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Insurance Reserve Intelligence Platform

Anugya A, Saket Mohanty, Abhilash Timmapur, Somya Rai

arXiv:2609.30765v1cs.AI

TL;DR

Repeated reserve calculations for sensitivity analysis, optimization, and large scenario evaluations can make classical Thiele-equation solving computationally expensive. This paper combines a classical solver with a PINN/KINN surrogate for term-life reserves. The final model reports R2 = 0.9887, MAE 785.48, and RMSE 1,212.76, alongside a 119.53x runtime speed-up over the classical solver in the measured benchmark.

  • Problem

    Repeated reserve calculations for sensitivity analysis, optimization, and large-scale scenario evaluation are computationally expensive with classical methods.

    Source: §1 · p. 2

  • Method

    The platform combines a classical Thiele-equation solver with a PINN/KINN neural surrogate for term-life reserve modelling.

    Source: §4.2 · p. 4, §15 · p. 16

  • Results

    R2 = 0.9887, MAE 785.48, and RMSE 1,212.76; the measured benchmark reports a 119.53x runtime speed-up over the classical solver.

    Source: §12 · p. 15, §15 · p. 16

  • Takeaways & Limitations

    The conclusion is that physics- and knowledge-informed networks can approximate classical reserve calculations effectively, with actuarial reliability requiring evaluation across physics consistency, boundary behavior, monotonicity, scenario response, sensitivity, optimization stability, runtime, and generalization.

    Source: §15 · p. 16

    Although knowledge-informed constraints improve reserve behavior, further work is required; monotonicity and generalization remain important limitations.

    Source: §13 · p. 15, §15 · p. 16

Abstract

from arXiv · show

Insurance reserve estimation is a fundamental actuarial task supporting premium pricing, solvency assessment, financial reporting, capital planning, and risk management. Classical reserve methods based on Thiele's differential equation provide a rigorous and interpretable foundation for life insurance valuation, but repeated reserve calculations become computationally expensive in sensitivity analysis, optimization, and large-scale scenario evaluation. This paper presents an Insurance Reserve Intelligence Platform for term-life reserve modelling that combines a classical Thiele-equation solver with a Physics-Informed Neural Network (PINN) enhanced by Knowledge-Informed Neural Network (KINN) losses. The framework includes synthetic policy generation, risk-adjusted premium calculation, classical reserve trajectory generation, reserve-ratio dataset construction, configurable neural training, validation diagnostics, sensitivity and elasticity analysis, prototype optimization workflows, and interest-rate scenario testing. A key refinement is the use of premium ratio and the explicit separation of pricing-time and scenario-time interest-rate semantics. The final model uses seven features: elapsed time, issue age, pricing interest rate, scenario interest rate, premium ratio, sum assured, and mortality intensity. It predicts a standardized reserve ratio instead of raw reserve values, improving numerical stability across policies with different sums assured. The model achieved an R2 of 0.9887, MAE of 785.48, and RMSE of 1212.76 on the test set. On 200 policies, PINN/KINN inference was approximately 119.53 times faster than the classical solver. Results show strong predictive accuracy, physics consistency, and boundary performance, while highlighting remaining limitations in monotonicity and out-of-distribution generalization.

1 Introduction

Repeated reserve calculations for trajectories, policy comparisons, sensitivity studies, and changed assumptions can make classical solving expensive. The project proposes a term-life platform combining classical reserve computation with neural prediction, evaluation, sensitivity analysis, and prototype optimization.

1 Introduction

  • Repeated calculations across policies, time points, scenarios, and optimization iterations can make reliable, interpretable classical reserve solving expensive.Actuaries need reserve trajectories over time, policy comparisons, sensitivity studies, and controlled experiments under changed assumptions.Source: §1 · p. 2
  • A differentiable neural surrogate trained against a classical solver can approximate the reserve function and provide rapid estimates after training.PINNs can include the governing actuarial equation in their training objective.Source: §1 · p. 2

2 Project Scope

  • The project develops a term-life reserve platform integrating classical computation with PINN/KINN prediction, evaluation, sensitivity analysis, and prototype optimization.Its scope includes synthetic policies, a Thiele solver, configurable losses, validation diagnostics, and prototype workflows.Source: §1 · p. 2, §2 · p. 2
  • Production deployment, real-data validation, complete baseline benchmarking, and a completed insurance digital twin are explicitly outside the paper’s claims.These are identified as future work.Source: §2 · p. 3

3 Contributions

The contributions combine a classical Thiele solver with a neural reserve surrogate and add physics-based and actuarial constraints to training. The framework also introduces synthetic policy generation and normalized reserve and premium representations.

3 Contributions

  • The classical Thiele solver supplies benchmark reserve trajectories, while the neural surrogate learns a differentiable approximation of the reserve surface.For term-life insurance, reserves depend on elapsed time, age, mortality, interest rate, premium, and sum assured; the solver integrates backward from maturity.Source: §3 · p. 3, §4.2 · p. 4, §4.1 · p. 4
  • The PINN objective includes the Thiele-equation residual, penalizing learned reserve surfaces that violate the actuarial differential equation.The equation describes reserve evolution using valuation interest, premium, mortality intensity, and sum assured.Source: §3 · p. 3, §4.2 · p. 4
  • The KINN objective adds actuarial knowledge beyond the PDE, including boundary behavior, monotonicity, reserve ceilings, solvency, smoothness, and scenario-oriented losses.The framework also identifies accurate, stable, interpretable reserves as important because under-reserving creates solvency risk while over-reserving reduces capital efficiency.Source: §3 · p. 3, §4.1 · p. 4
  • Synthetic term-life policies are generated using issue age, term, interest rate, mortality, premium, sum assured, and underwriting assumptions.Source: §3 · p. 3
  • The model predicts standardized reserve ratio V/S rather than raw monetary reserves, improving numerical stability across heterogeneous policy sizes.Source: §3 · p. 3
  • Replacing raw premium with premium ratio P/S lets premium and sum assured be treated more independently during sensitivity analysis.Source: §3 · p. 3

5 Dataset and Experimental Setup

The experimental setup uses synthetic term-life policies and checks their reserve variation across actuarial factors and policy duration. It represents policies with distinct pricing and scenario interest rates and uses a standardized reserve-ratio target.

5.1 Synthetic Dataset

  • Because real insurance datasets were unavailable, the project uses synthetic policies; Table 1 summarizes the dataset and policy-generation configuration.Reserve distribution statistics from the generated dataset are shown in Table 2.Source: §5.1 · pp. 4–5, Table 1 · p. 4, Table 2 · p. 5
  • The synthetic portfolio’s exploratory plot checks how peak reserve varies with issue age and policy term.The colour scale represents term, showing age and contract duration jointly.Source: §5.1 · p. 5, Figure 1 · p. 5

5.2 Feature Representation

  • The feature representation distinguishes pricing and scenario rates: the pricing rate sets the premium at issue, while the scenario rate values reserves under changed economic assumptions.The seven features are elapsed duration, issue age, pricing rate, scenario rate, premium ratio, sum assured, and mortality intensity.Source: §5.2 · p. 5

5.3 Reserve-Ratio Target

  • The model predicts a standardized reserve ratio rather than raw reserve, and the reserve is subsequently recovered using the stated transformation.Table 3 summarizes feature and target normalization statistics used in the runtime-analysis run.Source: §5.3 · p. 6, Table 3 · p. 6

6 Model Architecture and Learning Framework

The reserve predictor combines a fully connected neural network with Thiele-equation learning and actuarial constraints. Its weighted loss framework trains for reserve accuracy as well as equation compliance and actuarial behavior.

6.1 Network Architecture and Training Configuration

  • A fully connected network serves as the reserve predictor, and Table 4 presents its architecture and training configuration.The tanh activation provides smooth derivatives for PINN losses; dropout is disabled to avoid disturbing derivative-based losses and sensitivity estimates.Source: §6.1 · pp. 6–7, Table 4 · p. 7

6.2 Physics-Informed Learning

  • The model differentiates predicted reserves over time and compares the derivative with Thiele’s equation’s right-hand side, using the discrepancy as a PDE residual loss.This makes the governing actuarial equation part of the optimization objective, rather than training only on reserve labels.Source: §6.2 · p. 7

6.3 Knowledge-Informed Learning

  • Training adds actuarial constraints for terminal boundaries, monotonicity, reserve bounds, smooth evolution, and scenario-based supervision.Combining supervised learning, physics constraints, and actuarial knowledge makes the framework a hybrid PINN/KINN.Source: §6.3 · p. 7

7 Loss Framework

  • The total loss is a weighted sum of enabled losses, so training targets reserve accuracy alongside compliance with the differential equation and actuarial behavior.Figure 2 presents the loss framework.Source: §7 · p. 8, Figure 2 · p. 8

8 Development Timeline

Development progressed from stabilizing reserve-ratio prediction to validation, sensitivity analysis, optimization workflows, and interest-rate scenario behavior. The framework connects the PINN/KINN model with losses, training, prediction, and evaluation.

8 Development Timeline

  • Runs 1–3 replaced raw reserve targets with standardized reserve-ratio targets after near-zero predictions and unstable training, while adding trajectory comparisons to monitor fit to classical behavior.Inputs were normalized and loss weights tuned.Source: §8 · p. 8
  • Runs 4–8 reformulated the PDE residual in reserve-ratio space after raw-currency residuals proved numerically unstable, and added validation dashboards and sum-assured monotonicity checks.Reserve dependency plots were also added for actuarial consistency and interpretability.Source: §8 · p. 8
  • Runs 9–15 replaced raw premium with premium ratio P/S after premium dominated the learned reserve response, and added fixed-reference one-variable-at-a-time sensitivities.Sensitivity and elasticity analysis covered interest rate, mortality, premium, and sum assured.Source: §8 · p. 9
  • Runs 16–21 added pricing-style, product-design-style, and portfolio-style optimization workflows, then introduced separate pricing-rate and scenario-rate semantics and interest-rate scenario and peak-reserve losses.The optimization experiments showed the trained model could be used inside iterative decision loops.Source: §8 · p. 9

9 Framework Architecture

  • Figure 3 depicts a framework that connects the PINN/KINN model to the loss system, training process, prediction workflow, and evaluation modules.This is presented as more integrated than a simple prediction pipeline.Source: §9 · p. 9, Figure 3 · p. 9

10.2 Validation Diagnostics

  • Validation evaluates accuracy, boundary satisfaction, PDE residual behavior, monotonicity, and generalization because predictive accuracy alone does not guarantee actuarially correct behavior under perturbations..Source: §10.2 · p. 10

10.3 Sensitivity and Elasticity

  • Sensitivity analysis measures finite-difference reserve responses to one feature at a time, while elasticity expresses sensitivities as comparable percentage effects.Source: §10.3 · p. 10

11 Results

The PINN/KINN model agrees strongly with the classical solver in aggregate and for representative reserve trajectories, while validation reveals weaker monotonicity and generalization. It also supports faster reserve evaluation and prototype optimization, but mortality response remains an unresolved sensitivity issue.

11.1 Prediction Performance

  • The test-set results indicate strong aggregate agreement with the classical actuarial solver.Table 5 reports final prediction performance; the supplied passages do not include its metric values.Source: Table 5 · p. 10, §11.1 · p. 10

11.2 Reserve Trajectory Comparison

  • Representative policies show close reserve-trajectory tracking, with mean errors of approximately 61, 153, 47, and 31 monetary units.Shorter-term and small-reserve policies remain more challenging.Source: §11.2 · p. 11, Figure 4 · p. 11

11.3 Validation Results

  • Validation shows strong accuracy, physics, and boundary scores but weak monotonicity and generalization.The results indicate that aggregate fit and local physics behaviour do not yet ensure reliability across all actuarial monotonicity and out-of-distribution requirements.Source: §11.3 · p. 11, Table 6 · p. 11, Figure 5 · p. 12

11.4 Runtime Benchmark

  • On the tested 200-policy workload, the trained PINN/KINN evaluates reserves substantially faster than repeated numerical solution of Thiele’s equation.The benchmark was run on an Ubuntu-based cloud environment with an NVIDIA T4 GPU; the speed advantage is specific to that hardware and software setting.Source: §11.4 · p. 12, Table 7 · p. 12

11.5 Sensitivity Results

  • The reference-policy sensitivity experiment still predicts a negative mortality response, contrary to the expected actuarial direction for term-life reserves.The sensitivity results are reported as raw partial derivatives in Table 8.Source: §11.5 · p. 13, Table 8 · p. 13, Figure 6 · p. 13

11.6 Elasticity Results

  • In the reference-policy elasticity experiment, premium ratio has the strongest relative effect, while mortality response remains the main unresolved sensitivity issue.Elasticities express sensitivities as unit-free percentage responses.Source: §11.6 · p. 13, Table 9 · p. 13, Figure 7 · p. 14

11.7 Optimization Results

  • Optimization plots show reserve-value convergence across pricing, product-design-style, and portfolio-style experiments, but their differing objectives and constraints make values incomparable.The optimization modules demonstrate use inside optimization loops; stronger claims require objective values, constraints, success flags, and comparisons against classical-solver optimization.Source: §11.7 · p. 14, Table 10 · p. 14

12 Discussion

The discussion reports strong aggregate prediction, physics, and boundary performance, alongside faster reserve evaluation, but identifies weak generalization and incorrect mortality response. The limitations section bounds the evidence to synthetic term-life data and describes optimization as proof of concept without comprehensive baseline comparisons or ablations.

12 Discussion

  • The model achieves strong aggregate prediction and physics and boundary behaviour, with reserve evaluation approximately 119.53x faster than the classical solver on the tested workload.The discussion also reports low monotonicity and weak generalization scores, with an incorrect mortality response in the reference-policy experiment.Source: §12 · pp. 14–15

13 Limitations

  • The model was developed and evaluated on synthetic term-life data; validation on real-world insurance portfolios remains future work.The current implementation also does not yet support more complex insurance products.Source: §13 · p. 15
  • Further work is needed to improve generalization and monotonicity across all policy variables.These remain limitations despite the use of knowledge-informed constraints.Source: §13 · p. 15
  • Optimization modules are proof-of-concept implementations, and evaluation lacks comprehensive baseline comparisons and ablation studies.The paper also describes the framework as a modular foundation for integrating classical actuarial modelling with physics-informed machine learning.Source: §13 · p. 15

14 Future Work

Future work calls for broader benchmarking, larger-scale runtime studies, real or industry-calibrated data, and richer product and economic scenarios, before a governed insurance digital twin is pursued. The conclusion reports strong test-set performance and a measured runtime speed-up, while emphasizing remaining monotonicity and generalization limitations.

14 Future Work

  • Future evaluation should add standard machine-learning baselines and ablations, then test runtime and scalability beyond the 200-policy benchmark.Suggested portfolio sizes are 1,000, 10,000, and 50,000 policies.Source: §14 · p. 15
  • Future versions should test real or industry-calibrated datasets, support more complex products and richer scenarios, and add uncertainty quantification.Suggested scenarios include lapse, inflation, stochastic mortality, and full interest-rate term structures; proposed uncertainty methods include ensembles, quantile regression, conformal prediction, and Bayesian neural networks.Source: §14 · p. 15
  • An insurance digital twin is a long-term extension, following stronger validation, broader benchmarking, and more formal model-governance review.The proposed direction includes continuous reserve monitoring, real-time scenario analysis, optimization, and AI-assisted actuarial decision support.Source: §14 · p. 16

15 Conclusion

  • The platform combines a classical Thiele solver with a PINN/KINN surrogate and includes synthetic policy generation, reserve-ratio data, diagnostics, sensitivity and elasticity analysis, runtime benchmarking, and prototype optimization.The PINN incorporates Thiele’s differential-equation residual; the KINN adds actuarial constraints including boundary behaviour, monotonicity, reserve ceilings, solvency, smoothness, and scenario-specific losses.Source: §15 · p. 16
  • The final model reports R2 = 0.9887, MAE 785.48, and RMSE 1,212.76, plus a 119.53x runtime speed-up over the classical solver in the measured benchmark.The conclusion identifies monotonicity and generalization as remaining limitations and calls for evaluation beyond aggregate prediction accuracy.Source: §15 · p. 16

A Variable Definitions

Appendix A provides a table of the paper’s main variables, while Appendix B lists the detailed loss formulas by loss type. The supplied passages identify the table and formula categories but do not provide variable entries or formula expressions.

A Variable Definitions

  • Table 11 is identified as the source for the paper’s main variable definitions.The supplied passage gives the table title but not its variable entries.Source: Table 11 · p. 17

B Detailed Loss Formulas

  • Appendix B names supervised data, Thiele reserve-ratio PDE, boundary, mortality and sum-assured monotonicity, interest-rate scenario and sensitivity, solvency, reserve ceiling, smoothness, peak-reserve, L2 regularization, and Huber losses.The supplied passages identify these formula categories but do not provide their expressions.Source: B Detailed Loss Formulas · pp. 17–18
  • The listed loss categories also include sum-assured monotonicity, interest-rate scenario, peak-reserve, L2 regularization, and Huber losses.These are named as detailed loss formulas in Appendix B.Source: B Detailed Loss Formulas · pp. 18–19

C Knowledge-Informed Constraints

The section identifies a table of knowledge-informed training constraints and a subsequent table summarizing development stages.

C Knowledge-Informed Constraints

  • A table summarizes knowledge-informed constraints incorporated during training.Source: Table 12 · p. 19

D Development Summary

  • A table summarizes the platform’s major development stages.Source: Table 13 · p. 19

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