Mortality Improvement Rates: Modeling, Parameter Uncertainty, and Robustness

Abstract

Rather than looking at mortality rates directly, a number of recent academic studies have looked at modeling rates of improvement in mortality when making mortality projections. Although relatively new in the academic literature, the use of mortality improvement rates has a long-standing tradition in actuarial practice when allowing for improvements in mortality from standard mortality tables. However, mortality improvement rates are difficult to estimate robustly, and models of them are subject to high levels of parameter uncertainty, because they are derived by dividing one uncertain quantity by another. Despite this, the studies of mortality improvement rates to date have not investigated parameter uncertainty due to the ad hoc methods used to fit the models to historical data. In this study, we adapt the Poisson model for the numbers of deaths at each age and year proposed in Brouhns, Denuit, and Vermunt to model mortality improvement rates. This enables models of improvement rates to be fitted using standard maximum likelihood techniques and allows parameter uncertainty to be investigated using a standard bootstrapping approach. We illustrate the proposed modeling approach using data for the England and Wales population. The methods used in this article are available in the R package iMoMo.

ACKNOWLEDGMENTS

A previous version of this article was presented at the 2017 International Symposium “Living to 100” organized by the Society of Actuaries in Orlando, FL. We thank Ward Kingkade and the other participants of the symposium for useful comments that have helped improve this article. We also thank the two anonymous referees for their very constructive comments.

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Notes

1 These age functions can be nonparametric (having form determined entirely by the data) or parametric (having a predefined functional form), as discussed in Hunt and Blake (2021a).

2 One drawback of using a Poisson model for the death counts, common to both models of mortality rates and improvement rates, is that it assumes that the variance of an observation is equal to its expectation. Such over dispersion can be dealt with by using an over-dispersed Poisson model in a generalized nonlinear modeling framework or by allowing for heterogeneity in the population via the use of the negative binomial distribution, such as in Delwarde, Denuit, and Partrat (2007), Dodd et al. (2021), and Li, Hardy, and Tan (2009). However, we do not investigate this further in this study.

3 In the case of Mitchell et al. (2013), least squares estimation was used to fit the improvement rates, whereas in Haberman and Renshaw (2012), an iterated generalized linear model procedure was used to allow for overdispersion in the observed improvement rates. However, in neither case were these distributions selected on the basis of providing an appropriate distribution for the observed death counts. Consequently, this means that many common methods for assessing parameter uncertainty are not appropriate, as discussed in Subsection 3.4.

4 Equation (12)(12) $$D_{x,t}\sim \mathit{Poisson}\left(e_{x,t}{\widehat{m}}_{x,t-1}\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\eta }_{x,t})\right)$$(12) follows from noting that under the Mitchell et al. (2013) form of an improvement rate model the expected number of deaths at age x in year t is $e_{x,t}{\widehat{m}}_{x,t-1}\hspace{0.17em}\text{exp}\hspace{0.17em}(-{\eta }_{x,t})$.

5 Allowing for similar issues as described in Hunt and Blake (2020a,b) in order to obtain “well-identified” projections that do not depend on the arbitrary identifiability constraints chosen when fitting the model.

6 Demographic significance is defined in Hunt and Blake (2021a) as the interpretation of the components of a model in terms of the underlying biological, medical, or socio-economic causes of changes in mortality rates that generate them.

7 We note that under the indirect approach one has the option of using the fitted mortality rates $m_{x,T}$ as the starting point (or jumping-off rates) for future mortality projections instead of the observed mortality rates ${\widehat{m}}_{x,T}.$ This is akin to the use of actual or fitted jumping-off rates as discussed in Booth, Hyndman, and Tickle (2006).

8 In their mortality rate forms, there are interesting connections between the primary period index, K_t, in models with and without constant improvement rates, with the period indexes of models with constant improvement rates behaving very similarly to the de-trended period index of the equivalent model without constant improvement rates. We refer the reader to Callot, Haldrup, and Kallestrup-Lamb (2016) for a very detailed discussion of this in the context of the LC and LC-CI models.

9 The calculation of AIC and BIC is given by $\mathit{AIC}=2\nu -2\mathcal{L}$ and $\mathit{BIC}=\nu \hspace{0.17em}\text{log}\hspace{0.17em}M-2\mathcal{L},$ respectively, with a lower value of AIC and BIC being preferable and M denoting the number of observations in the data.

Additional information

Funding

This project has received funding from the ARC Center of Excellence in Population Ageing Research (Grant CE170100005) and from the Society of Actuaries Center of Actuarial Excellence Research Grant 2017–2020: Longevity Risk: Actuarial and Predictive Models, Retirement Product Innovation, and Risk Management Strategies.