Mortality projection: formula sheet

Key formulas

Mortality improvement rate

Improvement rate = 1 − q(x, t+1) ÷ q(x, t)

Proportional fall in the rate at age x from year t to t+1. A positive value means improvement. Defined for q(x, t) > 0.

Reduction factor over n years with constant annual improvement

q(x, t+n) = q(x, t) × (1 − r)ⁿ

Applies only if the improvement rate r is the same each year. Real improvement is rarely constant, so state this as an assumption.

Types of uncertainty in projections

Model risk, parameter risk, stochastic (random) risk

Remember these three. Model risk is the wrong structure, parameter risk is poorly estimated inputs, and stochastic risk is random variation in actual deaths.

Groups of drivers

Medical, public health, lifestyle, socio-economic

Use this list to structure descriptive answers.

Extrapolative idea (annual improvement)

Improvement rate = 1 − q(x, t+1) ÷ q(x, t)

A simple measure of year-on-year fall in the mortality rate at age x. Assuming it stays constant is a basic extrapolative method.

Projected rate with constant improvement

q(x, t+n) = q(x, t) × (1 − r)^n

Valid only if the improvement rate r is assumed constant at age x. State this assumption.

Three approaches

Extrapolative | Explanatory (process-based) | Expectation (expert opinion)

Know the one-line definition, a strength and a limitation of each.

Lee-Carter model

log m(x,t) = a_x + b_x k_t + ε(x,t)

m(x,t) is the central rate of mortality at age x in year t. ε has mean 0 and constant variance.

Constraints

Σ_x b_x = 1 and Σ_t k_t = 0

Needed so the parameters are identifiable. Other constraint pairs exist, so use the ones the question states.

Estimate of a_x

â_x = (1 ÷ T) Σ_t log m̂(x,t)

Holds under Σ k_t = 0. It is the average log rate at age x over the T years.

SVD step

Z(x,t) = log m̂(x,t) − â_x ≈ s₁ u₁(x) v₁(t)

u₁ and v₁ are the first left and right singular vectors, s₁ is the largest singular value.

Rescaling after SVD

b̂_x = u₁(x) ÷ Σ_x u₁(x); k̂_t = s₁ v₁(t) × Σ_x u₁(x)

The product b̂_x k̂_t is unchanged. Check Σ k̂_t = 0, which holds because each row of Z averages to 0.

Random walk with drift for k_t

k_t = k_{t-1} + d + e_t, e_t ~ N(0, σ²)

d is the drift. A negative d means improving mortality.

Drift estimate

d̂ = (k_T − k_1) ÷ (T − 1)

This is the mean of the T − 1 annual differences. Estimate σ² by the sample variance of those differences.

Point forecast and variance

E[k_{T+h}] = k_T + h d̂; Var(k_{T+h}) = h σ²

The variance ignores uncertainty in d̂, a_x and b_x. Say so if asked.

Forecast mortality

log m(x, T+h) = a_x + b_x k_{T+h}

Take the exponential to get the rate. This gives a central estimate on the log scale.

Number of free parameters

2A + T − 2

For A ages and T years: A values of a_x, A − 1 of b_x and T − 1 of k_t, after the two constraints.

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.

Lee-Carter model

ln m(x,t) = a(x) + b(x) × k(t) + ε(x,t)

Usual constraints: Σ b(x) = 1 and Σ k(t) = 0. k(t) is the period index, often projected as a random walk with drift.

Random walk with drift for k(t)

k(t+1) = k(t) + μ + σ × Z(t+1), with Z ~ N(0,1) independent

μ is the drift, σ the volatility of the period index. Parameter risk arises because μ and σ are estimated.

Projected k(t) after s years

k(T+s) = k(T) + s × μ + σ × √s × Z

Mean is k(T) + sμ and variance is sσ². Uncertainty grows with the square root of the horizon.

Total variance decomposition

Var(Y) = E[Var(Y | θ)] + Var(E[Y | θ])

Here θ is the parameters. The first term is process-type variation, the second is parameter uncertainty.

Simulated estimate

Estimate = (1/N) × Σ Y(i), for i = 1 to N

Standard error of the mean is s ÷ √N, where s is the sample standard deviation of the simulated values.

Percentile from simulations

Sort the N values; the p-th percentile is about the (p × N)-th smallest value

Used to quote a 95th or 99.5th percentile annuity value.

Quick revision

• 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.

Common mistakes

• Listing drivers with no explanation of how they affect mortality. Fix: Write a short reason for each point, such as 'vaccination reduces deaths from infectious disease at young ages'.
• Assuming improvement is the same at all ages and causes of death. Fix: Say that improvement varies by age, sex, cause of death, period and cohort.
• Calling Lee-Carter an explanatory model. Fix: Lee-Carter fits age and time patterns to past data and extrapolates them. It is extrapolative. It does not model causes of death.
• Listing strengths and limits without linking to the purpose. Fix: End each answer with a sentence saying which approach suits the stated use, for example long-term annuity reserving, and why.
• Forgetting the constraints, or stating them wrongly as Σ b_x = 0 or Σ k_t = 1. Fix: Remember it as: b sums to 1 (shares of improvement), k sums to 0 (centred index). Then a_x is the plain average of the log rates.
• Estimating a_x from the raw rates m(x,t) instead of the log rates. Fix: The model is on the log scale. Average log m(x,t) over t, and only exponentiate at the end.
• Treating the CBD model as a model of ln m(x,t) like Lee-Carter. 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. Fix: Always compute x − x̄ first. Then κ1 is the logit at the mean age, which is how the model is defined.
• Confusing parameter risk with stochastic risk. Fix: Parameter risk is about estimation error in fitted values. Stochastic risk is chance variation in outcomes even when the parameters are known exactly.
• Saying that a larger portfolio removes longevity risk. Fix: A larger portfolio reduces idiosyncratic risk only. Trend risk affects all lives together and stays.

Exam tips

• Structure descriptive answers by driver group and add reasons, not only labels.
• Always link drivers to a business consequence such as longevity risk for annuities and pensions.
• In calculations, state that improvement is proportional and assumed constant, then show the power step.
• Mention that improvement differs by age, sex, cause of death and cohort. Examiners reward this detail.
• Include at least one factor that may slow improvement to show balance in 'discuss' questions.
• Use the exact terms: extrapolative, explanatory (process-based) and expectation. Examiners mark against these.
• For 'discuss' questions, give both strengths and limitations. One-sided answers lose marks.
• Tie your conclusion to the purpose, for example annuity pricing or pension funding.