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Continuous-time modeling and bootstrap for chain-ladder reserving

Nicolas Baradel

TL;DR

The paper develops a Brownian-motion–driven, continuous-time stochastic model for aggregated chain-ladder claims, preserving Mack's moment structure while enabling exact, nonnegative reserve simulations via a Gamma-based bootstrap. By bridging Mack's discrete framework with a continuous-time diffusion (a non-reverting Feller-type process with branching properties), the authors provide a robust bootstrap methodology that captures asymmetry and non-negativity without ad hoc residuals. The proposed approach yields reserve distributions that align with traditional Mack-based results on regular data and demonstrates resilience to negative-value artifacts inherent in some discrete-time bootstrap schemes. Empirical results on Mack datasets illustrate the method's practical viability for risk assessment and capital planning in non-life insurance reserving.

Abstract

We revisit the famous Mack's model which gives an estimate for the conditional mean squared error of prediction of the chain-ladder claims reserves. We introduce a stochastic differential equation driven by a Brownian motion to model the accumulated total claims amount for the chain-ladder method. Within this continuous-time framework, we propose a bootstrap technique for estimating the distribution of claims reserves. It turns out that our approach leads to inherently capturing asymmetry and non-negativity, eliminating the necessity for additional assumptions. We conclude with a case study and comparative analysis against alternative methodologies based on Mack's model.

Continuous-time modeling and bootstrap for chain-ladder reserving

TL;DR

The paper develops a Brownian-motion–driven, continuous-time stochastic model for aggregated chain-ladder claims, preserving Mack's moment structure while enabling exact, nonnegative reserve simulations via a Gamma-based bootstrap. By bridging Mack's discrete framework with a continuous-time diffusion (a non-reverting Feller-type process with branching properties), the authors provide a robust bootstrap methodology that captures asymmetry and non-negativity without ad hoc residuals. The proposed approach yields reserve distributions that align with traditional Mack-based results on regular data and demonstrates resilience to negative-value artifacts inherent in some discrete-time bootstrap schemes. Empirical results on Mack datasets illustrate the method's practical viability for risk assessment and capital planning in non-life insurance reserving.

Abstract

We revisit the famous Mack's model which gives an estimate for the conditional mean squared error of prediction of the chain-ladder claims reserves. We introduce a stochastic differential equation driven by a Brownian motion to model the accumulated total claims amount for the chain-ladder method. Within this continuous-time framework, we propose a bootstrap technique for estimating the distribution of claims reserves. It turns out that our approach leads to inherently capturing asymmetry and non-negativity, eliminating the necessity for additional assumptions. We conclude with a case study and comparative analysis against alternative methodologies based on Mack's model.
Paper Structure (9 sections, 7 theorems, 46 equations, 3 figures, 4 tables)

This paper contains 9 sections, 7 theorems, 46 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mack's model
  3. A continuous-time model
  4. The bootstrap methodology
  5. Examples
  6. Mack's model with a parameterized Log-normal distribution
  7. Mack's model with Bootstrap
  8. Time series with Bootstrap
  9. Comparison and conclusion

Key Result

Lemma 2.2

For all $1 \leq i \leq n$ and $s \leq j \leq n$,

Figures (3)

  • Figure 1: Estimated conditional distributions of the total reserve with the dataset from Table \ref{['triangle1']}.
  • Figure 2: Estimated distributions of $C_{2}^{n}$ in our continuous-time model, $\mathbb{P}(C_{2}^n = 0)$ is neglected.
  • Figure 3: Estimated conditional distributions of the total reserve with the dataset from Table \ref{['triangle2']}.

Theorems & Definitions (19)

  • Lemma 2.2
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • proof
  • Corollary 3.7
  • Remark 3.8
  • ...and 9 more