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Constructing dynamic life tables with a single-factor model

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Abstract

The paper deals with the mortality risk evolution and presents a one-factor model explaining the dynamics of all mortality rates. The selected factor will be the mortality rate at the key age, and an empirical study involving males and females in France and Spain reveals that the present approach is not outperformed by more complex factor models. The key age seems to reflect several advantages with respect to other factors available in the literature. Actually, it is totally observable, and the methodology may be easily extended so as to incorporate more factors (more key ages), a cohort effect, specific mortality causes or specific ages. Furthermore, the choice of a key age as an explanatory factor is inspired by former studies about the dynamics of interest rates which allows us to draw on the model in order to address some longevity risk-linked problems. Indeed, one only has to slightly modify some interest rate-linked methodologies. Illustrative examples will be given.

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Notes

  1. In the line of the Capital Asset Pricing Model, for instance.

  2. V@R and CV@R are very important risk measures in the insurance industry. Both regulatory (Solvency II) and internal (corporate) risk management constrains often impose the use of V@R and CV@R. See for instances Hardy (2003).

  3. The model can be easily generalized for two or more factors, in order to yield multifactor life table models.

  4. We decide to implement the model with logit mortality rates, \(\text {logit}\left( q_{x,t}\right) =\text {ln}\left( \frac{q_{x,t}}{1-q_{x,t}}\right) \). Furthermore, we use logit mortality rates to avoid estimations of \(m_{x,t}\) greater than one (Lee 2000) and also, the log-odds function is also chosen because of the historical ties with the early actuarial work of Perks (1932). The model could be implemented using logarithm death rates, \(\text {log}\left( m_{x,t}\right) \), or logarithm mortality rates \(\text {log}\left( q_{x,t}\right) \) with similar results.

  5. See for more details Navarro and Nave (2001).

  6. An alternative approach to determine the key age \(x^*\) would be based on maximizing the likelihood function or its logarithm. In any case, under the hypothesis of normality and homoscedasticity, the estimates of \(\alpha \) and b not change. We eventually decided to follow the original paper of Elton et al. (1990).

  7. We have not adjusted any function to describe the behavior of \({\hat{\alpha }}_{x,x^{*}}\), although this could be easily done.

  8. See, for instance, Keele (2008) for further details about these functions and their properties.

  9. Following the methodology of Debón et al. (2010) to estimate the parameters.

  10. This fact has been highlighted in Felipe et al. (2002), Guillen and Vidiella-i Anguera (2005) and Debón et al. (2008) where it is documented the particularly acute impact of this disease in the Spanish population in comparison with other European countries.

  11. Other types of splines could be applied. For instance, penalized splines (see Eilers and Marx 1996) have been used for smoothing and forecasting mortality rates as in Currie et al. (2004b).

  12. This is the reason why Spain displays a change in the shape of mortality trend, as shown in the papers of Felipe et al. (2002) Guillen and Vidiella-i Anguera (2005) and Debón et al. (2008). In fact, this epidemic (AIDS) covers the range of ages between 20 and 40 and the peak was reached at early nineties. Meanwhile, the number of deaths for France population caused by AIDS in these period was by far less intense (Casabona et al. 2001).

  13. In the SFM, if we followed the same line of reasoning as in Girosi and King (2007), the innovations in logit mortality rates would have two sources: the random term of the key mortality rate and the error term of (1). The resulting structure of the innovations if we do not assume the random walk hypothesis but other ARIMA processes would be by far more complex than those described in Girosi and King (2007) and would be out of the scope of this paper, although it opens a very interesting line of research.

  14. See for instance, Brouhns et al. (2002), Renshaw and Haberman (2006) and Cairns et al. (2009).

  15. According to Lee (2000), the logit version avoids estimates of \(m_{x,t}\) greater than one.

  16. Following Debón et al. (2008), where the authors compared the calibration of Lee–Carter model to the Spanish mortality data applying different methods: SVD, maximum likelihood and GLM, the latter providing the best outcomes.

  17. This library provides a tool for fitting stochastic mortality models.

  18. See, for instance, Haberman and Renshaw (2009).

  19. It is possible to obtain a different tendency to forecast the mortality rates for specific range of ages, see for details Giordano et al. (2019).

  20. See Hyndman and Khandakar (2008) for further details.

  21. Other comparison criteria can be seen in Cairns et al. (2009), Plat (2009) and Haberman and Renshaw (2011).

  22. All the errors for each model explained in Sect. 3 are calculated similarly to (26) and (27).

  23. Recall that we have estimated the logit version of each model as explained in Sect. 3.

  24. In this case, we apply the Parametric Adjustment for \({\hat{b}}_{x,x*}\) as explained in Sect. 2.2.1.

  25. In this case, we apply Spline Adjustment for \({\hat{b}}_{x,x*}\) as explained in Sect. 2.2.2.

  26. See, for instances Chen et al. (2009).

  27. See, for instance Booth et al. (2006).

  28. See, for instance Felipe et al. (2002), Debón et al. (2010), D’Amato et al. (2012) and Wang et al. (2018).

  29. The approach of \({\hat{b}}_{x,x*}\) which produces the best result for SSE, MAE and MSE is the parametric approach for Spanish male and female. In the French case, they are the parametric adjustment for male and the spline adjustment for female population. Anyway, the rest of alternatives produce very similar outcomes.

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Acknowledgements

The authors are indebted to the anonymous referees whose suggestions improved the original manuscript. The authors are indebted to Ana Debón, María Durbán José Garrido, Pietro Millossovich, Arnold Séverine and Carlos Vidal for their suggestions. This work was partially supported by a grant from MEyC (Ministerio de Economía y Competitividad, Spain Project ECO2017-89715-P). The research by David Atance has been supported by a grant from the University of Alcalá.

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  1. Departamento de Economía y Dirección de Empresas, Universidad de Alcalá, Madrid, Spain

    David Atance & Eliseo Navarro

  2. Departamento de Economía de la Empresa, Universidad Carlos III de Madrid, Madrid, Spain

    Alejandro Balbás

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  1. David Atance
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Atance, D., Balbás, A. & Navarro, E. Constructing dynamic life tables with a single-factor model. Decisions Econ Finan 43, 787–825 (2020). https://doi.org/10.1007/s10203-020-00308-5

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