Cairns-Blake-Dowd Model

Definition

The Cairns-Blake-Dowd (CBD) model (Cairns, Blake, and Dowd 2006) is a stochastic mortality model designed for higher (pensioner and annuitant) ages, in which the logit of the one-year probability of death is expressed as a low-order polynomial in age with two or more period factors that evolve as a multivariate random walk with drift. It contrasts with the Lee-Carter Model, which decomposes the log central death rate using a single age-modulated period factor. By giving mortality both a level factor and an age-slope factor, CBD lets the age gradient of mortality improvement change over time — something the single-factor Lee-Carter structure cannot represent — at the cost of being applicable only over the roughly linear-in-logit adult age range (about ages 40–100).

Key Ideas

Logit-mortality scale: CBD models $\text{logit}\, q_{x,t} = \ln\!\big(q_{x,t}/(1-q_{x,t})\big)$ rather than the log death rate. At older ages the logit of $q_{x,t}$ is close to linear in age, which motivates the parametric age structure.
Two-factor original (M5): $\text{logit}\, q_{x,t} = \kappa_t^{(1)} + \kappa_t^{(2)}(x-\bar{x})$ , where $\bar{x}$ is the mean age in the fitting range. $\kappa_t^{(1)}$ is the level (overall mortality) and $\kappa_t^{(2)}$ is the slope (age gradient). A rising $\kappa_t^{(2)}$ means mortality is falling faster at the younger old-ages than at the oldest ages.
No age-response estimation: unlike Lee-Carter's $b_x$ , the age dependence is imposed parametrically (linear, or linear-plus-quadratic), so there are no age-specific sensitivity parameters to estimate — only the period (and optional cohort) factors.
The M1–M8 taxonomy: Cairns et al. (2009) catalogue a family of related models. M1 is Lee-Carter; M5 is the original two-factor CBD; M6 and M7 add a cohort (year-of-birth) effect, and M7 additionally adds a quadratic age term. The Bayesian multi-population layer in Multi-Population Mortality Modeling uses the quadratic M7-type form $\text{logit}(q_{x,t}) = \kappa_t^{0} + \kappa_t^{x}(x-x_0) + \kappa_t^{x^2}(x-x_0)^2$ .
Cohort effects: UK data in particular show strong year-of-birth patterns; the M6/M7 cohort term $\gamma_{t-x}$ captures generations (e.g., the UK "golden cohort") that experienced unusually fast mortality improvement.
Two-dimensional uncertainty: because the period factor is a vector, CBD produces a genuinely two-dimensional fan of forecasts — uncertainty in both the level and the slope — avoiding the degenerate single-factor uncertainty of basic Lee-Carter.

How It Works

Fit period factors year by year. For each calendar year $t$ , regress $\text{logit}\, q_{x,t}$ on age across the fitting range; the intercept estimates $\kappa_t^{(1)}$ and the slope estimates $\kappa_t^{(2)}$ (M5). Cohort and quadratic terms are added analogously in M6/M7.
Model the factors as a random walk with drift. The factor vector follows $\boldsymbol{\kappa}_t = \boldsymbol{\kappa}_{t-1} + \boldsymbol{\mu} + C\boldsymbol{\epsilon}_t$ — a multivariate random walk with drift $\boldsymbol{\mu}$ and innovation covariance $CC'$ . Any cohort process is modelled separately (often an ARIMA).
Forecast and invert. Simulate the factor process forward, reconstruct $\text{logit}\, q_{x,t}$ , invert the logit to recover $q_{x,t}$ , and build the projected life table. Repeating the simulation yields the predictive distribution used for longevity-risk capital.

Because the age structure is parametric, CBD is parsimonious and numerically stable at high ages, but it deliberately does not fit younger ages, where the logit of mortality is far from linear in age.

Why It Matters

Pension and annuity longevity risk: CBD was built for the ages that dominate pension and annuity liabilities, and is a workhorse of actuarial longevity modelling and Solvency II capital assessment.
Building block for multi-population models: the Multi-Population Mortality Modeling BMPMP framework (Hahn and Christiansen 2016) stacks CBD parameter vectors across populations and links them with a vector error correction model, using CBD as its Stage 1 mortality layer.
Complement to Lee-Carter: the two models occupy different niches — Lee-Carter for the full age range and aggregate $e_0$ work, CBD for high-age annuitant mortality with explicit slope and cohort dynamics. See SSA Mortality Forecasting and Period Mortality.
Model risk is explicit: the M1–M8 menu makes the choice among specifications (cohort? quadratic?) a visible, comparable source of forecast uncertainty rather than a hidden assumption.

Open Questions

Which specification? Choosing among M5–M8 (whether to include a cohort term, a quadratic age term, or age-dependent cohort decay) can materially change tail forecasts; principled selection criteria remain debated.
Cohort-effect identification: cohort terms are statistically fragile and can be confounded with period and age effects.
Age-range restriction: CBD is inapplicable below roughly age 40, so combining it with a separate model for younger ages without calibration discontinuities is unresolved (see Multi-Population Mortality Modeling).