A partial internal model for longevity risk.

*(English)*Zbl 1398.91334Summary: This paper proposes a simple partial internal model for longevity risk within the solvency 2 framework. The model is closely linked to the mechanisms associated with the so-called Danish longevity benchmark, where the underlying mortality intensity and the trend is estimated yearly based on mortality experience from the Danish life and pension insurance sector, and on current data from the entire Danish population. Within this model, we derive an estimate for the \(99.5\%\) percentile for longevity risk, which differs from the longevity stress of \(20\%\) from the standard model. The new stress explicitly reflects the risk associated with unexpected changes in the underlying population mortality intensity on a one-year horizon and with a \(99.5\%\) confidence level. In addition, the model contains a component, which quantifies the unsystematic longevity risk associated with a given insurance portfolio. This last component depends on the size of the specific portfolio.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

91D20 | Mathematical geography and demography |

##### Keywords:

Solvency 2; mortality; longevity stress; Danish longevity benchmark; systematic and unsystematic risk##### Software:

Human Mortality
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\textit{S. F. Jarner} and \textit{T. Møller}, Scand. Actuar. J. 2015, No. 4, 352--382 (2015; Zbl 1398.91334)

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##### References:

[1] | Bauer, D., Börger, M. & Russ, J. (2010). On the pricing of longevity-linked securities. Insurance: Mathematics and Economics46, 139-149. · Zbl 1231.91142 |

[2] | Biatat, V. D. & Currie, I. D. (2010). Joint models for classification and comparison of mortality in different countries. Proceedings of 25rd International Workshop on Statistical Modelling, Glasgow, pp. 89-94. |

[3] | Booth, H., Hyndman, R. J., Tickie, L. & Jong, P. (2006). Lee-Carter mortality forecasting: a multi-country comparison of variants and extensions. Demographic Research15, 289-310. |

[4] | Börger, M. (2010). Deterministic shock vs. stochastic Value-at-Risk — an analysis of the Solvency II standard model approach to longevity risk. Blätter DGVFM31, 225-259. · Zbl 1232.91341 |

[5] | Brouhns, N., Denuit, M. & Vermunt, J. K. (2002). A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics31, 373-393. · Zbl 1074.62524 |

[6] | Cairns, A. J. (2011). Modelling and management of longevity risk: approximations to survival functions and dynamic hedging. Insurance: Mathematics and Economics49, 438-453. · Zbl 1230.91068 |

[7] | Cairns, A. J., Blake, D. & Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. Journal of Risk and Insurance73, 687-718. |

[8] | Cairns, A. J., Blake, D., Dowd, K., Coughlan, G. D. & Khalaf-Allah, M. (2011). Bayesian stochastic mortality modelling for two populations. ASTIN Bulletin41, 29-59. · Zbl 1228.91032 |

[9] | CEIOPS. (2010). Solvency II Calibration Paper, CEIOPS-SEC-40-10. |

[10] | Currie, I. D., Durban, M. & Eilers, P. H. C. (2004). Smoothing and forecasting mortality rates. Statistical Modelling4, 279-298. · Zbl 1061.62171 |

[11] | Dahl, M. (2004). Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insurance: Mathematics and Economics35, 113-136. · Zbl 1075.62095 |

[12] | Dahl, M. & Møller, T. (2006). Valuation and hedging of life insurance liabilities with systematic mortality risk. Insurance: Mathematics and Economics39, 193-217. · Zbl 1201.91089 |

[13] | Jong, P. & Tickle, L. (2006). Extending Lee-Carter mortality forecasting. Mathematical Population Studies13, 1-18. · Zbl 1151.91742 |

[14] | DSA. (2012). Solvency 2: Longevity Stress and the Danish Longevity Benchmark. Published on the homepage of the Danish Society of Actuaries. http://www.aktuarforeningen.dk/English/Publications/tabid/92/Default.aspx. |

[15] | EIOPA. (2011). EIOPA Report on the fifth Quantitative Impact Study (QIS 5) for Solvency II, EIOPA-TFQIS5-11/001. |

[16] | EIOPA. (2012). Revised Technical Specifications for the Solvency II valuation and Solvency Capital Requirements calculations (Part I), EIOPA-DOC-12/467. https://eiopa.europa.eu/consultations/qis/insurance/long-term-guarantees-assessment. |

[17] | EIOPA. (2013). Technical Specification on the Long Term Guarantee Assessment (Part I), EIOPA-DOC-13/061. https://eiopa.europa.eu/consultations/qis/insurance/long-term-guarantees-assessment. |

[18] | European Commission. (2009). Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II). http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2009:335:0001:0155:EN:PDF. |

[19] | European Commission. (2011). Draft Implementing measures Solvency II. unpublished. |

[20] | Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). http://www.mortality.org. |

[21] | Jarner, S. F. & Kryger, E. M. (2011). Modelling adult mortality in small populations: the SAINT model. ASTIN Bulletin41, 377-418. · Zbl 1239.91128 |

[22] | Lee, R. D. & Carter, L. R. (1992). Modeling and forecasting of U.S. mortality. Journal of the American Statistical Association87, 659-675. · Zbl 1351.62186 |

[23] | Lee, R. D. & Miller, T. (2001). Evaluating the performance of the Lee-Carter method for forecasting mortality. Demography38, 537-549. |

[24] | Li, J. S.-H. & Hardy, M. R. (2011). Measuring basis risk in longevity hedges. North American Actuarial Journal15, 177-200. · Zbl 1228.91042 |

[25] | Li, J. S.-H., Hardy, M. R. & Tan, K. S. (2009). Uncertainty in mortality forecasting: an extension to the classical Lee-Carter approach. ASTIN Bulletin39, 137-164. · Zbl 1203.91113 |

[26] | Li, N. & Lee, R. (2005). Coherent mortality forecasts for a group of populations: an extension of the Lee-Carter method. Demography42, 575-594. |

[27] | Nielsen, L. H. (2010). Assessment of longevity risk under Solvency II. Life & Pension Risk, November issue, pp. 41-44. |

[28] | Olivieri, A. & Pitacco, E. (2009). Stochastic mortality: the impact on target capital. ASTIN Bulletin39, 541-563. · Zbl 1179.91108 |

[29] | Pitacco, E., Denuit, M., Haberman, S. & Olivieri, A. (2009). Modelling longevity dynamics for pensions and annuity business. Oxford: Oxford University Press. · Zbl 1163.91005 |

[30] | Plat, R. (2009). Stochastic portfolio specific mortality and the quantification of mortality basis risk. Insurance: Mathematics and Economics45, 123-132. · Zbl 1231.91226 |

[31] | Plat, R. (2011). One-year Value-at-Risk for longevity and mortality. Insurance: Mathematics and Economics49, 462-470. |

[32] | Renshaw, A. E. & Haberman, S. (2003). Lee-Carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics33, 255-272. · Zbl 1103.91371 |

[33] | Renshaw, A. E. & Haberman, S. (2006). A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics38, 556-570. · Zbl 1168.91418 |

[34] | Richards, S. J., Currie, I. D. & Ritchie, G. P. (2012). A Value-at-risk framework for longevity trend risk. Discussion paper presented to The Institute and Faculty of Actuaries. |

[35] | Tuljapurkar, S., Li, N. & Boe, C. (2000). A universal pattern of mortality decline in the G7 countries. Nature405, 789-792. |

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