Lee-Carter Stochastic Mortality Projection Engine

Every pension plan, annuity book and life insurer carries one enormous unhedged assumption: how fast mortality will keep improving. Lee-Carter (1992) turned that assumption into a statistical model with genuine forecast uncertainty, and it remains the benchmark every newer mortality model is measured against.

How To Use This Model

Reading This Tool

This tool simulates fifty years of mortality experience for ages 40 to 90 with improvement dynamics you control, then fits the Lee-Carter model exactly as the original paper did: take log death rates, centre by age, and extract the dominant time signal by singular value decomposition.

The fitted mortality index k_t is then projected forward as a random walk with drift, the drift and its standard error estimated from the fitted series itself, and the whole projection is translated into the number that actually matters to a pension plan: life expectancy at 65, with uncertainty bands.

Mortality Experience & Projection

Annual Mortality Improvement 1.5%

Improvement Concentration At Younger Ages Moderate

Year-To-Year Mortality Volatility Moderate

Projection Horizon 30 yrs

Random Seed 7

The simulated surface uses a Gompertz-style age profile calibrated to modern Canadian-scale mortality levels, with your improvement rate driving the hidden k_t process. The fit sees only the noisy rates, never the true parameters, exactly the estimation problem an actuary faces with real data.

Fitted Model & Longevity Projection

Fit: –

Estimated Drift Of k_t

Variance Explained By 1st SVD Factor

Life Expectancy At 65, Today

Projected e₆₅ At Horizon

Mortality Index k_t: Fitted History & Random-Walk-With-Drift Forecast Fan

Where The Improvement Lands

b_x At Age 45

b_x At Age 65

b_x At Age 85

Current m₆₅ (Fitted)

Age Sensitivity b_x: Which Ages Respond To The Mortality Index

Baseline Log Mortality a_x By Age

Actuarial Science & Longevity Risk

The Model & Estimation

ln m_x,t = a_x + b_xk_t + ε_x,t
â_x = (1/T)Σ_t ln m_x,t, (b_x, k_t) from the leading singular vectors of the centred surface
Identification: Σ_xb_x = 1, Σ_tk_t = 0
k_t+1 = k_t + δ + e_t, e_t ~ N(0, σ²_k), forecast se(k_T+h) ≈ σ_k√h

Estimation follows Lee and Carter (1992) exactly: centre log rates by age, then extract the rank-one structure by singular value decomposition (computed here by power iteration, verified to recover known parameters to machine precision on noiseless surfaces). The mortality index is forecast as a random walk with drift, the specification the original authors found adequate for U.S. data and which remains the default in practice. Life expectancy is computed from projected rates under a constant force of mortality within each age, q_x = 1−e^−m_x, with the standard half-year adjustment.

When To Actually Use This Model

Producing mortality improvement assumptions with an uncertainty band, rather than a single deterministic improvement scale, for pension valuations and annuity pricing sensitivity work.
Quantifying longevity risk exposure: the fan on this page is the raw material of longevity value-at-risk and the pricing of longevity swaps and buy-ins.
Teaching the decomposition at the heart of every modern stochastic mortality model, Renshaw-Haberman cohort extensions and the Cairns-Blake-Dowd family are all reactions to this baseline.

Key Assumptions & Limitations

A single time factor: every age improves in lockstep proportion b_x, so the model cannot capture cohort effects (generations with persistently distinct mortality), the Renshaw-Haberman extension adds these.
The random-walk-with-drift forecast extrapolates the historical trend indefinitely; structural changes, pandemics, medical breakthroughs, break it, as 2020-2021 experience demonstrated in every national dataset.
This page fits simulated experience for exposition. Real Canadian work uses CIA improvement scales (MI-2017 and successors) and Human Mortality Database inputs, with the same machinery underneath.

Foundational References

Lee, R. D., & Carter, L. R. (1992). Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association, 87(419), 659-671.

Brouhns, N., Denuit, M., & Vermunt, J. K. (2002). A Poisson Log-Bilinear Regression Approach to the Construction of Projected Lifetables. Insurance: Mathematics and Economics, 31(3), 373-393.

Cairns, A. J. G., Blake, D., & Dowd, K. (2006). A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty. Journal of Risk and Insurance, 73(4), 687-718.

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