Prediction Uncertainty in the Bornhuetter-Ferguson Claims Reserving - PowerPoint PPT Presentation

Prediction Uncertainty in the Bornhuetter-Ferguson Claims Reserving Method Daniel Alai Michael Merz Mario W¨ uthrich ETH Zurich August 1, 2009 D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 1 / 18

Overview Data and notation. Model considerations. The Bornhuetter-Ferguson predictor. Maximum likelihood estimation of the model parameters. Prediction uncertainty. Numerical example. D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 2 / 18

The Data Let X i , j denote the incremental claims of accident year i ∈ { 0 , 1 , . . . , I } and development year j ∈ { 0 , 1 , . . . , I } . At time I , we have observations D I = { X i , j , i + j ≤ I } . We predict the corresponding lower triangle { X i , j , i + j > I } . Define C i , j to be the cumulative claims of accident year i up to development year j , j � C i , j = X i , k . k =0 D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 3 / 18

The Data Let X i , j denote the incremental claims of accident year i ∈ { 0 , 1 , . . . , I } and development year j ∈ { 0 , 1 , . . . , I } . At time I , we have observations D I = { X i , j , i + j ≤ I } . We predict the corresponding lower triangle { X i , j , i + j > I } . Define C i , j to be the cumulative claims of accident year i up to development year j , j � C i , j = X i , k . k =0 D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 3 / 18

The Data Let X i , j denote the incremental claims of accident year i ∈ { 0 , 1 , . . . , I } and development year j ∈ { 0 , 1 , . . . , I } . At time I , we have observations D I = { X i , j , i + j ≤ I } . We predict the corresponding lower triangle { X i , j , i + j > I } . Define C i , j to be the cumulative claims of accident year i up to development year j , j � C i , j = X i , k . k =0 D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 3 / 18

The Data Let X i , j denote the incremental claims of accident year i ∈ { 0 , 1 , . . . , I } and development year j ∈ { 0 , 1 , . . . , I } . At time I , we have observations D I = { X i , j , i + j ≤ I } . We predict the corresponding lower triangle { X i , j , i + j > I } . Define C i , j to be the cumulative claims of accident year i up to development year j , j � C i , j = X i , k . k =0 D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 3 / 18

A Visual Representation accident development year j year i 0 . . . . . . j I 0 realizations of . . r.v. X i , j , i + j ≤ I . i . . . predicted r.v. X i , j , i + j > I I Figure: Claims development triangle. D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 4 / 18

Model Assumptions (ODP) The incremental claims X i , j are independent overdispersed Poisson distributed (ODP) with E [ X i , j ] = m i , j = µ i γ j , Var ( X i , j ) = φ m i , j , and � I j =0 γ j = 1. ν k are independent unbiased estimators of the expected ultimate ˆ claim µ k = E [ C k , I ] for all k ∈ { 0 , . . . , I } . X i , j and ˆ ν k are independent for all i , j , k . Remark: For MSEP considerations, an estimate of the uncertainty of the ν k is required. We assume that a prior variance estimate � ˆ Var (ˆ ν i ) is given exogenously. D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 5 / 18

Model Assumptions (ODP) The incremental claims X i , j are independent overdispersed Poisson distributed (ODP) with E [ X i , j ] = m i , j = µ i γ j , Var ( X i , j ) = φ m i , j , and � I j =0 γ j = 1. ν k are independent unbiased estimators of the expected ultimate ˆ claim µ k = E [ C k , I ] for all k ∈ { 0 , . . . , I } . X i , j and ˆ ν k are independent for all i , j , k . Remark: For MSEP considerations, an estimate of the uncertainty of the ν k is required. We assume that a prior variance estimate � ˆ Var (ˆ ν i ) is given exogenously. D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 5 / 18

Model Assumptions (ODP) The incremental claims X i , j are independent overdispersed Poisson distributed (ODP) with E [ X i , j ] = m i , j = µ i γ j , Var ( X i , j ) = φ m i , j , and � I j =0 γ j = 1. ν k are independent unbiased estimators of the expected ultimate ˆ claim µ k = E [ C k , I ] for all k ∈ { 0 , . . . , I } . X i , j and ˆ ν k are independent for all i , j , k . Remark: For MSEP considerations, an estimate of the uncertainty of the ν k is required. We assume that a prior variance estimate � ˆ Var (ˆ ν i ) is given exogenously. D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 5 / 18

Model Assumptions (ODP) The incremental claims X i , j are independent overdispersed Poisson distributed (ODP) with E [ X i , j ] = m i , j = µ i γ j , Var ( X i , j ) = φ m i , j , and � I j =0 γ j = 1. ν k are independent unbiased estimators of the expected ultimate ˆ claim µ k = E [ C k , I ] for all k ∈ { 0 , . . . , I } . X i , j and ˆ ν k are independent for all i , j , k . Remark: For MSEP considerations, an estimate of the uncertainty of the ν k is required. We assume that a prior variance estimate � ˆ Var (ˆ ν i ) is given exogenously. D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 5 / 18

The Bornhuetter-Ferguson Predictor In practice, the Bornhuetter-Ferguson (BF) predictor relies on the data D I for the loss development pattern and on external data or expert opinion for the expected ultimate claims E [ C i , I ]. The ultimate claim C i , I of accident year i is predicted by � � C BF = C i , I − i + ˆ ν i ˆ γ j , i , I j > I − i where ˆ γ j is an estimator for γ j . D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 6 / 18

Maximum Likelihood Estimators for ODP „ 1 « X l D I ( µ i , γ j , φ ) = φ ( X i , j log( µ i γ j ) − µ i γ j ) + log c ( X i , j ; φ ) i + j ≤ I j < I „ 1 I − 1 I − 1 » ”– « “ “ ” X X + φ ( X 0 , I log 1 − 1 − + log c ( X 0 , I ; φ ) µ 0 γ n − µ 0 γ n , n =0 n =0 where c ( · , φ ) is the suitable normalizing function. The development pattern obtained, ˆ γ j , is identical to that produced by the chain ladder method, γ CL γ j = ˆ ˆ . j D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 7 / 18

Maximum Likelihood Estimators for ODP „ 1 « X l D I ( µ i , γ j , φ ) = φ ( X i , j log( µ i γ j ) − µ i γ j ) + log c ( X i , j ; φ ) i + j ≤ I j < I „ 1 I − 1 I − 1 » ”– « “ “ ” X X + φ ( X 0 , I log 1 − 1 − + log c ( X 0 , I ; φ ) µ 0 γ n − µ 0 γ n , n =0 n =0 where c ( · , φ ) is the suitable normalizing function. The development pattern obtained, ˆ γ j , is identical to that produced by the chain ladder method, γ CL γ j = ˆ ˆ . j D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 7 / 18

Dispersion Parameter Estimation To estimate the dispersion parameter φ , one could use MLE. We do not. Instead, due to ease of implementation we use Pearson residuals, given by � m i , j ) 2 φ = 1 ( X i , j − ˆ ˆ , d m i , j ˆ i + j ≤ I where d = ( I +1)( I +2) − 2 I − 1 is the degrees of freedom of the model 2 and ˆ m i , j = ˆ µ i ˆ γ j . D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 8 / 18

Dispersion Parameter Estimation To estimate the dispersion parameter φ , one could use MLE. We do not. Instead, due to ease of implementation we use Pearson residuals, given by � m i , j ) 2 φ = 1 ( X i , j − ˆ ˆ , d m i , j ˆ i + j ≤ I where d = ( I +1)( I +2) − 2 I − 1 is the degrees of freedom of the model 2 and ˆ m i , j = ˆ µ i ˆ γ j . D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 8 / 18

Mean Square Error of Prediction The (conditional) mean square error of prediction (MSEP) of the BF predictor � C BF for single accident years i ∈ { 1 , . . . , I } is given by i , I � 2 � �� � � msep C i , I |D I ( � � � C BF C BF i , I ) = E i , I − C i , I � D I � � � 2 � � � 2 � � ν i ) + µ 2 = Var ( X i , j ) + ˆ Var (ˆ γ j − ˆ γ j γ j . i j > I − i j > I − i j > I − i j > I − i We treat the three terms separately. D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 9 / 18

Development Pattern Uncertainty We estimate � � � 2 γ j − γ j ) (ˆ j > I − i by the unconditional expectation �� � � �� �� � 2 �� � E (ˆ γ j − γ j ) = E ˆ γ j − γ j γ k − γ k ˆ . j > I − i j > I − i k > I − i Neglecting that MLEs have a possible bias term we make the following approximation: �� �� �� � � ˆ γ j − γ j ˆ γ k − γ k ≈ Cov (ˆ γ j , ˆ γ k ) . E j > I − i j > I − i k > I − i k > I − i D. H. Alai (ETH Zurich) Prediction Uncertainty in the BF Method August 1, 2009 10 / 18