The over-dispersed Skellam (ODS) reserving model: stochastic reserving for incurred triangles with negative increments

Chain-ladder often gives a sensible best estimate on an incurred claims triangle that has negative increments, yet the stochastic model most teams put around it, the over-dispersed Poisson bootstrap, cannot produce a negative cell. This research note introduces a model that can: the over-dispersed Skellam (ODS) reserving model. It keeps the chain-ladder mean, allows negative increments, and comes with a parametric bootstrap that is several orders of magnitude faster than the usual approach. The original paper is also available as a PDF. The work comes from the actuaries who build TP Tool, SolvencyTool’s software for technical provisions.

Abstract

We introduce a new stochastic reserving model tailored to incurred claims triangles, particularly suited for lines of business with long settlement delays. In addition, we propose a parametric bootstrap procedure that delivers substantial computational efficiency gains compared to standard approaches.

Introduction

Interest in uncertainty quantification has grown steadily in recent years, driven both by compliance requirements and by the need for effective risk management. This development has also encouraged a wider adoption of stochastic reserving methods to quantify the uncertainty of future claims in insurance portfolios.

Let $(X_{ij})_{(i,j)\in\{1,\ldots,I\}\times\{0,\ldots,J\}}$ denote the incremental claim amounts and

denote the observed incremental claim amounts at the end of calendar period $I$. Note that the $X_{ij}$’s can be e.g. claim payments or incurred claims.

It is of interest to find a modeling framework for the predictive distribution of the remaining incremental claim amounts $X_{ij}$ after calendar period $I$:

Such a framework is often desired for the standard case where $X_{ij}\geq 0$ for all $(i,j)$. Here, it can be suitable to use the over-dispersed Poisson (ODP) reserving model from England and Verrall (2002) and apply the non-parametric bootstrap approach implemented by e.g. Gesmann et al. (2025) in the ChainLadder::BootChainLadder function.

In practice, however, the reserving actuary may often experience $X_{ij}<0$ for some lines of business. This can be the case for accident insurance when considering the incremental incurred claims triangle even though chain-ladder might provide a completely reasonable actuarial best estimate. This has been our motivation for introducing a new reserve uncertainty modeling framework consistent with the underlying triangle of such a best estimate.

The ODS reserving model

Model 1. Given $\theta\in\mathbb{R}_{\geq 0}^{I+2J+2}$ defined by

and $\varphi\in(1,\infty)$, assume that $(X_{ij})_{(i,j)\in\{1,\ldots,I\}\times\{0,\ldots,J\}}$ are mutually independent with

Remark 1. Defining $\beta_j:=\beta_j^+-\beta_j^-$ for $j=0,\ldots,J$, the ODS model satisfies

for $(i,j)\in\{1,\ldots,I\}\times\{0,\ldots,J\}$.

Remark 2. The ODP model appears as a special case of the ODS model obtained by setting $\beta_j^-=0$ for $j=0,\ldots,J$. Equivalently, an ODS random variable can be constructed as the difference of two independent ODP random variables with the same dispersion parameter.

Remark 3. For simplicity in this report, the dispersion parameter $\varphi$ is estimated using Pearson residuals applied to the absolute values of both the observed $X_{ij}$ and the corresponding predictions.

Results

We apply the parametric bootstrap implementation of the ODS model to the synthetic triangle in the appendix. The ODS parameters were calibrated to match the chain-ladder estimates for comparability.

For reference, we compare the numerical results to those obtained from the BootChainLadder function from the R package ChainLadder, the standard bootstrap implementation of the over-dispersed Poisson chain-ladder model (England and Verrall), which is widely applied in practice.

Table 1. Summary statistics of simulated reserves.

The two approaches yield very similar reserve estimates and percentiles. Importantly, however, the ODS model remains consistent with the negative increments $X_{ij}$ for $j \in \{1,2\}$ observed in the triangle, while the reference ODP reserving model cannot accommodate such values.

Table 2. Benchmark results.

It appears evident that one can improve the computational performance remarkably by applying this new parametric bootstrap method.

Appendix: the synthetic triangle

References

England, P. D. and Verrall, R. J. (2002). Stochastic claims reserving in general insurance. British Actuarial Journal, 8(3), pp. 443-518. doi:10.1017/S1357321700003809

Gesmann, M. et al. (2025). ChainLadder: Statistical Methods and Models for Claims Reserving in General Insurance. R package. CRAN