Paid–incurred chain claims reserving method

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Abstract

We present a novel stochastic model for claims reserving that allows us to combine claims payments and incurred losses information. The main idea is to combine two claims reserving models (Hertig’s (1985) model and Gogol’s (1993) model ) leading to a log-normal paid–incurred chain (PIC) model. Using a Bayesian point of view for the parameter modelling we derive in this Bayesian PIC model the full predictive distribution of the outstanding loss liabilities. On the one hand, this allows for an analytical calculation of the claims reserves and the corresponding conditional mean square error of prediction. On the other hand, simulation algorithms provide any other statistics and risk measure on these claims reserves.

Section snippets

Cumulative payments

Our first observation is that, given Θ, cumulative payments Pi,j 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 Θ, we have for j0logPi,jPi,j1|{Bj1P,Θ}N(Φj,σj2), where we have set Pi,1=1. This gives the CL property (see also Lemma 5.2 in Wüthrich and Merz, 2008) E[Pi,j|Bj1P,Θ]=Pi,j1exp{Φj+σj2/2}. 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 Θ. However, in general, they are not known and need to be estimated from the observations. Assume that we are at time J and that we have observations (see also Table 1) DJP={Pi,j:i+jJ},DJI={Ii,j:i+jJ}andDJ=DJPDJI. 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 σ0,,σJ and τ0,,τJ1 and ΦmN(ϕm,sm2)for {0,,J},ΨnN(ψn,tn2)for n{0,,J

Prediction uncertainty

The ultimate loss Pi,J=Ii,J is now predicted by its conditional expectations E[Pi,J|DJP],E[Pi,J|DJI]orE[Pi,J|DJ], depending on the available information DJP, DJI or DJ (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 σj and τj 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

  • it combines two different channels of information to get a unified ultimate loss prediction;

  • for claims payments observation the CL structure is preserved using credibility weighted

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