Mortality Projection for IAI CS2: Study Guide

What this chapter covers

Mortality projection: topics in the order to study them

1. 1Mortality Trends and Drivers of ImprovementStart with the real-world picture of falling mortality, since every model is built to capture these patterns.
2. 2Approaches to Mortality ProjectionNext learn the broad methods and their pros and cons, which gives you a frame for judging each specific model.
3. 3Lee-Carter ModelThis is the core one-factor model, and you need it before the extensions that build on its structure.
4. 4Age-Period-Cohort and CBD ModelsStudy these after Lee-Carter so you can compare them, especially on cohort effects and age structure.
5. 5Parameter Uncertainty and Stochastic ProjectionsFinish with uncertainty and simulation, which apply to every model you have learned.

How to prepare Mortality projection

1. Read the trends and drivers topic once and write a short list of causes of improvement and risks to them, such as medical advances, lifestyle change and lower infant deaths.
2. Make a one-page comparison of extrapolation, explanation and expectation approaches, with one strength and one weakness each.
3. Write out the Lee-Carter equation in standard notation, say what each term means, and note the identifiability constraints. Practise stating them without notes.
4. Learn how the period index is projected, typically as a time series such as a random walk with drift, and what that implies for forecast variance.
5. Compare APC and CBD models in a table of your own: structure, number of parameters, treatment of cohorts, and weaknesses.
6. Work through sources of uncertainty: parameter, process and model risk. Practise explaining how simulation produces a fan of projections.
7. Do past-paper written questions under time, then repeat any R fitting tasks so you can produce output and comment on it.

Common mistakes in Mortality projection

• Writing the Lee-Carter equation without defining each term or stating constraints.

  Fix: Write the equation, then one line for a(x), b(x) and k(t), and state the constraints that make the model identifiable.

• Treating projection as a single forecast with no uncertainty.

  Fix: Always mention parameter, process and model risk and say how simulation shows the spread of outcomes.

• Confusing the age-period-cohort model with Lee-Carter and attributing cohort effects to the wrong model.

  Fix: Remember that basic Lee-Carter has age and period terms only, while APC adds an explicit cohort term. Keep a short comparison note.

• Giving a list of mortality drivers with no judgement about their future effect.

  Fix: For each driver, say whether it may continue, slow or reverse, and how that affects the choice of projection method.

• Ignoring the reliability of the time series chosen for the period index.

  Fix: State the time series model, its assumptions and what it implies for forecast variance over the horizon.

• Giving only output in Paper B without comment.

  Fix: Check residuals, comment on fit and plausibility, and state the assumptions behind the projection.

Last-day revision: Mortality projection

• ✓Mortality improvement means death rates fall over calendar time, and the pace can differ by age and cohort.
• ✓Three broad approaches: extrapolation, explanation and expectation. Extrapolation is the most used in practice.
• ✓Lee-Carter: ln(m(x,t)) = a(x) + b(x) × k(t) + error, with a(x) the average age pattern.
• ✓In Lee-Carter, b(x) shows how much each age responds to change in k(t), and k(t) is the period index.
• ✓Lee-Carter needs constraints to be identifiable, such as Σb(x) = 1 and Σk(t) = 0.
• ✓k(t) is often projected as a random walk with drift, so uncertainty widens as the horizon grows.
• ✓Lee-Carter has no cohort effect in its basic form, which can leave patterns in residuals.
• ✓APC models add a cohort term γ(t − x) to age and period effects.
• ✓The CBD model is designed for older ages and uses two period indices, one for level and one for slope with age.
• ✓Parameter uncertainty comes from estimation, process risk from random deaths, and model risk from choosing the wrong form.
• ✓Stochastic projection by simulation gives a distribution of future rates, not a single best estimate.
• ✓Always state assumptions, such as the fitting period, age range and the time series model used for the period index.

Mortality projection practice questions

• Which of the following is a recognised driver of past mortality improvement at older ages in developed countries that is most directly linke…
• In the Age-Period-Cohort (APC) model log m(x,t) = a(x) + b(t−x) + c(t) style formulation, which identifiability problem arises when the age,…
• A Lee-Carter model is fitted with k(t) following a random walk with drift: k(t+1) = k(t) + d + sigma Z. The fitted drift is d = -1.5, and k(…
• Which of the following is an example of an 'explanatory' (cause-based) approach to mortality projection, as opposed to extrapolative or proc…
• Which of the following is a known identifiability issue in the age-period-cohort model log m(x,t) = a(x) + b(t) + c(t-x), and how is it usua…
• In a Lee-Carter model fitted by singular value decomposition, the matrix of centred log death rates Z(x,t) = ln m(x,t) - a(x) has first sing…
• A projection uses mortality rates q(x,t) = q(x,0) × R(x,t), with R(x,t) = [1 − f(x)]^t where f(65) = 2% per year. Rates at age 65 in the bas…
• In a Lee-Carter model, k(t) is modelled as a random walk with drift: k(t+1) = k(t) + d + e(t). Fitted k(2020) = -10 and the estimated drift …