Algebraic Topology and Topological Data Analysis: A New Frontier for Actuarial Claim Reserving

Claim reserving is one of the most important tasks in non-life insurance, as it directly affects solvency assessment, financial reporting, and risk management. Traditional reserving methods often assume a relatively homogeneous claim-development process and may fail to capture hidden structures within complex insurance portfolios. This paper introduces a novel reserving framework that integrates Topological Data Analysis (TDA) with both aggregate and micro-level reserving methodologies. Using a portfolio of motor insurance claim payments, we employ topological techniques to identify latent claim-development regimes and portfolio heterogeneity. The extracted topological information is subsequently incorporated into an Inverse Probability Weighting (IPW) reserving framework and a TDA-enhanced Chain-Ladder (CL) methodology. The empirical results suggest that the proposed TDA-based approaches may improve reserve estimation accuracy relative to their traditional counterparts. Both TDA-IPW and TDA-CL produce reserve estimates that are remarkably close to realized future claim payments. The findings suggest that topological structures contain valuable information for actuarial reserving and that Topological Data Analysis may provide a promising new direction for the development of reserving methodologies.