Stochastic mortality: the impact on target capital.

*(English)*Zbl 1179.91108Summary: We take the point of view of an insurer dealing with life annuities, which aims at building up a (partial) internal model in order to quantify the impact of mortality risks, namely process and longevity risk, in view of taking appropriate risk management actions. We assume that a life table, providing a best-estimate assessment of annuitants’ future mortality is available to the insurer; conversely, the insurer has no access to data sets and the methodology underlying the construction of the life table. Nonetheless, the insurer is aware that, in the presence of mortality risks, a stochastic approach is required. The (projected) life table, which provides a deterministic description of future mortality, should then be used as the basic input of a stochastic model.

The model we propose focuses on the annual number of deaths in a given cohort, which we represent allowing for a random mortality rate. To this purpose, we adopt the widely used Poisson model, first assuming a Gamma-distributed random parameter, and second introducing time-dependence in the parameter itself. Further, we define a Bayesian-inferential procedure for updating the parameters to experience in some situations. The setting we define does not demand advanced analytical tools, while allowing for process and longevity risk in a rigorous way.

The model is then implemented for capital allocation purposes. We investigate the amount of the required capital for a given life annuity portfolio, based on solvency targets which could be adopted within internal models. The outcomes of such an investigation are compared with the capital required according to some standard rules, in particular those proposed within the Solvency 2 project.

The model we propose focuses on the annual number of deaths in a given cohort, which we represent allowing for a random mortality rate. To this purpose, we adopt the widely used Poisson model, first assuming a Gamma-distributed random parameter, and second introducing time-dependence in the parameter itself. Further, we define a Bayesian-inferential procedure for updating the parameters to experience in some situations. The setting we define does not demand advanced analytical tools, while allowing for process and longevity risk in a rigorous way.

The model is then implemented for capital allocation purposes. We investigate the amount of the required capital for a given life annuity portfolio, based on solvency targets which could be adopted within internal models. The outcomes of such an investigation are compared with the capital required according to some standard rules, in particular those proposed within the Solvency 2 project.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91B70 | Stochastic models in economics |

91D20 | Mathematical geography and demography |

##### Keywords:

life annuities; random fluctuations; systematic deviations; process risk; longetivity risk; solvency; insurance risk management; internal models
PDF
BibTeX
XML
Cite

\textit{A. Olivieri} and \textit{E. Pitacco}, ASTIN Bull. 39, No. 2, 541--563 (2009; Zbl 1179.91108)

Full Text:
DOI

##### References:

[1] | Insurance risk models (1992) |

[2] | Transactions of the 24th International Congress of Actuaries, MontrĂ©al, Canada 3 pp 209– (1992) |

[3] | Transactions of the 27th International Congress of Actuaries, Cancun (Mexico) (2002) |

[4] | Technical Specifications (2008) |

[5] | Technical Specifications. Part I: Instructions (2007) |

[6] | Bayes and empirical Bayes methods for data analysis (2000) · Zbl 1017.62005 |

[7] | ASTIN Bulletin 36 pp 79– (2006) |

[8] | Mathematical methods in risk theory (1970) · Zbl 0209.23302 |

[9] | Point processes and queues. Martingale dynamics (1981) |

[10] | DOI: 10.1007/s10203-006-0061-5 · Zbl 1160.91366 |

[11] | Insurance: Mathematics & Economics 37 pp 443– (2005) |

[12] | Getting to grips with fair value (2002) |

[13] | Belgian Actuarial Bulletin 6 pp 23– (2006) |

[14] | Insurance: Mathematics & Economics 29 pp 231– (2001) |

[15] | Stochastic mortality models (2004) |

[16] | Transactions of the 26th International Congress of Actuaries, Birmingham 6 pp 453– (1998) |

[17] | The Encyclopedia of Quantitative Risk Assessment and Analysis pp 1535– (2008) |

[18] | Life insurance mathematics (1995) |

[19] | Insurance: Mathematics & Economics 39 pp 193– (2006) |

[20] | Insurance: Mathematics & Economics 35 pp 113– (2004) |

[21] | Stochastic projection methodologies: Further progress and P-Spline model features, example results and implications (2006) |

[22] | An interim basis for adjusting the ”92” Series mortality projections for cohort effects (2002) |

[23] | DOI: 10.1017/S1357321700002762 |

[24] | DOI: 10.2307/2061224 |

[25] | Stochastic mortality modelling (2005) |

[26] | Solvency. Models, assessment and regulation (2006) · Zbl 1124.91041 |

[27] | Monte Carlo statistical methods (2004) · Zbl 1096.62003 |

[28] | DOI: 10.1017/S1474747203001276 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.