A Marked Cox Model for the Number of Ibnr Claims: Estimation and Application

Abstract

Incurred but not reported (IBNR) loss reserving is of great importance for Property & Casualty (P&C) insurers. However, the temporal dependence exhibited in the claim arrival process is not reflected in many current loss reserving models, which might affect the accuracy of the IBNR reserve predictions. To overcome this shortcoming, we proposed a marked Cox process and showed its many desirable properties in Badescu et al. (2016).In this paper, we consider the model estimation and applications. We first present an expectation–maximization (EM) algorithm which guarantees the efficiency of the estimators unlike the moment estimation methods widely used in estimating Cox processes. In addition, the proposed fitting algorithm can be implemented at a reasonable computational cost. We examine the performance of the proposed algorithm through simulation studies. The applicability of the proposed model is tested by fitting it to a real insurance claim data set. Through out-of-sample tests, we find that the proposed model can provide realistic predictive distributions.

Modeling the Occurrence of Events Subject to a Reporting Delay via an EM Algorithm

“…The EM framework is however compatible with latent underlying processes affecting the occurrence of events such as hidden Markov models or shot noise process (see e.g. Badescu et al, 2019;Avanzi et al, 2016). Another promising approach would be to investigate how time series models for counts (see Jung and Tremayne, 2011, for an overview) could be introduced in this setting.…”

“…Standard statistical methods for left-truncated data, such as the Cox proportional hazard model (Cox, 1972), can then be used to model the reporting of events. Badescu et al (2016Badescu et al ( , 2019 and Avanzi et al (2016) follow a strategy similar to Antonio and Plat (2014), but model the event occurrence process as a marked Cox process to allow for overdispersion and serial dependency. Along this strand Verrall and Wüthrich (2016) decouple the full likelihood in (1) by considering a plug-in estimate for the weekly periodic occurrence pattern of insurance claims, followed by estimating parametric distributions for the reporting delay.…”

Joint Model Prediction and Application to Individual-Level Loss Reserving

“…The MPP represents events, such as claims or claim payments, as a collection of time points on a timeline with some additional features (called marks) measured at each point. The marked Cox process provides an extension to allow for overdispersion and serial dependence (Avanzi et al, 2016;Badescu et al, 2016b and2016a). Another family of research using individual-level data employs generalized linear models (GLMs) in conjunction with survival analysis to incorporate settlement time as a predictor for ultimate claims (Taylor and Campbell, 2002;McGuire, 2004 andTaylor et al, 2008).…”

Functional Profile Techniques for Claims Reserving

“…To be frank on the last point, we have to say that we compare the proposed method to few, but standard ones; more extensive comparison would be considerably beyond the scope of this paper at this time. In addition to those listed above, the existing methods include also semi-parametric or non-parametric smoothing techniques (England and Verrall, 2001); Bayesian approaches, like that of Bornhuetter and Ferguson (1972), and the Cape-Cod method (Bühlmann, 1983), trying to incorporate also prior information, or utilizing some claim information for reporting delays (Jewell, 1989(Jewell, , 1990; for different perspective, see Clark, 2016); methods that emerged after reviews of Taylor (2000), England and Verrall (2002), and Wüthrich and Merz (2008) include those based on stochastic processes (Pigeon et al, 2014;Godecharle and Antonio, 2015;Badescu et al, 2019), generalized estimating equations (Hudecová and Pešta, 2013), generalized linear mixed models (Gerthofer and Pešta, 2017), copula modeling (Zhao and Zhou, 2010;Pešta and Okhrin, 2014), micro reserving methods based on individual claim developments (Antonio and Plat, 2014;Maciak et al, 2021), and machine learning techniques (Kim et al, 2008;Wüthrich, 2018;Delong et al, 2021).…”