Risk Modelling and Survival Analysis · Mortality projection

Age-Period-Cohort and Cairns-Blake-Dowd Mortality Models

Updated 11 October 2026 · Fact-checked

Age-period-cohort (APC) models project mortality by adding a cohort term (year of birth) to age and period effects. The Cairns-Blake-Dowd (CBD) model fits logit q(x,t) as a linear function of age, using two period indices κ1 and κ2, projected with a time series such as a bivariate random walk with drift.

Understand Age-Period-Cohort and CBD Models

Mortality changes with three things: the age of the person, the calendar year (period), and the year of birth (cohort). Age effects are the main driver. Period effects capture improvements shared by all ages in a given year, such as better medicine. Cohort effects follow a group born in the same years, for example people whose early life or lifestyle differed from those before and after.

The Lee-Carter model has an age shape and one period index: ln m(x,t) = a(x) + b(x)κ(t). It has no cohort term. If a cohort effect exists, it shows up as diagonal patterns in the residuals. An age-period-cohort (APC) model adds a cohort term, written ln m(x,t) = a(x) + b1(x)κ(t) + b2(x)γ(t−x). The term γ(t−x) depends only on year of birth t−x.

The Cairns-Blake-Dowd (CBD) model is built for older ages, usually about 60 to 90 and above. It models the one-year death probability q(x,t) and assumes the logit of q is linear in age: logit q(x,t) = κ1(t) + κ2(t)(x − x̄). Here x̄ is the average of the ages in the fit. κ1 is the overall level of mortality and κ2 is the slope with age. A fall in κ1 means mortality improves at all ages. A rise in κ2 means improvements are larger at younger ages than older ones.

The CBD family has extensions. Adding a cohort term gives M6: logit q(x,t) = κ1(t) + κ2(t)(x − x̄) + γ(t−x). Adding a quadratic age term gives M7: κ1(t) + κ2(t)(x − x̄) + κ3(t)((x − x̄)² − σ²) + γ(t−x), where σ² is the average of (x − x̄)² over the ages fitted. Cohort indices need their own projection, often an ARIMA process.

To project, you fit the model to historical data, treat the period indices as time series, project them forward, and rebuild q(x,t). Because the indices are random, you get stochastic projections and can measure uncertainty. Models are compared by fit, parameter count, biological reasonableness and robustness. Note that adding parameters improves fit but risks overfitting, so penalised measures such as AIC or BIC are used.

Key rules to remember

Lee-Carter model: ln m(x,t) = a(x) + b(x)κ(t) + ε(x,t); One period index, no cohort term. Usual constraints: Σ b(x) = 1 and Σ κ(t) = 0.
APC model: ln m(x,t) = a(x) + b1(x)κ(t) + b2(x)γ(t−x); Cohort index γ depends on year of birth c = t − x. Needs constraints to be identifiable.
CBD model (M5): logit q(x,t) = ln[q(x,t) ÷ (1 − q(x,t))] = κ1(t) + κ2(t)(x − x̄); Used for older ages. κ1 is the level, κ2 is the slope. x̄ is the mean age in the fit.
Inverse logit: q(x,t) = exp(L) ÷ (1 + exp(L)), where L = logit q(x,t); Use this to turn a fitted or projected logit back into a death probability.
CBD with cohort (M6): logit q(x,t) = κ1(t) + κ2(t)(x − x̄) + γ(t−x); Cohort year c = t − x.
CBD with quadratic term and cohort (M7): logit q(x,t) = κ1(t) + κ2(t)(x − x̄) + κ3(t)((x − x̄)² − σ²) + γ(t−x); σ² is the mean of (x − x̄)² over the fitted ages.
Bivariate random walk with drift for CBD: κ(t+1) = κ(t) + μ + C Z(t+1); κ = (κ1, κ2)ᵀ, μ is a drift vector, C comes from the Cholesky decomposition of the covariance matrix, Z is a vector of independent standard normals.
Cohort index rule: Cohort = year of birth = t − x; People of age x in year t are on the same cohort diagonal if t − x is the same.

How to solve Age-Period-Cohort and CBD Models questions

Use this order for most written questions on APC and CBD models, whether you are describing, calculating or comparing.

1Identify the model asked about and write its equation in standard notation. State whether it models m(x,t) or q(x,t), and which link (log or logit) is used.
2Define every term: age effect, period index, cohort term, and the cohort year c = t − x. Say what each parameter means in words.
3State the assumptions: the age range, the error structure, and the constraints needed so parameters are identifiable.
4For a calculation, find the index values for the year asked, work out the logit, and apply the inverse logit to get q. Keep all decimals until the last step.
5For projection, state the time series model for κ (and for γ if present), then project the indices and rebuild q(x,t).
6For comparison, link your points to features: age range suited, number of parameters, cohort treatment, fit and robustness, and uncertainty.
7Finish with a short conclusion that answers the exact question, with any limits such as extrapolation risk.

Quickest way: Logit-to-probability shortcut for CBD questions

When to use it: When you are given κ1, κ2 and an age and must find a death probability or compare two years.

Write L = κ1 + κ2(x − x̄) and add γ if a cohort term is given.
Compute L as a number, watching the sign of x − x̄.
Use q = e^L ÷ (1 + e^L). For negative L, q is below 0.5, and for older ages q is small.
Sense-check: q should be between 0 and 1 and rise with age if κ2 is positive.
For a comparison between years, compare the L values first. A lower L means a lower q, so you often do not need the full calculation.

Common mistakes in Age-Period-Cohort and CBD Models

Treating the CBD model as a model of ln m(x,t) like Lee-Carter.
Both are called mortality projection models and the equations look alike.
Fix: Remember CBD uses logit q(x,t). Write the link function first every time.
Forgetting to centre age by x̄ in the CBD formula.
Students plug in x directly because it is the age given.
Fix: Always compute x − x̄ first. Then κ1 is the logit at the mean age, which is how the model is defined.
Reading the logit as the probability itself.
The number is in the middle of the calculation and looks like a result.
Fix: Finish with q = e^L ÷ (1 + e^L). A negative logit gives a probability below 0.5.
Using age and year of birth wrongly for the cohort term.
Students use t + x or just x.
Fix: The cohort is c = t − x. Check with a simple case: a person aged 70 in 2020 was born in 1950.
Claiming CBD works well at all ages or that more parameters always means a better model.
Over-generalising from a single feature of the model.
Fix: CBD suits older ages only. When comparing, mention fit, parsimony (AIC or BIC), robustness and whether the projections are biologically reasonable.
Ignoring identifiability constraints for APC and cohort terms.
The constraint seems like a technical detail.
Fix: Age, period and cohort are linked by c = t − x, so you cannot estimate them all freely. State that constraints are needed and name the ones used.

Worked examples

Example 1

A CBD model (M5) is fitted to ages 60 to 89 with x̄ = 74.5. For a given year, κ1 = −2.80 and κ2 = 0.10. Calculate q for a life aged 79 in that year, to 4 decimal places.

Show the solution

Centre the age: x − x̄ = 79 − 74.5 = 4.5.
Compute the logit: L = −2.80 + 0.10 × 4.5 = −2.80 + 0.45 = −2.35.
Apply the inverse logit: q = e^(−2.35) ÷ (1 + e^(−2.35)).
e^(−2.35) = 0.09537 (to 5 decimal places).
q = 0.09537 ÷ 1.09537 = 0.0871.

Answer: q(79) ≈ 0.0871

Example 2

Explain two differences between the Lee-Carter model and the CBD model, and describe how adding a cohort term changes the CBD model.

Show the solution

Difference 1, link and quantity modelled: Lee-Carter models ln m(x,t) as a(x) + b(x)κ(t). CBD models logit q(x,t) as κ1(t) + κ2(t)(x − x̄).
Difference 2, age structure and range: Lee-Carter has a free age shape a(x) and b(x) with one period index, so it can be used across all ages. CBD has two period indices and a simple linear age effect, so it is designed for older ages where the logit is close to linear in age.
Related point: CBD lets improvement rates differ across ages through κ2. Lee-Carter has a single index, so improvements at all ages move together with fixed relative sensitivities b(x).
Cohort term: the M6 model adds γ(t−x), giving logit q(x,t) = κ1(t) + κ2(t)(x − x̄) + γ(t−x). This captures patterns that follow year of birth, which the basic CBD model leaves in the residuals as diagonal stripes.
Consequence: the cohort index needs its own projection, usually an ARIMA model, and constraints are needed for identifiability, since c = t − x. Fit improves, but there are more parameters and a risk of overfitting.

Answer: Lee-Carter uses ln m with one period index and free age shape. CBD uses logit q with two indices and linear age effect, suited to older ages. Adding γ(t−x) captures cohort effects but needs constraints, a separate projection and care over overfitting.

Exam tips

Write the model equation with the link function at the start. Examiners award marks for correct notation and clear definition of terms.
In comparison questions, give points tied to features (age range, parameters, cohort, uncertainty) and not just a list of names.
For numerical CBD questions, show L and then q. Partial marks depend on visible working.
Mention limits when projecting: extrapolating indices assumes past trends continue, and parameter and model uncertainty remain.
In Paper B, check that fitted parameters are in the order the code expects and that ages are centred the same way as in the question.

Practice questions from Mortality projection

Age-Period-Cohort and CBD Models: frequently asked questions

What is a cohort effect in mortality projection?

It is a pattern in mortality that depends on year of birth rather than on age or calendar year alone. People born in the same period share early-life or lifestyle conditions that can make their mortality different from neighbouring cohorts. It appears as diagonal patterns in residuals.

What is the main difference between Lee-Carter and CBD models?

Lee-Carter models ln m(x,t) with an age shape and one period index across all ages. CBD models logit q(x,t) as linear in age with two period indices and is aimed at older ages. Basic versions of both lack a cohort term.

Why does the CBD model use a logit?

The logit maps a probability between 0 and 1 onto the whole real line. At older ages the logit of q is close to linear in age, so a simple linear model fits well. Inverting the logit always gives a valid probability.

How are the period indices in the CBD model projected?

They are treated as a time series. A common choice is a bivariate random walk with drift for (κ1, κ2). The cohort index, if present, is often projected with an ARIMA model. Simulating these gives stochastic projections of q.