4.5875 | −4.5894 | 101 | −0.6286 | −0.6345 |

30 | −7.3880 | −7.3899 | 66 | −4.5141 | −4.5163 | 102 | −0.5389 | −0.5481 |

31 | −7.3594 | −7.3607 | 67 | −4.4558 | −4.4580 | 103 | −0.4581 | −0.4679 |

32 | −7.3008 | −7.3012 | 68 | −4.3774 | −4.3802 | 104 | −0.4095 | −0.4185 |

33 | −7.2536 | −7.2544 | 69 | −4.2982 | −4.3005 | 105 | −0.4625 | −0.4652 |

34 | −7.1764 | −7.1771 | 70 | −4.2170 | −4.2209 | |||

35 | −7.1106 | −7.1110 | 71 | −4.1381 | −4.1415 |

Beta | ||||||||
---|---|---|---|---|---|---|---|---|

Age | Model Studied | LC Reference Model | Age | Model Studied | LC Reference Model | Age | Model Studied | LC Reference Model |

0 | 0.0033 | 0.0033 | 36 | 0.0120 | 0.0120 | 72 | 0.0070 | 0.0071 |

1 | 0.0115 | 0.0115 | 37 | 0.0115 | 0.0115 | 73 | 0.0070 | 0.0071 |

2 | 0.0114 | 0.0114 | 38 | 0.0123 | 0.0123 | 74 | 0.0075 | 0.0077 |

3 | 0.0118 | 0.0117 | 39 | 0.0119 | 0.0118 | 75 | 0.0090 | 0.0091 |

4 | 0.0130 | 0.0131 | 40 | 0.0129 | 0.0129 | 76 | 0.0094 | 0.0094 |

5 | 0.0121 | 0.0118 | 41 | 0.0134 | 0.0135 | 77 | 0.0097 | 0.0097 |

6 | 0.0097 | 0.0102 | 42 | 0.0140 | 0.0140 | 78 | 0.0096 | 0.0097 |

7 | 0.0145 | 0.0146 | 43 | 0.0137 | 0.0137 | 79 | 0.0099 | 0.0098 |

8 | 0.0119 | 0.0121 | 44 | 0.0135 | 0.0136 | 80 | 0.0136 | 0.0129 |

9 | 0.0126 | 0.0127 | 45 | 0.0132 | 0.0133 | 81 | 0.0139 | 0.0134 |

10 | 0.0144 | 0.0138 | 46 | 0.0136 | 0.0137 | 82 | 0.0141 | 0.0137 |

11 | 0.0114 | 0.0118 | 47 | 0.0128 | 0.0127 | 83 | 0.0142 | 0.0137 |

12 | 0.0165 | 0.0162 | 48 | 0.0122 | 0.0122 | 84 | 0.0115 | 0.0107 |

13 | 0.0143 | 0.0142 | 49 | 0.0108 | 0.0108 | 85 | 0.0083 | 0.0073 |

14 | 0.0151 | 0.0151 | 50 | 0.0106 | 0.0106 | 86 | 0.0065 | 0.0057 |

15 | 0.0155 | 0.0151 | 51 | 0.0099 | 0.0099 | 87 | 0.0059 | 0.0054 |

16 | 0.0180 | 0.0182 | 52 | 0.0093 | 0.0093 | 88 | 0.0056 | 0.0052 |

17 | 0.0171 | 0.0171 | 53 | 0.0093 | 0.0093 | 89 | 0.0057 | 0.0054 |

18 | 0.0185 | 0.0187 | 54 | 0.0095 | 0.0094 | 90 | 0.0051 | 0.0049 |

19 | 0.0183 | 0.0180 | 55 | 0.0093 | 0.0092 | 91 | 0.0049 | 0.0048 |

20 | 0.0170 | 0.0169 | 56 | 0.0082 | 0.0080 | 92 | 0.0044 | 0.0046 |

21 | 0.0158 | 0.0158 | 57 | 0.0082 | 0.0081 | 93 | 0.0039 | 0.0041 |

22 | 0.0146 | 0.0146 | 58 | 0.0067 | 0.0066 | 94 | 0.0026 | 0.0030 |

23 | 0.0156 | 0.0156 | 59 | 0.0061 | 0.0059 | 95 | 0.0023 | 0.0027 |

24 | 0.0132 | 0.0134 | 60 | 0.0050 | 0.0049 | 96 | 0.0016 | 0.0024 |

25 | 0.0126 | 0.0127 | 61 | 0.0042 | 0.0041 | 97 | 0.0008 | 0.0014 |

26 | 0.0113 | 0.0115 | 62 | 0.0039 | 0.0038 | 98 | −0.0005 | 0.0004 |

27 | 0.0111 | 0.0112 | 63 | 0.0040 | 0.0039 | 99 | −0.0013 | −0.0006 |

28 | 0.0107 | 0.0108 | 64 | 0.0042 | 0.0042 | 100 | −0.0002 | −0.0001 |

29 | 0.0088 | 0.0088 | 65 | 0.0038 | 0.0038 | 101 | 0.0018 | 0.0022 |

30 | 0.0103 | 0.0102 | 66 | 0.0046 | 0.0046 | 102 | 0.0044 | 0.0047 |

31 | 0.0111 | 0.0110 | 67 | 0.0050 | 0.0051 | 103 | 0.0046 | 0.0048 |

32 | 0.0107 | 0.0106 | 68 | 0.0047 | 0.0047 | 104 | 0.0046 | 0.0053 |

33 | 0.0105 | 0.0104 | 69 | 0.0059 | 0.0059 | 105 | 0.0046 | 0.0051 |

34 | 0.0106 | 0.0105 | 70 | 0.0055 | 0.0055 | |||

35 | 0.0114 | 0.0113 | 71 | 0.0063 | 0.0064 |

Kappa | |||||
---|---|---|---|---|---|

Age | Model Studied | LC Reference Model | Age | Model Studied | LC Reference Model |

2000 | 24.4565 | 24.4761 | 2030 | −43.8008 | −43.8043 |

2001 | 24.4627 | 24.5265 | 2031 | −45.9908 | −45.9945 |

2002 | 21.2073 | 21.2070 | 2032 | −48.1809 | −48.1847 |

2003 | 18.4710 | 18.4083 | 2033 | −50.3709 | −50.3750 |

2004 | 10.9651 | 10.9513 | 2034 | −52.5610 | −52.5652 |

2005 | 9.4399 | 9.4231 | 2035 | −54.7510 | −54.7554 |

2006 | 6.0386 | 6.0366 | 2036 | −56.9410 | −56.9456 |

2007 | 3.1599 | 3.1578 | 2037 | −59.1311 | −59.1358 |

2008 | 1.4649 | 1.4631 | 2038 | −61.3211 | −61.3260 |

2009 | 1.6981 | 1.6984 | 2039 | −63.5112 | −63.5163 |

2010 | −1.1863 | −1.1865 | 2040 | −65.7012 | −65.7065 |

2011 | −4.7123 | −4.7140 | 2041 | −67.8912 | −67.8967 |

2012 | −6.6593 | −6.6691 | 2042 | −70.0813 | −70.0869 |

2013 | −8.5651 | −8.5639 | 2043 | −72.2713 | −72.2771 |

2014 | −12.6304 | −12.6243 | 2044 | −74.4614 | −74.4673 |

2015 | −9.8526 | −9.8338 | 2045 | −76.6514 | −76.6575 |

2016 | −13.5803 | −13.5743 | 2046 | −78.8414 | −78.8478 |

2017 | −15.8335 | −15.8386 | 2047 | −81.0315 | −81.0380 |

2018 | −15.6887 | −15.6650 | 2048 | −83.2215 | −83.2282 |

2019 | −16.4993 | −16.4493 | 2049 | −85.4116 | −85.4184 |

2020 | −16.1564 | −16.2296 | 2050 | −87.6016 | −87.6086 |

2021 | −24.0904 | −24.0924 | 2051 | −89.7916 | −89.7988 |

2022 | −26.2805 | −26.2826 | 2052 | −91.9817 | −91.9891 |

2023 | −28.4705 | −28.4728 | 2053 | −94.1717 | −94.1793 |

2024 | −30.6606 | −30.6630 | 2054 | −96.3618 | −96.3695 |

2025 | −32.8506 | −32.8532 | 2055 | −98.5518 | −98.5597 |

2026 | −35.0406 | −35.0435 | 2056 | −100.7418 | −100.7499 |

2027 | −37.2307 | −37.2337 | 2057 | −102.9319 | −102.9401 |

2028 | −39.4207 | −39.4239 | 2058 | −105.1219 | −105.1304 |

2029 | −41.6108 | −41.6141 | 2059 | −107.3120 | −107.3206 |

2060 | −109.5020 | −109.5108 |

## Notes

1 | It should be remembered that Lee and Carter’s initial model is not a probabilistic model, and simply proposes a parsimonious decomposition of interactions between age and year in the structure of mortality rates across a country. |

2 | https://cran.r-project.org/web/packages/Rsolnp/index.html (accessed on 31 December 2023). |

3 | https://cran.r-project.org/web/packages/demography/index.html (accessed on 31 December 2023). |

4 | https://www.ressources-actuarielles.net/C1256F13006585B2/0/39B54166464089AFC12572B0003D88C2/$FILE/20230921_FP.pdf?OpenElement (accessed on 31 December 2023). |

5 | https://actudactuaires.typepad.com/laboratoire/2021/03/mortalit%C3%A9-en-france-en-2020-suite.html (accessed on 31 December 2023). |

6 | The SCR is the minimum capital required to control the probability of ruin at one year in the sense of the economic balance sheet at the level of 0.5%. |

7 | EU Delegated Regulation n°2015/35: https://eur-lex.europa.eu/legal-content/FR/TXT/?uri=CELEX:32015R0035 (accessed on 31 December 2023). |

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**Figure 6.**Mortality force ratio over the entire age range and prospective analysis period (2021 to 2060).

**Figure 12.**Evolution of prospective residual life expectancies from age 65 to 105 for selected years, with a new volatility coefficient.

m | p | |
---|---|---|

Model studied | −2.19 | 4401.98 |

Lee–Carter reference model | −2.19 | 4402.33 |

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## Share and Cite

**MDPI and ACS Style**

Planchet, F.; Gautier de La Plaine, G.
Adding Shocks to a Prospective Mortality Model. *Risks* **2024**, *12*, 57.
https://doi.org/10.3390/risks12030057

**AMA Style**

Planchet F, Gautier de La Plaine G.
Adding Shocks to a Prospective Mortality Model. *Risks*. 2024; 12(3):57.
https://doi.org/10.3390/risks12030057

**Chicago/Turabian Style**

Planchet, Frédéric, and Guillaume Gautier de La Plaine.
2024. "Adding Shocks to a Prospective Mortality Model" *Risks* 12, no. 3: 57.
https://doi.org/10.3390/risks12030057