Zhou et al. — Modeling Mortality of Multiple Populations with Vector Error Correction Models

Summary

Zhou et al. extend the two-population mortality modeling framework of Cairns et al. (2011) by replacing the asymmetric RWAR process (random walk for dominant population + AR on spread) with two symmetric alternatives — a vector autoregression (VAR) and a vector error correction model (VECM) — that require no subjective assumption about which population is dominant. Applied to English and Welsh males from the Human Mortality Database (HMD) and UK insured lives from the Continuous Mortality Investigation (CMI) Bureau over ages 60–84 from 1961–2005, VECM outperforms VAR and RWAR on goodness-of-fit measured by the Bayesian Information Criterion (BIC), residual adequacy, in-sample coverage, out-of-sample tracking, and robustness to sample window. A Solvency II application shows that switching from a single-population model to a multi-population model has negligible impact on best-estimate annuity values (<3%) but large impact on the risk margin (the regulatory capital requirement for longevity risk), with RWAR producing a 47% risk-margin increase and VECM a 27% increase versus single-population.

Key Claims

• Dominance assumption is not always justifiable: In the England-Wales / UK insured-lives pair, the smaller insured-lives population (260k exposures) leads the larger general population (4.7M exposures) — cross-correlation analysis shows $\kappa^{(1)}$ (the Lee-Carter period mortality index for population 1) is significantly correlated with $\kappa^{(2)}$ lagged −3 and −13 years. Assuming England-Wales is dominant (as Cairns et al. do) gives the wrong causal direction.
• Swapping the assumed dominant population changes RWAR forecasts by 17%: The 50-year forecast of $\kappa$ for England-Wales males changes from −21.9 to −22.3 depending on which population is declared dominant — a visible instability that VAR and VECM avoid entirely by construction.
• VECM best by all criteria: BIC = 174.6 (VECM) vs. 176.4 (VAR) vs. 179.2 (RWAR). Standardized residuals from RWAR and VAR show significant downward trends; VECM residuals are trend-free. In-sample: VECM's 95% confidence interval (CI) covers all true 1991–2005 values; RWAR fails after 2000. Out-of-sample: VECM tracks recent acceleration in $\kappa$ decline most closely. Robustness: VECM forecasts change least across 45-year vs. 30-year windows and across three rolling recalibration periods.
• Longevity basis risk is a risk-margin issue, not a best-estimate issue: Best-estimate annuity values shift <3% from single-pop to multi-pop (RWAR +0.23%, VAR +0.85%, VECM +3.16%). But risk margins shift dramatically: RWAR +47%, VECM +27%, VAR +9% over single-population. The uncertainty about subpopulation divergence is captured in the tail (99.5th-percentile scenario), not the mean.
• Model choice matters for capital: RWAR, VAR, and VECM produce substantially different risk margins for identical contracts. Practitioners must validate their process choice — not just their point estimates.
• Known limitation: All three models assume mean-reversion of the $\kappa$ spread begins immediately at the forecast origin. Recent evidence of widening socioeconomic mortality differentials (Waldron 2007; Mackenbach et al. 2003) calls this assumption into question. Future work might permit mean-reversion to begin at a random future date.

Concepts Introduced or Extended

• Multi-Population Mortality Modeling
• Lee-Carter Model
• SSA Mortality Forecasting

Entities Mentioned

• Johnny S.-H. Li
• Ken Seng Tan
• Hilary Waldron (Waldron 2007 cited for widening socioeconomic mortality differentials)

My Take

The VECM framing is elegant and econometrically principled — Granger's Representation Theorem guarantees the non-divergence property automatically whenever the two $\kappa$ series are cointegrated. The dominance-assumption critique is empirically decisive: the cross-correlation finding is easy to check and the 17% forecast sensitivity to population labeling is a clear indictment of the RWAR approach. For this wiki's focus, the most important implications are: (1) subpopulation mortality modeling (e.g., Disability Insurance (DI) beneficiaries vs. general population) should account for potential lead-lag relationships rather than simply assuming the larger population drives the smaller; (2) longevity basis risk is primarily a capital/tail-risk issue, not a pricing issue — a distinction directly relevant to the Social Security Administration's (SSA) approach to modeling beneficiary mortality.