Zhou et al. (2013) Modeling Mortality of Multiple Populations with Vector Error Correction Models: Applications to Solvency II

Summary

Zhou et al. demonstrate that multi-population stochastic mortality models typically require a subjective and often unjustifiable assumption about which population is "dominant," and propose replacing the asymmetric Random Walk AutoRegression (RWAR) baseline (Cairns et al. 2011) with symmetric Vector Autoregression (VAR) and Vector Error Correction Model (VECM) specifications for the Lee-Carter time-varying mortality factors $\kappa_t^{(i)}$. Applied to English & Welsh (E&W) males and UK insured lives (1961–2005), VECM dominates on Bayesian Information Criterion (BIC), residual diagnostics, and all three robustness tests. A Solvency II application shows that the choice of multi-population model has only a small impact on best-estimate annuity values but a large impact (+9% to +47%) on the risk-adjusted price (risk margin).

Key Claims

• Dominance assumption is empirically unjustifiable. In the UK data, cross-correlations show the smaller population (insured lives) leads the larger (E&W males) at lags $-3$ and $-13$, contradicting the standard assumption.
• VAR is symmetric. Modeling $(\Delta\kappa_t^{(1)}, \Delta\kappa_t^{(2)})$ as a VAR(1) in differences avoids the dominance assumption. Non-divergence requires imposing $\phi_0/(1-\phi_1-\phi_2) = \theta_0/(1-\theta_1-\theta_2)$ as an explicit constraint.
• VECM is preferred. Adding the error-correction term $\rho^{(i)}(\kappa_{t-1}^{(1)} - \kappa_{t-1}^{(2)})$ enforces non-divergence automatically, provides explicit long-run equilibrium interpretation, and improves fit. Estimated adjustment speeds: $\rho^{(1)} = -0.208$, $\rho^{(2)} = -0.105$ — both populations correct toward equilibrium, but asymmetrically.
• BIC selects VECM. VECM (174.6) < VAR (176.4) < RWAR (179.2). RWAR and VAR residuals show significant downward trends; VECM residuals are flat.
• VECM is most robust. In all three robustness tests — look-back window, rolling sample period, swap of dominant population — VECM produces the smallest forecast deviations. Swapping the dominant population in RWAR changes the 50-year $\kappa$ forecast by ~17%; VAR and VECM are invariant by construction.
• Risk margin is model-sensitive; best estimate is not. Switching from single-population to multi-population changes best-estimate annuity values by only 0.2–3.2%, but risk margin by 9–47%. Model choice within the multi-population class matters: RWAR +47%, VECM +27%, VAR +9% above single-population base.
• Variance drives the risk margin gap. At $t = 2025$, $\sigma(\kappa_t^{(2)}) = 2.95$ (RWAR), $2.70$ (VECM), $2.16$ (VAR). The 99.5th-percentile Solvency Capital Requirement (SCR) is driven by tail volatility, so higher variance $\to$ higher risk margin regardless of model bias.

Concepts Introduced or Extended

• Lee-Carter Model — foundational stochastic mortality model; shared $\beta_x$ extension for two populations
• Longevity Basis Risk — risk from mortality divergence between general and sub-population; measured via Solvency II risk margin
• Cointegration — VECM applied to mortality factors; non-divergence as cointegration constraint; BIC model selection
• Vector Autoregression — VAR and VECM for joint $\kappa$ dynamics

Entities Mentioned

• Johnny S.-H. Li

Quotes

"As we are going to demonstrate in Section 2, this assumption is not always justifiable, and if the opposite assumption is used, the resulting projections can be very different."

"We note that our stochastic scenarios do not include parameter uncertainty or model risk, and therefore likely understate the risk margin. Nevertheless, the relative increase in risk margin already reflects the need to model the smaller population trend explicitly."

"RWAR, VAR and VECM yield quite different risk margins. This result suggests that practitioners should be careful in choosing a process of the time-varying factors in a two-population mortality model."

My Take

The paper's contribution to time-series methodology is modest — VECM for two I(1) series with one cointegrating vector is textbook Engle-Granger (1987). The novelty is the domain translation: showing that non-divergence in mortality modeling is precisely the cointegration hypothesis, and that the RWAR baseline is an asymmetric special case of VECM. The Solvency II application makes the stakes concrete: ignoring multi-population dynamics underestimates risk margin by a factor of 1.5–2x, and the specific model choice within the multi-population class shifts the risk margin by 40 percentage points. A limitation: parameter uncertainty and model risk are excluded from the stochastic scenarios, which the authors acknowledge would substantially raise all risk margins.