Mortality Projection: Lee-Carter and CBD

Life-contingent valuation that uses a single fixed mortality table treats future mortality as known and constant. Real cohort mortality has trended downward for over a century: U.S. female life expectancy at birth has roughly doubled since 1900. A pricing or reserving model that ignores this systematically underprices annuities and overprices term insurance, and it understates the longevity risk in a pension book.

Mortality projection models replace one fixed table with a family of tables indexed by calendar year. The two reference models on ALTAM are Lee-Carter and Cairns-Blake-Dowd. Both project one or two time factors forward and apply them to age-specific mortality.

Lee and Carter (1992) proposed a log-bilinear model for central mortality rates m_{x,t}: log m_{x,t} = a_x + b_x κ_t + ε_{x,t}. The age effect a_x captures the average log-mortality at age x. The age-sensitivity b_x captures how strongly age x responds to the time index. The time index κ_t captures the year. The residual ε_{x,t} is independent noise.

Identification: without constraints, the model is overparameterized by a scale and a shift in κ_t. The standard constraints are sum over t of κ_t equals zero and sum over x of b_x equals one. Then a_x is recovered as the time-average of log m_{x,t}.

After subtracting a_x from each row, the centered log-mortality matrix Z_{x,t} = log m_{x,t} − a_x is approximated by the rank-one product b_x κ_t. The best rank-one approximation in Frobenius norm comes from the singular value decomposition: Z = U Σ V^T, and the leading singular triplet (σ_1, u_1, v_1) gives b_x proportional to u_1 and κ_t proportional to σ_1 v_1. Normalization to the identification constraints fixes the constants.

Lee and Carter ran a second-stage calibration after SVD: they re-fit each κ_t so that the implied total deaths in year t match observed total deaths. This step preserves total counts and is what distinguishes the original 1992 estimator from a plain SVD. Both versions appear in practice.

After estimation, κ_t is treated as a time series and projected forward. The model most commonly used on ALTAM is a random walk with drift: κ_t equals κ_{t−1} plus a drift d plus a Gaussian innovation. The drift d̂ is estimated as the mean year-on-year change in the fitted κ. Forecasts κ_{T+h} are normal with mean κ_T plus h d̂ and variance h σ_ε^2.

Mortality forecasts then use log m_{x,T+h} = a_x + b_x κ_{T+h}. Forecast uncertainty has two components: parameter uncertainty in (a_x, b_x) and innovation uncertainty in κ_{T+h}. The κ-uncertainty dominates at long horizons, which is the key longevity-risk message.

Cairns, Blake, and Dowd (2006) proposed a two-factor logit-survival model for older-age mortality probabilities: logit q_{x,t} = κ^{(1)}_t + (x − x̄) κ^{(2)}_t. The level factor κ^{(1)}_t shifts mortality up or down at all ages. The slope factor κ^{(2)}_t tilts the age curve. The age x̄ is the central age in the fitting range.

CBD is fitted year by year by logistic regression of the empirical q_{x,t} on age, which gives (κ^{(1)}_t, κ^{(2)}_t) directly. The bivariate sequence is then projected forward as a correlated random walk with drift. CBD is normally fitted over a high-age range (often 60-89) and is the reference model in pension and annuity longevity-risk work.

Lee-Carter is the default for whole-age-range mortality, especially for population mortality. Its single time index makes it parsimonious and easy to communicate.

CBD was designed for older-age mortality where the rectangularization pattern matters and where the age slope itself changes over time. Pension funds and annuity writers use CBD because it captures the way the gradient of mortality across older ages can flatten or steepen separately from the overall level.

Toy data, log central mortality rates at ages 60 and 70 for three years. Year 2020: log m_{60} = -3.95, log m_{70} = -2.70. Year 2021: -4.00, -2.80. Year 2022: -4.05, -2.90.

Row averages give a_60 = -4.00 and a_70 = -2.80. Centered matrix Z has row Z_60 = (0.05, 0, -0.05) and row Z_70 = (0.10, 0, -0.10). This matrix is exactly rank one. SVD recovers b proportional to (1, 2) and κ proportional to (1, 0, -1). Imposing sum of b equal to one gives b_60 = 1/3, b_70 = 2/3, and then κ = (0.15, 0, -0.15) so that the constraint sum of κ equals zero holds.

Year-on-year changes in κ are -0.15 and -0.15, so the random-walk drift estimate is d̂ = -0.15. Forecast κ_{2023} = -0.15 + d̂ = -0.30. Predicted log m_{60, 2023} = a_60 + b_60 κ_{2023} = -4.00 + (1/3)(-0.30) = -4.10, so m_{60, 2023} = exp(-4.10) versus m_{60, 2022} = exp(-4.05). The implied year-on-year mortality improvement at age 60 is about 5%, and at age 70 it is exactly twice that on the log scale.

Take central age x̄ = 65 and current-year parameters κ^{(1)} = -3.5, κ^{(2)} = 0.10. Logit of q_{60} = -3.5 + (60 − 65)(0.10) = -4.0, so q_{60} = exp(-4) / (1 + exp(-4)) = 0.0180. Logit of q_{70} = -3.5 + (70 − 65)(0.10) = -3.0, so q_{70} = exp(-3) / (1 + exp(-3)) = 0.0474.

Now suppose the joint random walk has drift d̂ = (-0.05, +0.005). Project one year forward: κ^{(1)} becomes -3.55 and κ^{(2)} becomes 0.105. Re-evaluating, logit q_{70, t+1} = -3.55 + 5(0.105) = -3.025, so q_{70, t+1} = 0.0463, an improvement of about 2.3% relative to q_{70, t}. At age 60 the slope and level contributions reinforce: logit q_{60, t+1} = -3.55 + (-5)(0.105) = -4.075, so q_{60, t+1} ≈ 0.0167, an improvement of about 7% relative to q_{60, t}. The slope drift partially offsets the level improvement above x̄ and reinforces it below x̄; that asymmetry across ages is the entire point of using a two-factor model rather than Lee-Carter.

Annuity at age 65, interest i = 0.04, payment of 1 at the end of each year for 20 years, single life. Static mortality fixes q_{65+k} = 0.012 for all k. Survival to age 65 + k is 0.988^k. Twenty-year survival is 0.988^{20} = 0.7847.

Apply 1% per year mortality improvement: q_{65+k, year k} = 0.012 × 0.99^k. Sum of log survival over 20 years is approximately −0.012 × (1 − 0.99^{20}) / 0.01 = −0.2185. Improved 20-year survival is exp(-0.2185) = 0.8038, versus the static 0.7847. The 20-year deferred unit increases in present value from (1/1.04)^{20} × 0.7847 = 0.358 to (1/1.04)^{20} × 0.8038 = 0.367, an increase of about 2.4%.

That 2.4% per single deferred payment compounds across an annuity stream. For a 20-year temporary annuity-immediate at age 65, the total cost increase under 1% per year compound improvement is in the low single digits as a percentage. For a whole-life annuity at 65 the increase is larger because more of the payments are at older ages where improvement has compounded longer.

Mortality improvement is the expected trend. Longevity risk is the residual uncertainty around that trend. A static table fully misstates the level but understates both expected improvement and uncertainty. A point-projected table fixes the level but ignores the variance of κ_T+h and so understates the capital needed against worse-than-expected longevity.

On CERA and ALTAM, longevity risk is treated as a stochastic-mortality scenario that the company runs alongside the central best-estimate projection. The capital charge for longevity in regulatory frameworks (Solvency II, U.S. life insurance economic capital models) reflects the variance of projected κ_T+h, not just its mean.

Both Lee-Carter and CBD estimate drift from the in-sample mean change in the time index. The drift estimate is sensitive to the fitting window. Fitting Lee-Carter to U.S. mortality 1950-2010 gives a steeper drift than fitting it to 1900-2010, because mid-century gains were faster than long-run averages.

Best practice is to report sensitivity to the base period. ALTAM problems will often ask you to recompute a forecast with a shorter base period and explain the direction of the change. The honest answer is that drift estimates are point estimates with their own uncertainty, and the choice of base period is a modeling judgment.

Original papers: Lee, R.D., and Carter, L.R. (1992). Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association, 87(419), 659-671. Cairns, A.J.G., Blake, D., and Dowd, K. (2006). A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. Journal of Risk and Insurance, 73(4), 687-718.

Textbook: Dickson, Hardy, and Waters, Actuarial Mathematics for Life Contingent Risks, 3rd edition, Chapter 17 (mortality projection). The same chapter develops the family of CBD extensions (M5 through M8) introduced by Cairns and co-authors in follow-up papers.

Exam topics: ALTAM longevity-risk content within topic 1; ILA 201 longevity-risk module (LRM); RET 201 retirement-system longevity; CERA longevity-risk content. The exam-prep emphasis is on understanding the model structure, recomputing simple projections by hand, and reading the direction of common sensitivities rather than running full stochastic simulations.