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**Modelling dependent data for longevity projections.**
*(English)*
Zbl 1285.91054

Summary: The risk profile of an insurance company involved in annuity business is heavily affected by the uncertainty in future mortality trends. It is problematic to capture accurately future survival patterns, in particular at retirement ages when the effects of the rectangularization phenomenon and random fluctuations are combined. Another important aspect affecting the projections is related to the so-called cohort-period effect. In particular, the mortality experience of countries in the industrialized world over the course of the twentieth century would suggest a substantial age-time interaction, with the two dominant trends affecting different age groups at different times. From a statistical point of view, this indicates a dependence structure. Also the dependence between ages is an important component in the modeling of mortality [P. Barrieu et al., Scand. Actuar. J. 2012, No. 3, 203–231 (2012; Zbl 1277.91073)]. It is observed that the mortality improvements are similar for individuals of contiguous ages [S. Wills and M. Sherris, “Integrating financial and demographic longevity risk models: an Australian model for financial applications”, Australian School of Business Research Paper ACTL05 (2008; doi:10.2139/ssrn.1139724)]. Moreover, considering the data subdivided by set by single years of age, the correlations between the residuals for adjacent age groups tend to be high (as noted in [F. T. Denton, C. H. Feaver and B. G. Spencer, “Time series analysis and stochastic forecasting: an econometric study of mortality and life expectancy”, J. Popul. Econ. 18, No. 2, 203–227 (2005; doi:10.1007/s00148-005-0229-2)]. This suggests that there is value in exploring the dependence structure, also across time, in other words the inter-period correlation. The aim of this paper is to improve the methodology for forecasting mortality in order to enhance model performance and increase forecasting power by capturing the dependence structure of neighboring observations in the population. To do this, we adapt the methodology for measuring uncertainty in projections in the Lee-Carter context and introduce a tailor-made bootstrap instead of an ordinary bootstrap. The approach is illustrated with an empirical example.

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91D20 | Mathematical geography and demography |

### Citations:

Zbl 1277.91073
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\textit{V. D'Amato} et al., Insur. Math. Econ. 51, No. 3, 694--701 (2012; Zbl 1285.91054)

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### References:

[1] | Alonso, A.M.; Pena, D.; Romo, J., Forecasting time series with sieve bootstrap, Journal of statistical planning and inference, 100, 1, 1-11, (2002) · Zbl 1007.62077 |

[2] | Amemiya, T., Generalized least square with an estimated auto-covariance matrix, Econometrica, 41, 723-732, (1973) · Zbl 0305.62046 |

[3] | Barrieu, P.; Bensusan, H.; El Karoui, N.; Hillairet, C.; Loisel, S.; Ravanelli, C.; Salhi, Y., Understanding, modelling and managing longevity risk: key issues and main challenges, Scandinavian actuarial journal, 2010, 1-29, (2011), iFirst Article |

[4] | Booth, H.; Maindonald, J.; Smith, L., Applying lee – carter under conditions of variable mortality decline, Population studies: A journal of demography, 56, 3, 325-336, (2002) |

[5] | Bose, A., Politis, D.N., 1993. A review of the Bootstrap for dependent samples, Department of Statistics Purdue University Technical Report, n. 93-4, pp. 1-19. |

[6] | Brouhns, N.; Denuit, M.; van Keilegom, I., Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scandinavian actuarial journal, 3, 212-224, (2005) · Zbl 1092.91038 |

[7] | Bühlmann, P., Sieve bootstrap for time series, Bernoulli, 3, 123-148, (1997) · Zbl 0874.62102 |

[8] | Butt, Z., Haberman, S., 2009. ilc: a collection of \(r\) functions for fitting a class of Lee-Carter mortality models using iterative fitting algorithms. Actuarial Research Paper, n. 190. |

[9] | Chang, Y.; Park, J.Y., A sieve bootstrap for the test of a unit root, Journal of time series analysis, 24, 379-400, (2003) · Zbl 1036.62070 |

[10] | Choi, E.; Hall, P., Bootstrap confidence regions computed from autoregressions of arbitrary order, Biometrika, 62, 461-477, (2000) · Zbl 0966.62027 |

[11] | Coughlan, G., 2011. Longevity as an asset class, Presentation at Longevity 7: Seventh International Longevity Risk and Capital Markets Solutions Conference. |

[12] | D’Amato, V.; Haberman, S.; Russolillo, M., The stratified sampling bootstrap: an algorithm for measuring the uncertainty in forecast mortality rates in the Poisson lee – carter setting, Methodology and computing in applied probability, 14, 1, 135-148, (2012) · Zbl 1362.62182 |

[13] | D’Amato, V.; Di Lorenzo, E.; Haberman, S.; Russolillo, M.; Sibillo, M., The Poisson log-bilinear lee – carter model: applications of efficient bootstrap methods to annuity analyses, North American actuarial journal, 15, 2, 315-333, (2011) |

[14] | Denton, F.T; Feaver, C.H.; Spencer, B.G., Time series analysis and stochastic forecasting: an econometric study of mortality and life expectancy, Journal of population economics, 18, 203-227, (2005) |

[15] | Efron, B.; Tibshirani, R.J., An introduction to the bootstrap, (1997), Chapman & Hall |

[16] | Fries, J.F., Aging, natural death and compression of morbidity, New england journal of medicine, 1980, 303, 130-135, (1980) |

[17] | Hardle, W.; Horowitz, J.; Kreiss, J.P., Bootstrap methods for time series, International statistical review, 71, 435-459, (2003) · Zbl 1114.62347 |

[18] | Hyndman, R.J.; Ullah, S., Robust forecasting of mortality and fertility rates: a functional data approach, Computational statistics & data analysis, 51, 10, 4942-4956, (2007) · Zbl 1162.62434 |

[19] | Koissi, M.C.; Shapiro, A.F.; Hognas, G., Evaluating and extending the lee – carter model for mortality forecasting: bootstrap confidence interval, Insurance: mathematics and economics, 26, 1-20, (2006) · Zbl 1098.62138 |

[20] | Kreiss, J., Bootstrap procedures for AR(1)-process, (), 107-113 |

[21] | Kunsch, H.R., The jackknife and the bootstrap for general stationary observations, Annals of statistics, 17, 3, 1217-1241, (1989) · Zbl 0684.62035 |

[22] | Lahiri, S.N., Resampling methods for dependent data, (2003), Springer · Zbl 1028.62002 |

[23] | Lee, R.D.; Carter, L.R., Modelling and forecasting US mortality, Journal of the American statistical association, 87, 659-671, (1992) |

[24] | Liu, X.; Braun, W.J., Investigating mortality uncertainty using the block bootstrap, Journal of probability and statistics, 2010, 15, (2010), Article ID 813583 |

[25] | LLMA, 2010. Longevity pricing framework, A framework for pricing longevity exposures developed by the LLMA (Life & Longevity Markets Association). |

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

[27] | Renshaw, A.E.; Haberman, S., Lee – carter mortality forecasting: a parallel generalised linear modelling approach for england and wales mortality projections, Applied statistics, 52, 119-137, (2003) · Zbl 1111.62359 |

[28] | Renshaw, A.E.; Haberman, S., On the forecasting of mortality reduction factors, Insurance: mathematics and economics, 32, 379-401, (2003) · Zbl 1025.62041 |

[29] | Renshaw, A.E.; Haberman, S., Lee – carter mortality forecasting with age specific enhancement, Insurance: mathematics and economics, 33, 255-272, (2003) · Zbl 1103.91371 |

[30] | Renshaw, A.E.; Haberman, S., A cohort-based extension to the lee – carter model for mortality reduction factors, Insurance: mathematics and economics, 38, 556-570, (2006) · Zbl 1168.91418 |

[31] | Renshaw, A.E.; Haberman, S., On simulation-based approaches to risk measurement in mortality with specific reference to Poisson lee – carter modelling, Insurance: mathematics and economics, 42, 797-816, (2008) · Zbl 1152.91598 |

[32] | Wills, S., Sherris, M., 2008. Integrating financial and demographic longevity risk models: an Australian model for financial applications. Pension Institute Discussion Paper, 0817, 1, p. 12. |

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