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Negative binomial models for development triangles of counts

Luis E. Nieto-Barajas, Rodrigo S. Targino

TL;DR

The paper tackles the problem of predicting outstanding claims in run-off triangles (IBNR) by proposing negative binomial marginal models that capture overdispersion in discrete claim counts. It develops an independent NB specification with parameters $\alpha_i$ and $\pi_j$, and augments it with a latent, moving-average type dependence across development years via a gamma–Poisson mixture controlled by $\gamma_j$ and lag $q$, preserving NB marginals. Bayesian inference is conducted with an augmented likelihood and an MCMC scheme (Gibbs with Metropolis–Hastings steps) and evaluated using LPML, BIAS, and PVAR, including model comparisons to identify the dependence lag. Applications to simulated data, general insurance, and automobile datasets show that incorporating development-year dependence improves predictive accuracy and can reduce reserve overestimation compared with chain-ladder approaches.

Abstract

Prediction of outstanding claims has been done via nonparametric models (chain ladder), semiparametric models (overdispersed poisson) or fully parametric models. In this paper, we propose models based on negative binomial distributions for the prediction of outstanding number of claims, which are particularly useful to account for overdispersion. We first assume independence of random variables and introduce appropriate notation. Later, we generalise the model to account for dependence across development years. In both cases, the marginal distributions are negative binomials. We study the properties of the models and carry out bayesian inference. We illustrate the performance of the models with simulated and real datasets.

Negative binomial models for development triangles of counts

TL;DR

The paper tackles the problem of predicting outstanding claims in run-off triangles (IBNR) by proposing negative binomial marginal models that capture overdispersion in discrete claim counts. It develops an independent NB specification with parameters and , and augments it with a latent, moving-average type dependence across development years via a gamma–Poisson mixture controlled by and lag , preserving NB marginals. Bayesian inference is conducted with an augmented likelihood and an MCMC scheme (Gibbs with Metropolis–Hastings steps) and evaluated using LPML, BIAS, and PVAR, including model comparisons to identify the dependence lag. Applications to simulated data, general insurance, and automobile datasets show that incorporating development-year dependence improves predictive accuracy and can reduce reserve overestimation compared with chain-ladder approaches.

Abstract

Prediction of outstanding claims has been done via nonparametric models (chain ladder), semiparametric models (overdispersed poisson) or fully parametric models. In this paper, we propose models based on negative binomial distributions for the prediction of outstanding number of claims, which are particularly useful to account for overdispersion. We first assume independence of random variables and introduce appropriate notation. Later, we generalise the model to account for dependence across development years. In both cases, the marginal distributions are negative binomials. We study the properties of the models and carry out bayesian inference. We illustrate the performance of the models with simulated and real datasets.
Paper Structure (10 sections, 1 theorem, 19 equations, 11 figures, 6 tables)

This paper contains 10 sections, 1 theorem, 19 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Models
  3. Independence model
  4. Dependence model
  5. Bayesian inference
  6. Numerical analyses
  7. Simulation study
  8. General insurance data
  9. Automobile data
  10. Concluding remarks

Key Result

Proposition 1

Let $\{X_{i,j}\}$ for $i,j=1,\ldots,n$ be a finite sequence whose probability law is described by equations eq:depmodel. Then,

Figures (11)

  • Figure 1: Graphical representation of dependence model \ref{['eq:depmodel']} for $q=2$.
  • Figure 2: Simulated data. Posterior estimates of parameters: $\alpha_i$, $i=1,\ldots,n$ (left) and $\pi_j$, $j=1,\ldots,n$ (right) with $n=10$. True value (dots) and 95% CI (lines).
  • Figure 3: Simulated data. Left: posterior estimates for parameters $\gamma_j$, $j=1,\ldots,n$ with $n=10$. True value (dots) and 95% CI (lines). Right: boxplots of posterior predicted aggregated number of claims $N_i$, for $i=2,\ldots,n$.
  • Figure 4: Simulated data. Posterior predictions for $X_{i,j}$, $i,j=1,\ldots,n$ with $n=10$. True value (dots) and 95% CI (lines). Within sample (solid lines) and out of sample (dotted lines).
  • Figure 5: General insurance data. Posterior estimates of parameters $\alpha_i$, $i=1,\ldots,n$ (left) and $\pi_j$, $j=1,\ldots,n$ (right) with $n=10$. Posterior mean (dots) and 95% CI (lines).
  • ...and 6 more figures

Theorems & Definitions (1)

  • Proposition 1