Projecting future mortality is one of the most delicate tasks in modern actuarial science. The price of a life annuity, the adequacy of a pension reserve and the solvency of a life insurance portfolio all depend on the quality of those projections. For almost three decades, the reference framework has been the Lee-Carter model, published in 1992. Today, however, a growing body of research shows that neural networks and deep learning can capture patterns that classical models miss.
The Lee-Carter model and its logic
Ronald Lee and Lawrence Carter's idea is elegant in its simplicity. The central mortality rate at age $x$ in year $t$ is modeled as:
$$\ln(m_{x,t}) = a_x + b_x \, k_t + \varepsilon_{x,t}$$
where $a_x$ describes the average age profile of mortality, $k_t$ is a time index that summarizes the overall evolution of mortality, and $b_x$ measures the sensitivity of each age to changes in that index. Once the parameters are estimated (usually by singular value decomposition), the index $k_t$ is projected with a time series model, typically a random walk with drift, to obtain future rates.
The model's success is explained by its parsimony, its transparency and its ability to generate confidence intervals. It is easy to communicate to regulators and technical committees, and it serves as the basis for later variants, such as the Renshaw-Haberman models with a cohort effect or the Cairns-Blake-Dowd family for advanced ages.
The limits of the classical approach
The limitations of Lee-Carter are well known. By assuming a bilinear structure, it forces the effect of time on each age to be proportional and stable, which rarely holds. The model treats each population in isolation, so analyzing both sexes, several countries or different subgroups requires separate models, losing shared information. It also poorly captures nonlinear interactions among age, period and cohort.
These restrictions matter because projection errors translate into longevity risk: the possibility that policyholders and pensioners systematically live longer than expected. For a pension fund or a life annuity insurer, underestimating mortality improvement can erode reserves for decades.
Enter neural networks
This is where the work of Ronald Richman and Mario Wuthrich has been decisive. In a series of papers, including their proposal of a neural network for multiple population mortality, they show that a deep learning architecture can learn the mortality of many populations jointly, sharing structure and improving predictive accuracy compared with Lee-Carter fitted population by population.
The conceptual advantages are several:
- Nonlinearity: networks approximate complex relationships among age, period, cohort and population characteristics without imposing a rigid functional form.
- Joint learning: through embeddings, the network represents each country, sex or subgroup as a learned vector, capturing latent similarities among populations.
- Data flexibility: they can incorporate additional covariates and temporal structures through recurrent or convolutional networks.
Richman and Wuthrich have documented that these models tend to reduce out of sample forecast error relative to classical ones, especially with multiple population data such as those in the Human Mortality Database. A recurring finding is that networks can reproduce Lee-Carter as a special case and then improve on it, which makes them a natural generalization rather than a break.
Why it matters for pensions and life insurance
The practical relevance is direct. A finer mortality projection translates into:
- Better calibrated reserves for life annuities and defined benefit plans.
- More accurate pricing in life insurance and longevity products.
- A more realistic assessment of longevity risk for capital and risk transfer, for example in longevity swaps.
With aging populations and pension systems under pressure, reducing systematic bias in projections has considerable economic value.
The interpretability challenge
The main cost of this power is interpretability. A model with thousands of parameters is hard to explain to a board of directors or to a regulator used to the transparency of $a_x$, $b_x$ and $k_t$. This raises governance questions: how to validate it, how to audit its behavior in extreme scenarios and how to ensure it does not overfit historical noise.
Research responds in several ways: designing architectures that preserve an interpretable structure by embedding the Lee-Carter form within the network, or applying explainable AI techniques to attribute predictions to specific variables. Wuthrich has stressed rigorous validation, control of overfitting and reproducibility as conditions for the responsible use of these models.
A transition, not a replacement
The most sensible reading is not that neural networks replace Lee-Carter, but that they expand the actuary's toolkit. The classical model remains an excellent starting point, a benchmark and a means of communication. Networks bring predictive power and joint learning across populations, in exchange for greater demands on validation and explainability.
For the Mexican actuary, becoming familiar with these methods is not an academic luxury. The combination of richer databases, greater computing capacity and regulatory frameworks that demand robust models means that deep learning in mortality is gradually moving into professional practice.
References
- Lee, R. and Carter, L. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association.
- Richman, R. and Wuthrich, M. V. A neural network extension of the Lee-Carter model to multiple populations. Annals of Actuarial Science.
- Richman, R. AI in actuarial science. Annals of Actuarial Science.
- Cairns, A., Blake, D. and Dowd, K. Stochastic mortality models.