Claims Reserving with a Robust Generalized Additive Model

Abstract

In the actuarial literature, many existing stochastic claims-reserving methods ignore the excessive effects of outliers. In practice, however, these outlying observations may occur in the upper triangle and can have a nontrivial and undesirable influence on the existing reserving models. In this article, we consider the situation when outliers of claims are present in the upper triangle. We demonstrate that the model fitting and prediction results of the classical chain-ladder method can be substantially affected by these outliers. To mitigate this negative effect, we propose a robust generalized additive model (GAM). An associated robust bootstrap based on stratified sampling is also developed to obtain a more reliable predictive bootstrap distribution of outstanding claims. Using both simulation examples and real data, we compare our proposed robust GAM with nonrobust counterparts and robust GLM. We demonstrate that the robust GAM provides comparable results with those of other models when outliers are not present and that the robust GAM demonstrates significant improvements in estimation accuracy and efficiency when outliers are present.

ACKNOWLEDGMENTS

We particularly thank the Editor and the anonymous referees for providing valuable and insightful comments on earlier drafts. The usual disclaimer applies.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

Notes

1 The number of knots is usually set at the value larger than the number of degrees of freedom one expects. As stated in Wood and Wood (2015), a specific choice of the number of knots is not critical in general. This number should be large enough of to incorporate the underlying “truth” reasonably well.

2 ${\mathit{D}}_1$ and ${\mathit{D}}_2$ are computed based on the basis functions $a_l(i)$ and $b_l(j),$ respectively. Please refer to equation (5.4) in Wood (2017). In general, the choice of penalty matrix aims to penalize the second derivative squared of splines s₁, s₂.

3 The model error may also exist here, which we do not consider in this study.

4 Another common type of residual, the deviance residuals, is less suitable for bootstrapping because the back-transforming of the resampled deviance residuals to incremental claims cannot be solved analytically.

5 The scale parameter can also be estimated using the sum of squared deviance residuals divided by the associated degrees of freedom. As discussed in England and Verrall (1999), the choice on the type of residuals usually makes little difference.

6 In the presence of over-dispersion, d can be adjusted by multiplying ${\tilde{\varphi }}^{\frac{1}{2}},$ where $\tilde{\varphi }$ is an initial guess of the dispersion parameter. Alternatively, the argument of ψ can be standardized as $r_{i,j}/{\varphi }^{1/2}$ and simultaneous robust estimation of mean and dispersion (Croux, Gijbels, and Prosdocimi 2012) can be used. However, this approach is computationally expensive (and challenging) and lacks theoretical support, and thus, we will leave it to our future research.

7 Without loss of generality, we use a common choice of $\gamma =5\%.$ Alternatives such as 1% trimming proportion will lead to similar results.

8 We also simulate for 10,000 run-off rectangles as a sensitivity test, and the results are very close to those presented in Table 2.

9 Other competing reserving models include the stochastic vector projection method by Portugal, Pantelous, and Assa (2018) and the generalized link ratio method by Portugal, Pantelous, and Verrall (2021) when the chain-ladder model assumption is not justified.

Additional information

Funding

Guangyuan Gao gratefully acknowledges the financial support from the MOE Project of Key Research Institute of Humanities and Social Sciences (22JJD910003).