On the use of case estimate and transactional payment data in neural networks for individual loss reserving

Authors: Benjamin Avanzi, Matthew Lambrianidis, Greg Taylor, Bernard Wong

Abstract

The use of neural networks trained on individual claims data has become increasingly popular in the actuarial reserving literature. We consider how to best input historical payment data in neural network models. Additionally, case estimates are also available in the format of a time series, and we extend our analysis to assessing their predictive power. In this paper, we compare a feed-forward neural network trained on summarised transactions to a recurrent neural network equipped to analyse a claim’s entire payment history and/or case estimate development history. We draw conclusions from training and comparing the performance of the models on multiple, comparable highly complex datasets simulated from SPLICE (Avanzi, Taylor and Wang, 2023). We find evidence that case estimates will improve predictions significantly, but that equipping the neural network with memory only leads to meagre improvements. Although the case estimation process and quality will vary significantly between insurers, we provide a standardised methodology for assessing their value.

Complexity vs Empirical Score

Math Complexity: 5.5/10
Empirical Rigor: 8.5/10
Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous empirical evaluation comparing feed-forward and recurrent neural networks for individual loss reserving using synthetic granular claim datasets. It provides high clarity and novel insights regarding the limited marginal value of recurrent memory structures relative to static feature aggregation and individual case estimates.

Research Flowchart

  flowchart TD
    A["Research Goal: Assess impact of payment history & case estimates on neural network reserving"] --> B["Data Inputs: Simulated complex claims datasets from SPLICE"]
    B --> C["Methodology: Compare Feed-Forward NN (summary data) vs Recurrent NN (full history)"]
    C --> D["Computational Process: Train NNs on transactional payments & time-series case estimates"]
    D --> E["Key Finding 1: Case estimates significantly improve reserving predictions"]
    D --> F["Key Finding 2: Equipping NNs with memory (RNN) yields only meagre improvements"]
    E & F --> G["Outcome: Standardised methodology to evaluate case estimate value in individual reserving"]

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ERM in an optimal control framework (2013)
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The Impact of Pension Funding Mechanisms on the Stability and Payoff from S wiss DC Pension Schemes: A Sensitivity Analysis (2016)
A Note on Realistic Dividends in Actuarial Surplus Models (2016)
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On the distribution of the excedents of funds with assets and liabilities in presence of solvency and recovery requirements (2016)
Optimal excedent distributions in an asset-liability surplus model under solvency considerations (2016)
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On optimal joint reflective and refractive dividend strategies in spectrally negative L'evy processes (2016)
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Optimal Dividends in the Dual Model under Transaction Costs (2014)
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On the optimal dividend strategy in a regime-switching diffusion model (2012)
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On optimal periodic dividend strategies with Erlang$(n)$ inter-decision times (2016 WIP)
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A Stochastic Model to Estimate the Amount of IBNR Claims Using Micro-data (2014)
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Tech Guru Riles the Industry By Seeking Huge Patent Fees (2008)
Stochastic Control in Insurance (2008)
Laplace Transforms of Probability Distributions and their Inversions are easy on Logarithmic Scales (2008)
Optimal dividend and issuance of equity policies in the presence of proportional costs (2008)
On optimality of the barrier strategy in de F inetti’s dividend problem for spectrally negative L 'evy processes. (2008)
The compound Poisson risk model with multiple thresholds (2008)
Optimal Dividends in Risk Models with Dependence (2008)
The Decompositions of the Discounted Penalty Functions and Dividends-penalty Identity in a Markov-modulated Risk Model (2008)
The Moments of the Present Value of Total Dividends in the Compound Binomial Model Under a Constant Dividend Barrier and Stochastic Interest Rates (2008)
The dividend function in the jump-diffusion dual model with barrier dividend strategy (2008)
Finite-time dividend-ruin models (2008)
Constant dividend barrier in a risk model with interclaim-dependent claim sizes (2008)
Randomized dividends in the compound binomial model with a general premium rate (2008)
Refracted L 'e vy processes and ruin (2008)
Optimal Dividend Strategies in a C ram 'e r- L undberg Model with Capital Injections (2008)
A risk model with paying dividends and random environment (2008)
Optimal financing and dividend control of the insurance company with proportional reinsurance policy (2008)
Some results on the dual model with dividend payments (2008)
Optimal dividends with incomplete information in the dual model (2008)
Methods for estimating the optimal dividend barrier and the probability of ruin (2008)
On risk model with dividends payments perturbed by a brownian motion - an algorithmic approach (2008)
Optimal Dividends in the Brownian Motion Risk Model with Interest (2009)
The classical risk model with constant interest and threshold strategy (2008)
On a compounding assets model with positive jumps (2008)
Constructing useful theory: The case of Six Sigma (2009)
Dividend Moments in the Dual Risk Model: Exact and Approximate Approaches (2008)
Moments of Discounted Dividends for a Threshold Strategy in the Compound P oisson Risk Model (2008)
Discussion on H. Albrecher, H. U. Gerber and H. Yang’s “A Direct Approach to the Discounted Penalty Function” (2010)
The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber-Shiu Discounted Penalty Function (2008)
Optimal dividends under reinsurance (2008)
Minimizing the Probability of Ruin when Consumption is Ratcheted (2008)
Optimizing venture capital investments in a jump diffusion model (2008)
Recursive Calculation of the Dividend Moments in a Multi-Threshold Risk Model (2008)
Optimal Dividends in the Dual Model with Diffusion (2008)
On Optimal Dividend Strategies: Review and Dual Model (2008)
Optimal Dividend Strategies for a Risk Process under Force of Interest (2008)
A L 'e vy insurance risk process with tax (2008)
On the dual risk model with tax payments (2008)
The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier (2007)
Optimal Dividends in the Dual Model and a Discussion of the Merger of Two Companies (2007)
Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion (2007)
Dividend maximization under consideration of the time value of ruin (2007)
Distribution of the present value of dividend payments in a L 'e vy risk model (2007)
Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs (2007)
On Dual Insurance Risk Models (2007)
On the expected discounted penalty function for a perturbed risk process driven by a subordinator (2007)
The expectation of aggregate discounted dividends for a Sparre Andersen risk process perturbed by diffusion (2007)
Time to ruin, insolvency penalties and dividends in a Markov modulated multi-risk model with common shocks (2007)
Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model (2007)
The Dividend Payments for General Claim Size Distributions under Interest Rate (2007)
Distributional study of De Finetti’s dividend problem for a general L 'e vy insurance risk process (2007)
Topics in Delayed Renewal Risk Models (2007)
Jump diffusion processes and their applications in insurance and finance (2007)
Optimal expected exponential utility of dividend payments in a Brownian risk model (2007)
Optimal Dividend Strategy in the compound Poisson Model with Constant Interest (2007)
Fitting combinations of exponentials to probability distributions (2007)
Optimal Dividend Policy with Mean-Reverting Cash Reservoir (2007)
Correspondence between lifetime minimum wealth and utility of consumption (2007)
A note on the compound binomial model with randomized dividend strategy (2007)
On the optimal dividend problem for a spectrally negative L 'e vy process (2007)
Optimal dividends in the dual model (2007)
Lundberg’s risk process with tax (2007)
On Exact Solutions for Dividend Strategies of Threshold and Linear Barrier Type in a Sparre Anderson Model (2007)
A Risk Model with Multilayer Dividend Strategy (2007)
Time dependent analysis of finite buffer fluid flows and risk models wih a dividend barrier (2007)
An exact and explicit solution for the valuation of American put options (2006)
Some Results behind Dividend Problems (2006)
On the first time of ruin in the bivariate compound Poisson model (2006)
On a discrete time risk model with delayed claims and a constant dividend barrier (2006)
The compound binomial model with randomized decisions on paying dividends (2006)
The compound Poisson risk model with a threshold dividend strategy (2006)
The Diffusion Perturbed Compound Poisson Risk Model with a Dividend Barrier (2006)
The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion (2006)
Enhancing insurer value through reinsurance optimization (2006)
Toward a Unified Approach to Fitting Loss Models (2006)
Maximizing Dividends Without Bankruptcy (2006)
On the Merger of Two Companies (2006)
On optimal dividends: From reflection to refraction (2006)
On Optimal Dividend Strategies in the Compound Poisson Model (2006)
A Note on the Dividends-Penalty Identity and the Optimal Dividend Barrier (2006)
On the Expected Discounted Penalty Function for L 'e vy Risk Processes (2006)
Optimal Dividends under a Ruin Probability Constraint (2006)
Renewal Risk Processes with Stochastic Returns on Investments - A Unified Approach and Analysis of the Ruin Probabilities (2006)
Discussion on X. Zhou’s “On a Classical Risk Model with a Constant Dividend Barrier” (2006)
Optimal Dividends in an O rnstein- U hlenbeck Type Model with Credit and Debit Interest (2006)
Classical and Impulse Stochastic Control for the Optimization of the Dividend and Risk Policies of an Insurance Firm (2006)
On a risk model with dependence between interclaim arrivals and claim sizes (2006)
Structured Credit Portfolio Analysis, Baskets and CDOs (2006)
Exponential behavior in the presence of dependence in risk theory (2006)
On the Non-Optimality of Horizontal Barrier Strategies in the Sparre-Andersen Model (2006)
On a Classical Risk Model with a Constant Dividend Barrier (2005)
Optimal investment for insurer with jump-diffusion risk process (2005)
On a Correlated Aggregated Claims Model with Thinning Dependence Structure (2005)
Weak convergence approach to compound Poisson risk processes perturbed by diffusion (2005)
On a martingale associated to generalized Ornstein–Uhlenbeck processes and an application to finance (2005)
Applied stochastic control of jump diffusions (2005)
The win-first probability under interest force (2005)
Introduction to Stochastic Calculus with Applications (2005)
Flow of Dividends under a Constant Force of Interest (2005)
On the expected time to ruin and the expected dividends when dividends are paid while the surplus is above a constant barrier (2005)
The expected time to ruin in a risk process with constant barrier via martingales (2005)
Payout policy in the 21st century (2005)
Multivariate Counting Processes: Copulas and Beyond (2005)
Optimal Reinsurance and Dividend Distribution Policies in the Cram'er-Lundberg Model (2005)
Representations of the first hitting time density of an O rnstein- U hlenbeck process (2005)
On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier (2005)
On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times (2005)
When does surplus reach a certain level before ruin? (2004)
Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin (2004)
On the martingale property of stochastic exponentials (2004)
De Finetti, Bruno (1906-1985) (2004)
A link between wave governed random motions and ruin processes (2004)
Smoothed Monte Carlo estimators for the time-in-the-red in risk processes (2004)
On a class of renewal risk models with a constant dividend barrier (2004)
An Exact Solution of the Risk Equation with a Step Current Reserve Function (2004)
Ruin probabilities for competing claim processes (2004)
Ruin probabilities and decompositions for general perturbed risk processes (2004)
Optimizing expected utility of dividend payments for a Brownian risk process and a peculiar nonlinear ODE (2004)
Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy (2004)
Monte Carlo Methods in Financial Engineering (2004)
Optimal Dividends: Analysis with B rownian Motion (2004)
Some Optimal Dividend Problems (2004)
Analysis of the ruin probability using Laplace transforms and Karamata-Tauberian theorem (2004)
Approximation of Optimal Reinsurance and Dividend Payout Policies (2004)
A ruin model with dependence between claim sizes and claim intervals (2004)
Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling (2003)
The classical risk model with a constant dividend barrier: analysis of the Gerber–Shiu discounted penalty function (2003)
Asymptotic Theory for a Risk Process with a High Dividend Barrier (2003)
A clarification note about hitting times densities for Ornstein-Uhlenbeck processes (2003)
Geometric Brownian Motion Models for Assets and Liabilities: From Pension Funding to Optimal Dividends (2003)
Pricing Discrete Dynamic Fund Protections (2003)
A note on the expected present value of dividends with a constant barrier in the discrete time model (2003)
Simulation methods in ruin models with non-linear dividend barriers (2003)
On a Correlated Aggregated Claims Model with Poisson and Erlang Risk Processes (2002)
Time in the red in a two state Markov model (2002)
A generalized defective renewal equation for the surplus process perturbed by diffusion (2002)
Optimal Financing of a Corporation Subject To Random Returns (2002)
Fourier/Laplace Transforms and Ruin Probabilities (2002)
On the time value of ruin in the discrete time risk model (2002)
Optimal Dynamic Premium Control in Non-life Insurance. Maximizing Dividend Pay-outs (2002)
Handbook of Brownian motion - facts and formulae (2002)
Risk Theory with a Nonlinear Dividend Barrier (2002)
Perspectives of Risk Sharing (2002)
Spectrally Negative L 'e vy Processes with Applications in Risk Theory (2001)
Continuous Martingales and Brownian Motion (2001)
Optimal risk control for a large corporation in the presence of returns on investments (2001)
Optimal risk and dividend distribution control models for an insurance company (2000)
Investing for Retirement: Optimal Capital Growth and Dynamic Asset Allocation (2000)
Discounted probabilities and ruin theory in the compound binomial model (2000)
Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation (2000)
On the convergence of a solution procedure for a risk model with Gamma-distributed claims (2000)
A process with stochastic claim frequency and a linear dividend barrier (1999)
Controlling Risk Exposure and Dividends Payout Schemes: Insurance Company Example (1999)
Dividend policy and ruin probability (1999)
Optimal risk and dividend control for a company with a debt liability (1998)
Incorporating the value of bankruptcy into the optimal risk/dividend control of financial corporation. (1998)
Discussion on H. U. Gerber and E. S. W. Shiu’s “On the Time Value of Ruin” (1998)
Double barrier hitting time distributions with applications to exotic options (1998)
Optimal proportional reinsurance policies for diffusion models (1998)
Pricing Perpetual Options for Jump Processes (1998)
On the Time Value of Ruin (1998)
Utility Functions: From Risk Theory to Finance (1998)
On the discounted penalty at ruin in a jump-diffusion and the perpetual put option (1998)
Risk Processes Perturbed by alpha-stable L'evy Motion (1998)
L 'e vy Processes (1998)
Optimal choice of dividend barriers for a risk process with stochastic return on investments (1997)
Discrete Multivariate Distributions (1997)
The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin (1997)
The effect of interest on negative surplus (1997)
Survival and growth with liability: Optimal portfolio strategies in continuous time (1997)
Actuarial Mathematics (1997)
Controlled diffusion models for optimal dividend pay-out (1997)
L "o sungsverfahren eines Risikomodells bei exponentiell fallender Schadensverteilung (1996)
Risk vs. profit potential: A model for corporate strategy (1996)
Firm behaviour under the threat of liquidation (1996)
Quelques martingales associ 'e es `a l’int 'e grale du processus d’Ornstein-Uhlenbeck. Application `a l’ 'e tude des premiers instants d’atteinte (1996)
Actuarial bridges to dynamic hedging and option pricing (1996)
Classical numerical ruin probabilities (1996)
Optimization of the flow of dividends (1995)
Optimal investment policies for a frm with a random risk process: Exponential utility and minimizing the probability of ruin (1995)
On some measures of the severity of ruin in the classical Poisson model (1994)
Some comments on the compound binomial model. (1994)
On the Stability of Recursive Formulas (1993)
How long is the surplus below zero? (1993)
Dividend Policy and Signaling by Insurance Companies (1993)
Numerical Solution of Stochastic Differential Equations (1992)
Brownian Motion and Stochastic Calculus (1991)
Risk Theory with the Gamma Process (1991)
Rational ruin problems—a note for the teacher (1991)
Risk theory for the compound P oisson process that is perturbed by diffusion (1991)
Recursive Calculation of Survival Probabilities (1991)
When does the surplus reach a given target? (1990)
Economics of Insurance (1990)
The probability of eventual ruin in the compound binomial model (1989)
Three Methods to Calculate the Probability of Ruin (1989)
Martingales and insurance risk (1989)
The B "u hlmann-Straub Model with the premium calculated according to the variance principle (1989)
Risk theory in a Markovian environment (1989)
On optimal dividend payments and related problems (1988)
Mathematical Fun with the Compound Binomial Process (1988)
Recursive calculation of finite-time ruin probabilities (1988)
Bemerkung zu einem versicherungsmathematischen Modell (1987)
On the Probability and Severity of Ruin (1987)
Optimal Consumption for General Diffusions with Absorbing and Reflecting Barriers (1984)
On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment (1984)
Discussion on M. Davis’ “Piecewise-deterministic Markov processes. A general class of non-diffusion stochastic models” (1984)
On convex principles of premium calculation (1985)
The W iener process with drift between a linear retaining and an absorbing barrier (1981)
On the Probability of Ruin in Presence of a Linear Dividend Barrier (1981)
A Characteristic Property of the Poisson Distribution (1979)
Optimal Reinsurance and Dividend Payment Strategies (1978)
On Optimal Dividend, Reinvestment, and Liquidation Policies for the Firm (1977)
Optimal Control of Jump Processes (1977)
Credibility Formulas of the Updating Type (1975)
The Dilemma Between Dividends and Safety and a Generalization of the L undberg- C ram 'e r Formulas (1974)
The Mathematical Theory of Insurance (1974)
Martingales in Risk Theory (1973)
The Pricing of Options and Corporate Liabilities (1973)
Games of Economic Survival with Discrete- and Continuous-Income Processes (1972)
The Valuation of Option Contracts and a Test of Market Efficiency (1972)
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1972)
Der E influss von Z ins auf die R uinwahrscheinlichkeit (1971)
An extension of the renewal equation and its application in the collective theory of risk (1970)
Mathematical Methods in Risk Theory (1970)
Stochastic Theory of a Risk Business (1969)
Entscheidungskriterien f "u r den zusammengesetzten P oisson- P rozess (1969)
Risk Theory (1969)
An Introduction to Probability Theory and Its Applications (1968)
Combinatorial Methods in the Theory of Stochastic Processes (1967)
A Bayesian Significance Test for Multinomial Distributions (1967)
First exit times from a square root boundary (1967)
The Theory of Risk (1967)
The Theory of Control Processes Applied to Insurance Companies (1967)
Discussion on K. Borch’s “The Theory of Risk” (1967)
One-Person Games of Economic Survival (1966)
The Theory of Stochastic Processes (1965)
Discounted Dynamic Programming (1965)
The Optimal Management Policy of an Insurance Company (1964)
A Remark on Economic Survival Game (1962)
An Economic Survival Game (1962)
The Investment, Financing, and Valuation of the Corporation (1962)
Equilibrium in a Reinsurance Market (1962)
The Random Walk Between a Reflecting and an Absorbing Barrier (1961)
Dividend Policy, Growth, and the Valuation of Shares (1961)
The utility concept applied to the theory of insurance (1961)
Topological Methods in Cardinal Utility Theory (1960)
Games of Economic Survival (1959)
On the collective theory of risk in a case of contagion between claims (1957)
Su un’impostazione alternativa della teoria collettiva del rischio (1957)
Dynamic Programming (1957)
Ruin probabilities for time-correlated claims in the compound binomial model (2001)
Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes. (1955)
Theory of games and economic behavior (1944)
The Theory of Investment Value (1938)
On the mathematical theory of risk (1930)
"U ber die T heorie der R "u ckversicherung (1909)
A study of cross-validation and bootstrap for accuracy estimation and model selection (1995)
Batch normalization: Accelerating deep network training by reducing internal covariate shift (2015)
Layer normalization (2016) — link
Prudential Standard GPS 340 Insurance Liability Valuation (2023)
Long short-term memory (1997)
Learning phrase representations using RNN encoder-decoder for statistical machine translation (2014) — link
Encoding neighbor information into geographical embeddings using convolutional neural networks (2020)
Convolutional neural network classification of telematics car driving data (2019)
Attention is all you need (2017)
Neural networks (2021)
Loss Reserving by CNNs (2021)
A tree-based algorithm adapted to microlevel reserving and long development claims (2019)
Individual loss reserving using a gradient boosting-based approach (2019)
Machine learning in individual claims reserving (2018)
Recurrent Neural Networks for Multivariate Loss Reserving and Risk Capital Analysis (2024) — link
Micro-level reserving for general insurance claims using a long short-term memory network (2023)
The elements of statistical learning: data mining, inference, and prediction (2009)
Gradient-based learning applied to document recognition (1998)
An introduction to statistical learning with applications in Python (2023)
Neural networks for the joint development of individual payments and claim incurred (2020)
Distributional Refinement Network: Distributional Forecasting via Deep Learning — → 2406.00998
Advancing loss reserving: A hybrid neural network approach for individual claim development prediction (2024)
Penalising unexplainability in neural networks for predicting payments per claim incurred (2019)
On the use of cross-validation for time series predictor evaluation (2012)
Smearing estimate: a nonparametric retransformation method (1983)
Recurrent Neural Networks for Multivariate Loss Reserving and Risk Capital Analysis (2025)
MS Windows NT Kernel Description
Bias regularization in neural network models for general insurance pricing (2020)
The balance property in neural network modelling (2022)
Autocalibration and Tweedie-dominance for insurance pricing with machine learning (2021)
Decoupled weight decay regularization (2017) — link
Individual Loss Reserving with Recurrent Neural Networks (2024)
Distributional Refinement Network: Distributional Forecasting via Deep Learning — → 2406.00998
Ensemble distributional forecasting for insurance loss reserving (2024)
Advancing loss reserving: A hybrid neural network approach for individual claim development prediction (2025)
Stochastic loss reserving using generalized linear models (2016)
An Interpretable Deep Learning Model for General Insurance Pricing (2025) — link
AI Tools for Actuaries (2025)
Finding structure in time (1990)

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