Macro vs. Micro Methods in Non-Life Claims Reserving (an Econometric Perspective)
1.1. Macro and Micro Methods
- Those models neglect a lot of information that is available on a micro-level (per individual claim). Some additional covariates can be used, as well as exposure, etc. In most applications, not only is that information available, but usually, it has a valuable predictive power. To use that additional information, one cannot simply modify macro-level models, and it is necessary to change the general framework of the model. It becomes possible to emphasize large losses and to distinguish them from regular claims, to get more detailed information about future payments, etc.
- As discussed in this paper, macro-level models on aggregated data can be seen as models on clusters and not on individual observations, as we will do with micro-level models. In the context of macro-level models for loss reserving,  mention that prediction errors can be large, because of the small number of observations used in run-off triangles and the fact that clusters are usually not homogeneous. Quantifying uncertainty in claim reserving methods is not only important in actuarial practice and to assess accuracy of predictive models, it is also a regulatory issue. Finally, a small sample size can cause a lack of robustness and a risk of over-parametrization for macro-level models.
2. Clustering in Generalized Linear Mixed Models
2.1. The Multiple Linear Regression Model
- no multicollinearity in the data matrix;
- exogeneity of the independent variables , , ; and
- homoscedasticity and nonautocorrelation of error terms with .
- The ordinary least-squares estimator for - from Model (1)—is defined as
2.2. The Quasi-Poisson Regression
- Maximum likelihood estimator of is the solution of
- The sum of predicted values is
- By using a similar argument, we have when n goes to infinity
- The property that variances are not equal is a direct consequence of classical results from the theory of generalized linear models (see ), since the covariance matrices of estimators are given by
- Since the MLE and the QLE share the same asymptotic distribution (see ), the proof is similar to Corollary 4(ii).
2.3. Poisson Regression with Random Effect
3. Clustering and Loss Reserving Models
3.1. The Quasi-Poisson Model for Reserves
3.1.2. Illustration and Discussion
- simulate the number of payments for each cluster assuming , ;
- for each cluster, simulate a vector of proportions assuming , ;
- for each cluster, define
- adjust Model C and Model D; and
- calculate the best estimate and the MSEP of the reserve by using Proposition 6.
- impossible to compute that average without individual data;
- discrete explanatory variables used in the micro-level model; and
- since claims reserve model have a predictive motivation, it is risky to project the value of an aggregated variable on future clusters.
3.2. The Mixed Poisson Model for Reserves
3.2.2. Illustration and Discussion
- see previous section;
- for each accident year, allocate randomly the source (t) of each payment;
- fit model G; and
- compute the best estimate and the MSEP of the reserve.
Conflicts of Interest
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|Inv. link func.|
|Model D||28655773||see Figure 1|
|Model E ()||28657364||see Figure 2|
|Model F ()||20514566||see Figure 2|
|mixed Poisson non-cond.||27930624||3297401|
|mixed Poisson cond.||25972947||2280902|
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Charpentier, A.; Pigeon, M. Macro vs. Micro Methods in Non-Life Claims Reserving (an Econometric Perspective). Risks 2016, 4, 12. https://doi.org/10.3390/risks4020012
Charpentier A, Pigeon M. Macro vs. Micro Methods in Non-Life Claims Reserving (an Econometric Perspective). Risks. 2016; 4(2):12. https://doi.org/10.3390/risks4020012Chicago/Turabian Style
Charpentier, Arthur, and Mathieu Pigeon. 2016. "Macro vs. Micro Methods in Non-Life Claims Reserving (an Econometric Perspective)" Risks 4, no. 2: 12. https://doi.org/10.3390/risks4020012