Paid–incurred chain claims reserving method
Section snippets
Cumulative payments
Our first observation is that, given , cumulative payments satisfy the assumptions of Hertig’s (1985) log-normal CL model (see also Section 5.1 in Wüthrich and Merz, 2008). That is, conditional on , we have for where we have set . This gives the CL property (see also Lemma 5.2 in Wüthrich and Merz, 2008) The tower property for conditional expectations (see, for example Williams, 1991, 9.7 (i)) then implies
Parameter estimation
So far, all consideration were done for known parameters . However, in general, they are not known and need to be estimated from the observations. Assume that we are at time and that we have observations (see also Table 1) We estimate the parameters in a Bayesian framework. Therefore we define the following model: Model Assumption 3.1 Bayesian PIC Model Assume Model Assumption 1.1 hold true with deterministic and and
Prediction uncertainty
The ultimate loss is now predicted by its conditional expectations depending on the available information , or (see (3.1), (3.4), (3.7)). With Theorem 3.2, Theorem 3.3, Theorem 3.4 all posterior distributions in the Bayesian PIC Model 3.1 are given analytically. Therefore any risk measure for the prediction uncertainty can be calculated with a simple Monte Carlo simulation approach. Here, we consider the conditional mean square error of
Example
We revisit the first example given in Dahms (2008) and Dahms et al. (2009) (see Table 10, Table 11). We do a first analysis of the data under Model Assumption 3.1 where we assume that and are deterministic parameters (using plug-in estimates). In a second analysis we also model these parameters in a Bayesian framework.
Conclusions
We have defined a stochastic PIC model that simultaneously considers claims payments information and incurred losses information for the prediction of the outstanding loss liabilities by assigning appropriate credibility weights to these different channels of information. The benefits of our method are that
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it combines two different channels of information to get a unified ultimate loss prediction;
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for claims payments observation the CL structure is preserved using credibility weighted
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Cited by (41)
Mack-Net model: Blending Mack's model with Recurrent Neural Networks
2022, Expert Systems with ApplicationsCitation Excerpt :The most relevant methods within this family are Munich Chain Ladder (Quarg & Mack, 2004), Double Chain Ladder (Martínez-Miranda, Nielsen, & Verrall, 2012) and Paid-Incurred Chain (Posthuma, Cator, Veerkamp, & Van Zwet, 2008). With regard to this last model, it is worth mentioning that Merz and Wüthrich (2010) developed a Bayesian implementation of it, while Happ, Merz, and Wüthrich (2012) and Happ and Wüthrich (2013) introduced methods strongly related to this approach. Finally, Halliwell (2009) and Venter (2008) used incurred and paid data to develop regression-based reserving models, while Antonio and Plat (2014), Martínez-Miranda, Nielsen, and Verrall (2013) and Pigeon, Antonio, and Denuit (2014) used both sources of information to estimate the expected ultimate cost.
Stochastic reserving with a stacked model based on a hybridized Artificial Neural Network
2021, Expert Systems with ApplicationsRobust Bayesian estimation and prediction of reserves in exponential model with quadratic variance function
2017, Insurance: Mathematics and EconomicsCash flow generalisations of non-life insurance expert systems estimating outstanding liabilities
2016, Expert Systems with ApplicationsCitation Excerpt :While incurred chain ladder probably makes good sense in a deterministic forecasting framework, the stochastic nature of the expert opinion in incurred data is not taken into account by this practice. There is a little literature acknowledging the added value of incurred data: the probably most famous Munich chain ladder approach by Quarg and Mack (2004), regression approaches by Halliwell (1997), Halliwell (2009), Venter (2008), and a paid-incurred chain reserving method by Posthuma, Cator, Veerkamp, and van Zwet (2008), Merz and Wüthrich (2010), Happ, Merz, and Wüthrich (2012) and Happ and Wüthrich (2013). These eight papers combine payment data and incurred data into one statistical model resulting in one reserve estimate.
SPLICE: A synthetic paid loss and incurred cost experience simulator
2023, Annals of Actuarial Science