The FCA’s January 2026 interim report on the pure protection market (MS24/1.4) contained a finding that should make mortality actuaries uncomfortable. Some firms, the FCA said, are carrying legacy mortality assumptions that have not been updated to capture recent improvements — with the problem concentrated in reinsurance pricing. Separately, CMI_2025 reported that all-age England & Wales mortality in 2025 was the lowest on record, roughly 2% below 2024. If your pricing mortality basis pre-dates COVID and has not been revisited since, you are exposed on two fronts: regulatory scrutiny and genuine economic error.
Into this context arrives Samuel Clark’s MDMx (arXiv:2603.20518, Ohio State, March 2026): a multi-dimensional mortality model that organises the entire Human Mortality Database as a four-way tensor and extracts structure from it via Tucker decomposition. It will not replace CMI_2025 as your projection model, and it will not replace Whittaker-Henderson smoothing of your own experience data. But it does offer something neither of those tools provides: a principled, internationally benchmarked view of where UK mortality sits relative to peer countries, and an automated mechanism for flagging years in which UK mortality deviated anomalously from expectations.
That is a narrower claim than many mortality papers make for themselves. It is also an honest one.
What Tucker decomposition is
Tucker decomposition is the higher-order generalisation of matrix SVD. Where SVD factors a matrix M into U Σ Vᵀ — left factors, singular values, right factors — Tucker factors a tensor into a dense core G multiplied along each mode by a factor matrix.
For a four-way tensor M ∈ ℝ^(S × A × C × T), Tucker decomposition gives:
M ≈ G ×₁ S_mat ×₂ A_mat ×₃ C_mat ×₄ T_mat
Here ×ₙ denotes the mode-n product: G is the core tensor of shape (r₁ × r₂ × r₃ × r₄), and each factor matrix maps the latent dimension back to the observed dimension along that mode. S_mat is (2 × r₁), A_mat is (A × r₂), C_mat is (C × r₃), T_mat is (T × r₄).
The rank tuple (r₁, r₂, r₃, r₄) controls compression. Lower ranks mean fewer parameters and a smoother, more constrained reconstruction. Higher ranks allow more flexibility but risk overfitting and lose the interpretability of the factors.
Tucker differs from CP decomposition (CANDECOMP/PARAFAC), which is the special case where the core is superdiagonal — effectively a sum of rank-1 terms with no interaction between modes. Tucker allows mode interactions through the full core, which matters here: the relationship between age patterns and country effects is not assumed to be independent.
The MDMx tensor
Clark organises the Human Mortality Database as:
- Mode 1 (sex): 2 categories, male and female
- Mode 2 (age): single-year ages from 0 to 110+
- Mode 3 (country): ~50 countries with HMD coverage
- Mode 4 (year): calendar years from 1751 to present
The response variable is logit(₁qₓ) — the logit of the one-year probability of death at age x — not log death rates. This is a deliberate choice. Logit transforms keep the response on the real line while respecting the [0,1] constraint on probabilities, and they produce better-behaved residuals in the tails where log-rate transforms can become unstable for very young or very old ages.
The total tensor contains roughly 2.9 million observed values. After applying Tucker decomposition with selected ranks r₁=2, r₂=4, r₃=8, r₄=7, the model has approximately 50,000 parameters — a compression ratio of about 58:1. The rank selection criterion is 99.99% variance retention: ranks are increased until adding another factor contributes less than 0.01% of total variance.
What the factor matrices encode:
- S_mat (2×2): trivially — sex differences in mortality level and age pattern
- A_mat (ages×4): four latent age patterns. The first resembles the standard mortality curve; subsequent factors capture curvature at different life stages
- C_mat (countries×8): eight latent country mortality profiles. Countries cluster by development level, healthcare system, and historical mortality transition stage
- T_mat (years×7): seven temporal patterns, including long-run improvement trends and abrupt shocks
The core tensor G ties these together. An entry G[s,a,c,t] gives the interaction weight between latent sex component s, age component a, country component c, and temporal component t.
Four tasks from one decomposition
The structural value of Tucker over a suite of separate models is that the decomposition is shared. All four of MDMx’s capabilities draw on the same estimated factor matrices.
1. Model life table clustering. Project each country’s mortality schedule into the factor space defined by C_mat. Apply Gaussian mixture modelling in that low-dimensional space. Countries in the same mixture component share a latent mortality regime. The UK sits in a Western European cluster with Germany, the Netherlands, and the Nordic countries. Within-cluster trajectories can be smoothed to give regime-consistent model life tables.
2. Life table fitting with disruption detection. Given a country’s observed mortality time series, fit it within the Tucker framework using a three-stage algorithm: grid search initialisation, Gauss-Newton refinement, then a Bayes-factor test on each year. Years where the Laplace-approximated Bayes factor exceeds a threshold are flagged as disrupted — genuine departures from the underlying trend that should not be fitted into the structural model. Clark curates event dictionaries of known shocks (major wars, the 1918 influenza pandemic, COVID-19); the Bayes-factor criterion provides an automated screen for anomalies the dictionaries might miss.
3. Summary indicator prediction. Clark reformulates his prior SVD-Comp model in tensor coordinates. Given only two summary mortality indicators — the under-5 mortality rate and the 45q15 adult mortality probability — MDMx recovers the full age schedule of logit(₁qₓ). This is valuable for countries with incomplete vital statistics. For UK insurers, the relevance is indirect: the mechanism confirms that MDMx’s age structure is internally coherent.
4. Forecasting. The approach proceeds in two stages. First, PCA is applied to the country-year slice of the core tensor — the G_ct matrices — to extract the dominant temporal dynamics. Second, a damped local linear trend Kalman filter projects each principal component forward, with hierarchical drift: 80% from HMD-wide trends and 20% from the individual country’s own history. Prediction intervals come from the Kalman uncertainty, which gives the 93.7% empirical coverage at nominal 95% seen in the cross-validation.
Accuracy
Clark reports rolling-origin cross-validation with six origins and a 15-year forecast horizon. We reproduce his headline table:
| Metric | MDMx | Hyndman-Ullah | Lee-Carter |
|---|---|---|---|
| e₀ MAE (years) | 1.44 | 1.44 | 1.73 |
| Sex-gap MAE (years) | 0.60 | 0.84 | 1.11 |
| PI coverage (95% nominal) | 93.7% | — | — |
The life expectancy MAE result is a tie with Hyndman-Ullah and a modest improvement over Lee-Carter. The sex-gap result is the clearest win. MDMx’s 0.60-year MAE on the male-female life expectancy differential represents a 46% improvement over Lee-Carter’s 1.11 years.
This is structural, not coincidental. In Lee-Carter, the age effect bₓ is a fixed vector and sex is handled by estimating separate models. The male-female differential evolves only through the interaction of two independent κₜ trajectories, with nothing in the model structure that enforces coherent joint dynamics. In MDMx, sex is mode 1 of the shared tensor, so male and female age patterns and temporal trends are jointly constrained by the same factor matrices. Sex coherence is built into the architecture.
Hyndman-Ullah (2007), the functional data approach using smoothed log mortality curves and product-ratio decomposition, gets close to MDMx on life expectancy but provides no prediction interval coverage comparison in Clark’s experiments — the Hyndman-Ullah forecasts are point estimates for this comparison.
What MDMx is not
We want to be unambiguous about two non-replacements.
MDMx does not replace CMI_2025. The CMI Mortality Projections Model is calibrated to England & Wales population mortality, governed by the IFoA, updated annually, and designed to produce the mortality improvement factors that UK insurers apply to their base tables. CMI_2025 is authoritative for UK population projection. MDMx is a research tool for understanding international mortality structure and forecasting across the HMD’s 50-country panel. Using MDMx instead of CMI for UK projection assumptions would be methodologically indefensible and likely ungovernable under Solvency II internal model standards.
MDMx does not replace Whittaker-Henderson smoothing or P-splines for experience analysis. Our insurance-whittaker library handles the core actuary’s task: smoothing an insurer’s own A/E mortality experience across age and duration cells to produce a credible insured-lives basis. This is a fundamentally different problem from fitting a population model to HMD data. An insured book has hundreds of thousands of exposure years at most; HMD has 2.9 million. The signal-to-noise ratio and the distributional assumptions differ accordingly.
Where MDMx is actually useful for UK insurers
The useful applications are real but specific.
International benchmarking. MDMx clusters the UK into a peer group of Western European countries based on its mortality structure. If your pricing team is trying to assess whether UK assured-lives mortality assumptions are moving in the direction consistent with international peer experience, the Tucker-derived country factors give you a principled answer. “The UK has remained in its long-run cluster since 1990 with no structural divergence” is a different governance statement from “we looked at some ONS charts.”
Disruption detection for COVID and governance. The Bayes-factor disruption detection is the most directly applicable piece. For a model validation exercise under the FCA’s current scrutiny, being able to quantify: “UK mortality in 2021 deviated X standard deviations from its Tucker-predicted path, which is consistent with the Bayes factor exceeding the disruption threshold” is more defensible than eyeballing excess deaths charts. The framework makes the anomaly classification auditable. See insurance-governance for the model validation infrastructure this would plug into.
FCA MS24/1 response. The FCA has given firms until 31 March 2026 to respond to the interim report. Any firm carrying pre-2020 mortality assumptions and hoping to address the FCA’s legacy-assumption concern will need to demonstrate both that they are aware of current mortality trajectories and that their assumptions have been benchmarked against them. MDMx provides the international benchmark; CMI_2025 provides the UK projection. Used together, they cover both angles.
Software and implementation
There is no MDMx Python package. Clark’s prior model (SVD-Comp) is available on CRAN as SVDMx and on GitHub at sinafala/svd-comp, but MDMx is not yet publicly released as software.
The Tucker algebra itself is straightforward in Python. TensorLy handles the decomposition cleanly:
import tensorly as tl
from tensorly.decomposition import tucker
# M has shape (2, n_ages, n_countries, n_years), entries are logit(q_x)
core, factors = tucker(M, rank=[2, 4, 8, 7], init='svd', random_state=42)
# factors[0]: sex (2×2), factors[1]: age (n_ages×4),
# factors[2]: country (n_countries×8), factors[3]: year (n_years×7)
The harder work is upstream: assembling the HMD data pipeline (the HMD requires registration for data access), computing logit(₁qₓ) from the period life tables, handling missing data for countries with incomplete historical coverage, and then implementing the three-stage fitting algorithm and Bayes-factor disruption detection. A faithful replication of the full MDMx methodology from the paper alone, without a reference implementation, is probably two to three weeks of careful work.
If Clark publishes an R package or Python implementation, the landscape changes significantly. We will cover it then.
The honest verdict
MDMx is a technically accomplished piece of work. The Tucker architecture solves a real problem — structural sex coherence in mortality forecasting — in an elegant way. The unified four-task framework means that model life tables, fitting, prediction from summaries, and forecasting all draw on the same decomposition, which enforces a consistency that separate models cannot provide.
For UK protection pricing actuaries, this is a frontier research awareness piece, not an immediate implementation target. The regulatory hook is genuine: if the FCA asks whether your mortality assumptions are consistent with international experience, MDMx gives you a principled framework for answering. But the primary tools remain CMI_2025 for projection and Whittaker-Henderson (or P-splines) for experience graduation. MDMx sits alongside them as a benchmark and governance aid, not in their place.
The paper is at arXiv:2603.20518.