An Analysis of Period and Cohort Mortality Shocks in International Data

Abstract

This article uses Bayesian maximum a posteriori (BMAP) estimation to fit a cohort-based mortality model that applies the Gompertz mortality law to fixed cohorts across different periods (rather than the more usual application to fixed periods across different cohorts). Period effects are then estimated as residuals. In this approach, cohort effects can be viewed as a proxy for causes of death with long latency, which have become relatively more important in recent decades in richer countries. We estimate the model independently using male and female adult population mortality data in 31 countries. We are able to associate historical events with many of the observed period and cohort shocks, most notably the 1918 flu epidemic, and find striking geographical and cultural correlations in the results. We find that after 1960, the variance of period mortality shocks has declined by an average factor of 5 in most of the countries we examine. Over the same period, cohort shocks appear to have become a more important factor causing changes in mortality than period shocks. We also find that period and cohort shocks appear to be driven by different underlying factors. Our results have important implications for stochastic mortality modeling, and may explain why stochastic mortality models that rely largely on period mortality shocks struggle to generate sufficient variation in mortality rates. Our results will also be useful to those who construct reinsurance portfolios, those who issue or trade longevity-linked securities, and those who study the origins of human mortality.

ACKNOWLEDGMENTS

The authors thank David Blake, Andrew Cairns, James Sefton, Andrew Hunt, Jeff Mulholland, and participants at the 2017 Longevity 13 conference in Taiwan and the 2018 American Risk and Insurance Association meeting in Chicago, IL, for providing valuable feedback on earlier versions of the article. Comments made by an anonymous referee greatly assisted us in improving the article. All remaining errors are our own.

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Notes

1 Public expenditure on old-age pensions alone averaged 8.2% of gross domestic product (GDP) across the Organization for Economic Cooperation and Development (OECD), up from 5.7% in 1980 (OECD 2015). Private pension assets in 22 major markets have grown at a rate of nearly 9% p.a. in local-currency terms over the last 10 years, and now total $36.4 trillion (Willis Towers Watson 2017), representing 62% of GDP in those countries. Assets in defined benefit (DB) plans are held in respect of liabilities that are explicitly longevity linked; those in defined contribution (DC) plans match future consumption streams that are, too. And globally, life insurance made up around 14% of personal financial assets, a 5% compound annual growth rate since 2012 (McKinsey 2014).

2 Solvency II—the European prudential regulatory regime applied to insurance companies—explicitly requires insurers to hold a buffer in respect of longevity risk. And the recently approved IORP II—the regulation applied to European occupational pension schemes—includes language requiring occupational pension funds to be aware of the implications of “biometric” risks (which include longevity) and to hold capital in excess of technical provisions against them—even if the proposed holistic balance sheet approach, which would have gone further, was shelved. In the United States, the National Association of Insurance Commissioners (NAIC) is actively investigating the extent to which longevity risk should be recognized in statutory reserves and/or risk-based capital. (European Union 2015, 2016; NAIC 2017).

3 For instance, ESA 2010 includes for the first time in its core accounts all private-sector occupational pension liabilities, irrespective of funding status. The supplementary table on pensions will further present all accrued-to-date pension liabilities, including unfunded government occupational and other pension schemes, as well as social security pensions (see Eurostat 2016).

4 For instance, in the United States, deaths due to cancer, heart disease, and stroke were 2.5 times more common than deaths due to accidents and influenza and pneumonia in 1933. By the early 1970s, this ratio was higher than 9, and in 2014 was still higher than 7 (NCHS 2018).

5 Li and Lee (2005) first proposed the concept of mortality coherence. This was developed by Cairns et al. (2011a; 2011b) and Dowd et al. (2011). Most recently, Li et al. (2017) introduced a weakened form of coherence, which they call semicoherence.

6 All the countries in the HMD except ex-Soviet Union countries were included. Without extra basis functions, the cohort-based model does not fit well to those countries because the magnitude of the posttransition mortality shock (1) was extremely large and (2) affected mainly younger adults.

7 The data are described in Table 2, shown later.

8 Fay and Herriott (1979) set ${\epsilon }_i\sim N(0,{\sigma }_A^2+{\omega }_i).$ Our alternative formulation makes little difference to parameter estimates for cohort and period shocks. Our specification does, however, lead to a different interpretation of the coefficient ${\sigma }_A^2$—it is a ratio—and is more tolerant of model misfit where data are sparse. We show why we set ${\omega }_i=1/D_i$ in the Appendix.

9 See Table 2, columns 4 and 5, for $c_{\min }$ and $c_{\max }.$

10 A formula for the Fisher likelihood is given in the Appendix.

11 We ignore the average residual across both period and age, as this is very close to zero.

12 $x_{\min }$ is 35 and $x_{\max }$ is 100. See Table 2, columns 2 and 3, for $t_{\min }$ and $t_{\max }.$

13 In our results, the average adjusted R-squared is greater than 99%.

14 Details of the standardization are shown in the Appendix.

15 The fitted values and estimates for all countries are available from the authors upon request.

16 Cairns et al. (2016) attribute the effect in the U.K. data to errors in exposure related to the demobilization of armed forces in 1918 and the subsequent baby boom. This factor may also underlie the strong 1918 cohort effect measured for other European countries heavily affected by the conflict. We discuss this and other data errors in a subsequent section.

17 Estimating period shocks in the same way as we estimate cohort shocks, that is, by fitting AR(1) models to ${\left\{{\widehat{u}}_t^{\gamma }\right\}}_{k,l}$ and defining period mortality shocks as the residuals of these regressions, has little effect on the results shown in Tables 4 and 5.

18 The effect is not sensitive to the cutoff year.

19 Note that a high value of ${\beta }_{1,c}$ means that a cohort will have lower mortality when young, but equally higher mortality when old; ${\beta }_{1,c}$ has no effect on the average mortality level of a cohort, which is fully captured by ${\beta }_{0,c}.$

20 The power of this test is small because of small sample sizes; we can only include cohorts born between the first period of the sample and 18 years before the final period. Further, only nine countries have data for calendar year 1918.

21 Under the assumption of unchanging correlations between countries, all estimates will be unbiased, but those from country pairs that have longer time series in common will be more precise.

22 Single linkage clustering defines the distance between two clusters as the smallest pairwise distance between items in each cluster. The average linkage considers the average of the pairwise cluster distances as the distance between the two clusters. Complete linkage clustering considers the maximum of pairwise cluster distances as the cluster distance, and Ward's minimum variance method minimizes the total within-cluster variance.

23 In the data, northerly countries were associated with negative values of the second principal component and southerly countries with positive ones. To align the map with geographic convention, we plotted the negative of the second principal component.

24 Vaupel et al. (1979) suggest that curvature may be the consequence of within-cohort mortality heterogeneity, even if the biological rate of aging is constant in all individuals in the cohort, because frail individuals die first, meaning that the cohort is increasingly dominated by less frail individuals as it ages.