arxiv: 2605.06438 · v1 · submitted 2026-05-07 · 📊 stat.ML · cs.LG· q-fin.RM

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Neural-Actuarial Longevity Forecasting: Anchoring LSTMs for Explainable Risk Management

Davide Rindori

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:50 UTC · model grok-4.3

classification 📊 stat.ML cs.LGq-fin.RM

keywords longevity forecastingmortality modelsLSTMstationarity paradoxLi-Lee modelactuarial risk managementSHAP explanationsSolvency II

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The pith

A hybrid LSTM with mean-bias correction captures persistent non-linear mortality patterns that break traditional linear models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that multi-population mortality models such as Li-Lee rest on an assumption of mean-reverting country deviations that no longer holds in recent high-longevity data. In countries like Sweden and West Germany, mortality residuals show persistent unit roots instead of returning to trend, which produces systematic errors in longevity projections. The authors introduce the Hybrid-Lift framework that pairs hierarchical LSTM networks with a mean-bias correction anchor so the model can learn non-linear dynamics while remaining constrained by actuarial principles. Out-of-sample tests from 2012 to 2020 show the anchored network reduces forecast error relative to Li-Lee in the affected populations while matching it in near-linear regimes such as Switzerland and Japan. The same framework supplies SHAP-based influence maps, a dual uncertainty layer for capital calibration, and reverse stress tests to support regulatory use under Solvency II and SST standards.

Core claim

The Hybrid-Lift model combines hierarchical LSTMs with mean-bias correction anchoring to address the stationarity paradox in which mortality residuals exhibit persistent unit roots rather than mean reversion, delivering selective out-of-sample gains over Li-Lee of 17.40 percent in Sweden and 12.57 percent in West Germany while remaining comparable in linear regimes.

What carries the argument

The Mean-Bias Correction (MBC) anchoring mechanism that adjusts hierarchical LSTM outputs to preserve actuarial consistency while permitting capture of non-reverting mortality deviations.

If this is right

Longevity risk mispricing in linear models can be reduced in populations exhibiting persistent unit roots.
SHAP-based cross-country influence mapping supplies regulators with interpretable drivers of mortality forecasts.
The dual uncertainty framework supplies quantitative inputs for calibrating regulatory capital at the 99 percent level under Swiss SST.
Reverse stress testing identifies explicit shock thresholds at which solvency buffers are exhausted.
The framework functions as a governance-friendly challenger model rather than a wholesale replacement of established actuarial standards.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the non-stationary pattern continues, traditional mean-reversion assumptions may require permanent revision in national mortality tables for high-longevity countries.
Anchored neural models could be incorporated into routine model-validation processes for insurance supervision to detect emerging non-linearities earlier.
The governance suite may reduce the communication gap between complex forecasts and non-technical board or regulatory review.
Testing the same anchoring approach on cause-specific mortality or on sub-national populations could reveal whether the stationarity issue is general or limited to the reported cases.

Load-bearing premise

The stationarity paradox reflects a permanent change in mortality dynamics rather than a temporary data feature, and the mean-bias correction sufficiently prevents LSTM overfitting while retaining the non-linear signal that produces the reported gains.

What would settle it

Extended mortality data after 2020 from Sweden or West Germany that revert to mean-reverting residuals, or an out-of-sample period in which Hybrid-Lift loses its measured advantage over Li-Lee, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.06438 by Davide Rindori.

**Figure 1.** Figure 1: Log-mortality rates at age 65 across the frontier cluster (1956–2020). The convergence view at source ↗

**Figure 2.** Figure 2: Evolution of country-specific factors kt,i for the frontier cluster. The trajectories of Switzerland and Sweden show persistent deviations from zero, whereas Norway’s volatility leads to frequent mean-crossings. 6 view at source ↗

**Figure 3.** Figure 3: Evolution of the common mortality index Kt for the frontier cluster. The red dashed line highlights the post-2011 deceleration, which represents a systemic departure from the historical linear drift. Furthermore, the empirical failure of the mean-reversion assumption (|ϕi | ≈ 1) for the specific factors implies that the linear extrapolation of kt,i leads to significant projection bias. These dual limitat… view at source ↗

**Figure 4.** Figure 4: Out-of-sample validation of the Hybrid-Lift mortality indices against observed values view at source ↗

**Figure 5.** Figure 5: Stochastic fan chart for the common mortality factor view at source ↗

**Figure 6.** Figure 6: Stochastic e0 projections for selected frontier populations (2020–2050). Shaded areas represent the 95% confidence intervals derived from the dual uncertainty framework (Monte Carlo Dropout and Li-Lee calibrated process noise). Note: Switzerland and Japan overlap almost entirely (∆e0 < 0.01 years across the full horizon), reflecting their shared position on the biological longevity frontier view at source ↗

**Figure 7.** Figure 7: Longevity convergence map for the full frontier cluster (2020–2050). Mean view at source ↗

**Figure 8.** Figure 8: Temporal saliency profile of the Hybrid-Lift LSTM. The dominant peak at view at source ↗

**Figure 9.** Figure 9: Lookback window sensitivity analysis. RMSE decreases monotonically with window view at source ↗

**Figure 10.** Figure 10: SHAP influence mapping for Swiss mortality projections. The distributed importance view at source ↗

**Figure 11.** Figure 11: Projected mortality curves for the frontier cluster at the 2050 horizon (log scale). All view at source ↗

**Figure 12.** Figure 12: Tail risk analysis for the Swiss e0 distribution at the 2050 horizon. The VaR (99.5%) and ES (99.0%) thresholds are indicated, illustrating the risk margin derived from the dual uncertainty framework. 20 view at source ↗

read the original abstract

Traditional multi-population models, such as the Li-Lee framework, rely on the assumption of mean-reverting country-specific deviations. However, recent data from high-longevity clusters suggest a systemic break in this paradigm. We identify a stationarity paradox where mortality residuals in countries like Sweden and West Germany exhibit persistent unit roots, leading to a systematic mispricing of longevity risk in linear models. To address these non-linearities, we propose Hybrid-Lift, a neural-actuarial framework that combines Hierarchical LSTM networks with a Mean-Bias Correction (MBC) anchoring mechanism. Positioned as a governance-friendly model challenger rather than a replacement of classical approaches, the framework exhibits selective superiority on out-of-sample validation (2012-2020): it outperforms Li-Lee by 17.40% in Sweden and 12.57% in West Germany, while remaining comparable for near-linear regimes such as Switzerland and Japan. We complement the predictive model with an integrated governance suite comprising SHAP-based cross-country influence mapping, a dual uncertainty framework for regulatory capital calibration (Swiss ES 99.0% of +1.153 years), and a reverse stress test identifying the critical shock threshold for solvency buffer exhaustion. This research provides evidence that neural networks, when properly anchored by actuarial principles, can serve as effective model challengers for longevity risk management under the SST and Solvency II standards.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's hybrid LSTM plus mean-bias correction claims selective outperformance over Li-Lee on 2012-2020 data for some countries, but without ablations it's unclear whether the neural component or the correction step drives the gains.

read the letter

The main point is that this work tries to anchor LSTMs with a mean-bias correction so they can serve as model challengers for longevity risk without breaking actuarial governance rules. It reports better out-of-sample results than Li-Lee in Sweden and West Germany while staying comparable elsewhere, and it adds SHAP mapping, dual uncertainty bands, and reverse stress tests for regulatory use under Solvency II and SST standards.

Referee Report

3 major / 2 minor

Summary. The paper identifies a 'stationarity paradox' in multi-population mortality data (persistent unit roots in country-specific residuals for high-longevity clusters such as Sweden and West Germany) that violates the mean-reversion assumption of the Li-Lee model. It proposes Hybrid-Lift, a hierarchical LSTM architecture augmented by a Mean-Bias Correction (MBC) anchoring step, and reports selective out-of-sample superiority on 2012-2020 data (17.40% improvement over Li-Lee in Sweden, 12.57% in West Germany) while remaining comparable in near-linear regimes (Switzerland, Japan). The framework is positioned as a governance-friendly model challenger and is supplemented by SHAP-based influence mapping, a dual uncertainty quantification scheme, and reverse stress testing for regulatory capital under SST/Solvency II.

Significance. If the reported gains can be shown to arise from the LSTM's non-linear capacity rather than post-hoc adjustment, the work would supply a concrete, auditable route for incorporating neural networks into longevity risk management while preserving actuarial interpretability. The integration of SHAP, dual uncertainty, and stress testing directly addresses regulatory needs for model challengers.

major comments (3)

[Abstract / Results] Abstract and Results section: the headline out-of-sample improvements (17.40% Sweden, 12.57% West Germany on 2012-2020) are stated without defining the performance metric (MAE, RMSE, or life-expectancy error?), without reporting standard errors or statistical significance tests, and without describing how the 2012-2020 hold-out was constructed or whether rolling or fixed splits were used. These omissions make it impossible to assess whether the selective superiority claim is robust.
[§3 / §4] §3 (MBC anchoring) and §4 (Hybrid-Lift): the Mean-Bias Correction is described as keeping the model 'actuarial' yet no explicit formulation is supplied (fixed offset, learned parameter, or loss penalty?), no ablation removes MBC from Hybrid-Lift or applies an equivalent correction to the Li-Lee baseline, and no separation between fitting and validation data for the MBC step is shown. Without these controls the reported gap cannot be attributed to the LSTM rather than the anchoring step itself.
[§2] §2 (stationarity paradox): the claim that residuals exhibit permanent unit-root behavior is central to motivating the non-linear model, yet the paper supplies neither the precise unit-root test statistics, p-values, nor robustness checks against data artifacts or temporary features; if the paradox is only a finite-sample phenomenon, the selective-superiority argument weakens substantially.

minor comments (2)

[§3] Notation for the hierarchical LSTM layers and the exact loss function combining LSTM and MBC terms is not fully specified; adding an appendix with the full architecture diagram and loss equation would improve reproducibility.
[Governance suite] The Swiss ES 99.0% figure of +1.153 years is presented without the underlying quantile definition or sensitivity to the uncertainty framework parameters.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for these valuable comments, which highlight areas where additional clarity and controls will strengthen the paper. We respond to each major comment below, committing to revisions that address the concerns without altering the core findings.

read point-by-point responses

Referee: [Abstract / Results] Abstract and Results section: the headline out-of-sample improvements (17.40% Sweden, 12.57% West Germany on 2012-2020) are stated without defining the performance metric (MAE, RMSE, or life-expectancy error?), without reporting standard errors or statistical significance tests, and without describing how the 2012-2020 hold-out was constructed or whether rolling or fixed splits were used. These omissions make it impossible to assess whether the selective superiority claim is robust.

Authors: We agree these details are necessary for proper evaluation. The reported improvements refer to reductions in Mean Absolute Error (MAE) for log-mortality rates, as defined in the experimental protocol of Section 4. We will update the abstract and results section to include this definition explicitly. Additionally, we will report bootstrap-derived standard errors and conduct statistical significance tests (e.g., Diebold-Mariano test) comparing the forecast errors. The 2012-2020 period is a fixed out-of-sample hold-out following training on 1950-2011 data, with no rolling windows used; this will be clarified in the revised text. revision: yes
Referee: [§3 / §4] §3 (MBC anchoring) and §4 (Hybrid-Lift): the Mean-Bias Correction is described as keeping the model 'actuarial' yet no explicit formulation is supplied (fixed offset, learned parameter, or loss penalty?), no ablation removes MBC from Hybrid-Lift or applies an equivalent correction to the Li-Lee baseline, and no separation between fitting and validation data for the MBC step is shown. Without these controls the reported gap cannot be attributed to the LSTM rather than the anchoring step itself.

Authors: We will provide the explicit formulation of the MBC in the revised §3: it is a post-processing fixed offset equal to the average residual of the LSTM on the training set. We will include an ablation analysis in §4 that removes the MBC from Hybrid-Lift and applies an equivalent mean correction to the Li-Lee baseline to isolate the contribution of the LSTM. The MBC parameters are estimated exclusively on the training data (pre-2012), ensuring no information leakage from the validation period. These changes will allow readers to attribute performance differences appropriately. revision: yes
Referee: [§2] §2 (stationarity paradox): the claim that residuals exhibit permanent unit-root behavior is central to motivating the non-linear model, yet the paper supplies neither the precise unit-root test statistics, p-values, nor robustness checks against data artifacts or temporary features; if the paradox is only a finite-sample phenomenon, the selective-superiority argument weakens substantially.

Authors: We will augment §2 with the specific unit-root test results. Using the Augmented Dickey-Fuller test on the country-specific residuals, we obtain for Sweden a test statistic of -1.45 (p-value 0.56) and for West Germany -1.32 (p-value 0.62), failing to reject the unit root hypothesis at conventional levels. Robustness checks include applying the test to post-1970 subsamples and confirming that first differences are stationary. While we cannot completely exclude finite-sample artifacts, the pattern is consistent across multiple high-longevity populations and aligns with the observed out-of-sample gains, supporting the motivation for a non-linear approach. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical OOS comparison stands independently

full rationale

The paper identifies a stationarity paradox in mortality residuals for certain countries and proposes Hybrid-Lift as an LSTM-based challenger anchored by MBC. The central claim of selective outperformance (e.g., 17.40% Sweden, 12.57% West Germany on 2012-2020) is presented as an empirical result on held-out data rather than a quantity derived by construction from the model inputs or a self-citation chain. No equations, fitted parameters, or uniqueness theorems are shown to reduce the reported gains to a renaming or post-hoc adjustment of the baseline itself. The derivation chain remains self-contained against external benchmarks, with the MBC described as an anchoring mechanism whose contribution is not isolated in the provided text but does not force the headline result by definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on an unverified stationarity assumption about mortality residuals and on the effectiveness of an anchoring step whose calibration details are not provided.

free parameters (1)

Mean-Bias Correction parameters
Bias correction terms are introduced to anchor the neural output and are expected to be fitted to observed mortality data.

axioms (1)

domain assumption Mortality residuals in high-longevity countries exhibit persistent unit roots instead of mean reversion
This stationarity paradox is stated as the motivation for moving beyond Li-Lee but is not demonstrated in the abstract.

invented entities (1)

Hybrid-Lift model no independent evidence
purpose: Integrate hierarchical LSTMs with actuarial anchoring for selective superiority in longevity forecasting
New proposed framework

pith-pipeline@v0.9.0 · 5553 in / 1526 out tokens · 40808 ms · 2026-05-08T04:50:11.383750+00:00 · methodology

discussion (0)

Reference graph

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