Multi-Population Mortality Modeling

Definition

Multi-population mortality modeling refers to stochastic frameworks that jointly forecast age-specific death rates for two or more related populations, enforcing the biological constraint that mortality in closely related groups should not diverge indefinitely. The central challenge is specifying the long-run equilibrium relationship between populations — and doing so without requiring the user to subjectively nominate which population "dominates" the others. The leading approach is the vector error correction model (VECM), which grounds non-divergence in the cointegration structure of the Lee-Carter time-varying factors and explicitly models the error-correction dynamics that pull deviating populations back toward long-run equilibrium.

Key Ideas

Why joint modeling is necessary: Mortality improvements in related populations — different genders of the same country, general population vs. insured lives, one country vs. another — are correlated. Separate single-population Lee-Carter models applied to each group produce forecasts that may diverge without bound, which is biologically implausible and creates spurious pricing differences in longevity-linked instruments.
The random-walk-plus-AR(1) (RWAR) baseline (Cairns et al. 2011): The dominant-population approach models the larger population's $\kappa$ as a random walk and the spread ( $\kappa^{(1)} - \kappa^{(2)}$ ) as a mean-reverting $\text{AR}(1)$ . Non-divergence is satisfied, but the model is asymmetric: it requires declaring one population dominant. Swapping the assumed dominant population can change 50-year forecasts by $\approx 17\%$ , revealing high sensitivity to an unverifiable assumption.
Dominance assumption often wrong: Zhou et al. (2014) demonstrate with English/Welsh males ( $4.7$ million) and UK insured lives ( $260{,}000$ ) that the smaller population leads the larger — cross-correlations show $\kappa^{(1)}$ is significantly correlated with lagged $\kappa^{(2)}$ at lags $-3$ and $-13$ , meaning insured-lives improvements today predict general-population improvements years later, not vice versa.
Vector autoregression (VAR) extension: Models $(\Delta\kappa^{(1)}, \Delta\kappa^{(2)})$ jointly with a symmetric vector autoregression; no dominant population assumed; captures cross-correlations. Non-divergence requires constraining the long-run drift to be equal across populations. Better than RWAR, but does not model the long-run equilibrium level explicitly.
VECM extension: Adds an error-correction term — the deviation of $\kappa^{(1)} - \kappa^{(2)}$ from its long-run mean — to the VAR. Grounded in Engle-Granger cointegration theory: if $\kappa^{(1)}$ and $\kappa^{(2)}$ are cointegrated (share a stochastic trend), a VECM representation exists that automatically enforces non-divergence. Adjustment coefficients $\alpha^{(1)}$ and $\alpha^{(2)}$ govern the speed at which each population's $\kappa$ responds to disequilibrium. The VECM interpretation is more intuitive: when population 2's mortality is temporarily too low relative to population 1, positive pressure pulls it back toward equilibrium. VECM wins on Bayesian Information Criterion (BIC), residual adequacy, in-sample coverage, out-of-sample tracking, and robustness in the Zhou et al. (2014) empirical evaluation.
Longevity basis risk: The financial risk arising from the mismatch between a hedging instrument calibrated to one population (e.g., a general population mortality index) and the actual mortality of the portfolio being hedged (e.g., an insurer's book of lives). Multi-population models are the tool for measuring basis risk: the difference between the mortality forecast for the hedged portfolio and the forecast for the index population. Basis risk is primarily a risk-margin (capital) issue rather than a best-estimate pricing issue — switching from single-pop to multi-pop models shifts best-estimate annuity values by $<3\%$ but shifts the Solvency II risk margin ( $99.5$ th-percentile capital requirement) by $+9\%$ to $+47\%$ , depending on the model chosen.
Cointegration and the biological constraint: The non-divergence requirement is a biological judgment — related human populations cannot permanently separate in mortality. In econometric terms, this is the cointegration constraint: $\kappa^{(1)}$ and $\kappa^{(2)}$ share a common stochastic trend and their difference is stationary. The VECM encodes this constraint as an identifying structural restriction rather than an ad hoc assumption about dominance.
Non-convergence as empirical norm (Hahn and Christiansen 2016): European national mortality rates do not converge; discrepancies between countries are stable or widening over observed history. The Li-Lee (2005) common-factor model assumes long-run convergence to a reference population and fails for heterogeneous country groupings (Hungary, Bulgaria, Russia cannot be jointly fit). The Bayesian Multi-Population Mortality Projection (BMPMP) framework accommodates non-convergence by nesting populations in a reference pool p* rather than requiring all populations to converge to a single dominant reference.
BMPMP model (Hahn and Christiansen 2016): Two-stage Bayesian Multi-Population Mortality Projection. Stage 1 — Cairns-Blake-Dowd (CBD) logit mortality layer: $\text{logit}(q_{x,t}) = \kappa^0_t + \kappa^x_t(x - x_0) + \kappa^{x^2}_t(x - x_0)^2$ for ages $40$ – $100$ per population (CBD is inapplicable outside this range). Stage 2 — VECM on the stacked CBD parameter vectors $(\kappa^0, \kappa^x, \kappa^{x^2})$ across populations, with cointegration rank $r$ and lag order $k$ as user-selected controls. Estimated via Bayesian Markov Chain Monte Carlo (MCMC): Gibbs sampler for VECM hyperparameters $H$ ; Metropolis-Hastings for latent CBD parameters $K$ . Key sensitivity finding: cointegration rank $r$ is critical ( $r=0$ fails residual adequacy; $r=5$ in main study); lag order $k$ is irrelevant ( $k=1$ and $k=2$ produce nearly identical forecasts).
Left-skewed predictive distributions: Higher cointegration rank produces left-skewed posterior predictive mortality distributions — more probability mass for extreme improvement than deterioration. This asymmetry reflects the empirical pattern of accelerating improvement in tightly cointegrated mortality systems and has direct implications for longevity risk capital requirements: the $99.5$ th-percentile tail is heavier on the improvement side than symmetric models imply.

How It Works

The VECM for Two Populations

Working within the Lee-Carter structure $\ln m_{x,t}^{(i)} = \alpha_x^{(i)} + \beta_x^{(i)} \kappa_t^{(i)}$ , with the constraint $\beta_x^{(1)} = \beta_x^{(2)}$ to ensure non-divergence, the VECM for the first differences $\Delta\kappa^{(1)}$ and $\Delta\kappa^{(2)}$ is:

$\Delta\kappa_t^{(1)} = \delta_0^{(1)} + \alpha^{(1)}(\kappa_{t-1}^{(1)} - \kappa_{t-1}^{(2)}) + \phi_1^{(1)}\Delta\kappa_{t-1}^{(1)} + \phi_2^{(1)}\Delta\kappa_{t-1}^{(2)} + \epsilon_t^{(1)}$ $\Delta\kappa_t^{(2)} = \delta_0^{(2)} + \alpha^{(2)}(\kappa_{t-1}^{(1)} - \kappa_{t-1}^{(2)}) + \phi_1^{(2)}\Delta\kappa_{t-1}^{(1)} + \phi_2^{(2)}\Delta\kappa_{t-1}^{(2)} + \epsilon_t^{(2)}$

The error-correction term $\kappa_{t-1}^{(1)} - \kappa_{t-1}^{(2)}$ is the equilibrium deviation. When it is positive (population 1 has improved more than population 2), $\alpha^{(1)} < 0$ and $\alpha^{(2)} > 0$ pull the system back. As $t \to \infty$ , the expected difference between $\kappa^{(1)}$ and $\kappa^{(2)}$ is a constant, satisfying non-divergence.

Model Comparison (England-Wales / UK Insured Lives, 1961–2005)

Model	BIC	In-sample $95\%$ confidence interval (CI) coverage	Dominant-pop sensitivity
RWAR	$179.2$	Fails after 2000	$17\%$ swing from swapping
VAR	$176.4$	Adequate	None (symmetric)
VECM	$174.6$	Full coverage	None (symmetric)

Why It Matters

DI beneficiary mortality: The general-population vs. Disability Insurance (DI)-beneficiary mortality gap (documented in Meseguer 2021) is exactly a two-population modeling problem. Single-population forecasts applied to DI beneficiaries may substantially understate the mortality gap if the subpopulation's trend diverges from the general population. Multi-population models with a VECM structure are the appropriate tool for measuring this basis risk. See DI Beneficiary Mortality.
Longevity hedging market: Pension schemes and insurers increasingly hedge longevity risk using instruments tied to general population mortality indices. The pricing of basis risk — how much extra capital a reinsurer must hold for writing such contracts — depends critically on the multi-population model used. Single-population models understate uncertainty; RWAR overstates it relative to VECM.
Solvency II / regulatory capital: The risk margin ( $6\%$ of the present value of future Solvency Capital Requirements (SCRs)) is the operative capital metric for longevity risk under European insurance regulation. Choosing RWAR vs. VECM changes the risk margin for a longevity swap by up to $20$ percentage points — a material difference for capital planning.
Methodological generality: The VECM approach applies to any Lee-Carter-style model (M1 through M7 in the Cairns taxonomy), not just the basic Lee-Carter (LC) structure. It can also accommodate more than two populations by extending to higher-dimensional cointegrated systems.

Open Questions

All current VECM implementations assume mean-reversion begins at the forecast origin. If socioeconomic mortality differentials continue widening (as Waldron 2007 and Mackenbach et al. 2003 document), a model that permits delayed or uncertain mean-reversion onset would be more realistic.
How should cointegration rank $r$ be selected? Hahn and Christiansen (2016) show $r$ is the dominant sensitivity parameter in BMPMP, but they select it by residual adequacy and expert judgment rather than a formal information criterion — a principled selection procedure for $r$ remains an open problem.
Does the VECM approach extend naturally to cause-specific mortality, where cointegration relationships may differ by cause?
The CBD framework underlying BMPMP is restricted to ages $40$ – $100$ . Extending multi-population Bayesian MCMC mortality projection to younger ages (e.g., using a Lee-Carter or Makeham layer for ages $0$ – $40$ ) without introducing separate calibration discontinuities is an unresolved methodological gap.