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Continuous-time modeling and bootstrap for Schnieper's reserving

Nicolas Baradel

Abstract

We revisit Schnieper's model, which decomposes incurred but not reported (IBNR) reserves into two components: reserves for newly reported claims (true IBNR) and reserves for changes over time in the estimated cost of already reported claims (IBNER). We propose a continuous-time stochastic model for the aggregate claims process, driven by a random Poisson measure for the arrival of newly reported claims and by Brownian motion for the cost fluctuations of reported claims. This framework is consistent with the key assumptions of Schnieper's original approach. Within this setting, we develop a bootstrap method to estimate the full predictive distribution of claims reserves. Our approach naturally accounts for asymmetry, ensures non-negativity, and respects intrinsic bounds on reserves, without requiring additional assumptions. We illustrate the method through a case study and compare it with alternative stochastic techniques based on Schnieper's model.

Continuous-time modeling and bootstrap for Schnieper's reserving

Abstract

We revisit Schnieper's model, which decomposes incurred but not reported (IBNR) reserves into two components: reserves for newly reported claims (true IBNR) and reserves for changes over time in the estimated cost of already reported claims (IBNER). We propose a continuous-time stochastic model for the aggregate claims process, driven by a random Poisson measure for the arrival of newly reported claims and by Brownian motion for the cost fluctuations of reported claims. This framework is consistent with the key assumptions of Schnieper's original approach. Within this setting, we develop a bootstrap method to estimate the full predictive distribution of claims reserves. Our approach naturally accounts for asymmetry, ensures non-negativity, and respects intrinsic bounds on reserves, without requiring additional assumptions. We illustrate the method through a case study and compare it with alternative stochastic techniques based on Schnieper's model.
Paper Structure (9 sections, 8 theorems, 71 equations, 3 figures, 5 tables)

This paper contains 9 sections, 8 theorems, 71 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Schnieper's model
  3. A continuous-time model
  4. The bootstrap methodology
  5. Numerical example
  6. Schnieper's model with a parameterized Log-normal distribution
  7. Schnieper'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: Illustration of the weighted linear regression for $\Sigma$
  • Figure 2: Estimated conditional distributions of the total reserve.
  • Figure 3: Comparison of $\sqrt{\widehat{MSEP}}$ and $Q(R; 99.5\%) - \widehat{R}$ as a function of $\mathbb{E}(Z)$.

Theorems & Definitions (19)

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