Development Pattern and Prediction Error for the Stochastic Bornhuetter-Ferguson Claims Reserving Method

Transcription

1 Development Pattern and Predcton Error for the Stochastc Bornhuetter-Ferguson Clams Reservng Method Annna Saluz, Alos Gsler, Maro V. Wüthrch ETH Zurch ASTIN Colloquum Madrd, June 2011

2 Overvew 1 Notaton 2 Bornhuetter-Ferguson Method (BF) 3 Normal Model 4 Estmaton of the Parameters and ther Correlatons 5 Predcton Uncertanty 6 Conclusons and Remarks A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

3 Notaton Accdent years, 0 I Table: Clams development trangle accdent development year j year 0... j... J 0. observatons D I. D c I I A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

4 Notaton Accdent years, 0 I Development years j, 0 j J I Table: Clams development trangle accdent development year j year 0... j... J 0. observatons D I. D c I I A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

5 Notaton Accdent years, 0 I Development years j, 0 j J I Incremental clams X,j Table: Clams development trangle accdent development year j year 0... j... J 0. observatons D I. D c I I A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

6 Notaton Accdent years, 0 I Development years j, 0 j J I Incremental clams X,j Cumulatve clams C,j Table: Clams development trangle accdent development year j year 0... j... J 0. observatons D I. D c I I A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

7 Notaton Accdent years, 0 I Development years j, 0 j J I Incremental clams X,j Cumulatve clams C,j Observed clams at tme I: D I = {X,j ; + j I} Table: Clams development trangle accdent development year j year 0... j... J 0. observatons D I. D c I I A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

8 Notaton Accdent years, 0 I Development years j, 0 j J I Incremental clams X,j Cumulatve clams C,j Observed clams at tme I: D I = {X,j ; + j I} Goal: Predct D c I = {X,j; + j > I, I, j J} Table: Clams development trangle accdent development year j year 0... j... J 0. observatons D I. D c I I A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

9 Bornhuetter-Ferguson Method (BF) BF predctor for the outstandng loss labltes (IBNR and IBNeR n case of ncurred clams data) R = C,J C,I at tme I where ˆµ a pror estmate for µ = E[C,J ] R = ˆµ (1 ˆβ I ), A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

10 Bornhuetter-Ferguson Method (BF) BF predctor for the outstandng loss labltes (IBNR and IBNeR n case of ncurred clams data) R = C,J C,I at tme I where ˆµ a pror estmate for µ = E[C,J ] R = ˆµ (1 ˆβ I ), 1 ˆβ I estmated stll to come factor at the end of development year I, ˆβ J = 1, (( ˆβ j ) j=0,...,j estmated cumulatve development pattern) A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

11 Bornhuetter-Ferguson Method (BF) BF predctor for the outstandng loss labltes (IBNR and IBNeR n case of ncurred clams data) R = C,J C,I at tme I where ˆµ a pror estmate for µ = E[C,J ] R = ˆµ (1 ˆβ I ), 1 ˆβ I estmated stll to come factor at the end of development year I, ˆβ J = 1, (( ˆβ j ) j=0,...,j estmated cumulatve development pattern) ˆγ 0 = ˆβ 0, ˆγ j = ˆβ j ˆβ j 1, 1 j J, (estmated ncremental development pattern) A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

12 Bornhuetter-Ferguson Method (BF) BF predctor for the outstandng loss labltes (IBNR and IBNeR n case of ncurred clams data) R = C,J C,I at tme I where ˆµ a pror estmate for µ = E[C,J ] R = ˆµ (1 ˆβ I ), 1 ˆβ I estmated stll to come factor at the end of development year I, ˆβ J = 1, (( ˆβ j ) j=0,...,j estmated cumulatve development pattern) ˆγ 0 = ˆβ 0, ˆγ j = ˆβ j ˆβ j 1, 1 j J, (estmated ncremental development pattern) Assumpton of a cross-classfed model E[X,j ] = µ γ j A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

13 Motvaton for the Estmaton of the Pattern Often the chan ladder (CL) development pattern s used for ˆγ j J 1 ˆγ j CL = k=j ˆf 1 k J 1 k=j 1 I k 1 ˆf 1 k, =0 C,k+1 ˆfk = I k 1. =0 C,k Issue: In the BF method we are gven a pror estmates of the µ. The CL pattern gnores any pror nformaton. Goal: Incorporate the a pror estmates n the estmaton of the development pattern. A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

14 Motvaton for the Estmaton of the Pattern Often the chan ladder (CL) development pattern s used for ˆγ j J 1 ˆγ j CL = k=j ˆf 1 k J 1 k=j 1 I k 1 ˆf 1 k, =0 C,k+1 ˆfk = I k 1. =0 C,k Issue: In the BF method we are gven a pror estmates of the µ. The CL pattern gnores any pror nformaton. Goal: Incorporate the a pror estmates n the estmaton of the development pattern. If the µ were known then the best lnear unbased estmate of γ j would be γ (0) j = I j A frst canddate s the raw estmate =0 X,j I j =0 µ, 0 j J. ˆγ (0) j = However, they do not sum up to 1. I j =0 X,j I j =0 ˆµ, A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

15 Intutve Estmaton of the Pattern If the full rectangle was known an obvous estmate s gven by I =0 ˆγ j = X,j I =0 C, 0 j J.,J A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

16 Intutve Estmaton of the Pattern If the full rectangle was known an obvous estmate s gven by I =0 ˆγ j = X,j I =0 C, 0 j J.,J If only the upper trapezod and the µ are known, replace the unknown X,j by predctors µ ˆγ j ˆγ j = I j =0 X,j + I =I j+1 µ ˆγ j I J =0 C,J + ( I =I J+1 C,I + ), 0 j J. J l=i +1 µ ˆγ l A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

17 Intutve Estmaton of the Pattern If the full rectangle was known an obvous estmate s gven by I =0 ˆγ j = X,j I =0 C, 0 j J.,J If only the upper trapezod and the µ are known, replace the unknown X,j by predctors µ ˆγ j ˆγ j = I j =0 X,j + I =I j+1 µ ˆγ j I J =0 C,J + ( I =I J+1 C,I + ), 0 j J. J l=i +1 µ ˆγ l In an over-dspersed Posson model the maxmum lkelhood estmates (MLE s) satsfy these equatons. A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

18 Normal Model Model Assumptons (Normal Model) N1 The X,j are ndependent and normally dstrbuted and there exst parameters µ 0, µ 1,..., µ I and γ 0, γ 1,..., γ J wth J j=0 γ j = 1 and σ0, 2..., σj 2 such that E[X,j ] = µ γ j, Var(X,j ) = µ σ 2 j, where σj 2 > 0, 0 j J. N2 The a pror estmates ˆµ for µ = E[C,J ] are unbased and ndependent of all X l,j. A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

19 Maxmum Lkelhood Estmaton (MLE) of the γ j s We calculate the MLE s assumng that the true µ s and σj 2 s are known and then replace the µ s by the a pror estmates ˆµ and the σj 2 s by estmates ˆσ j 2. A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

20 Maxmum Lkelhood Estmaton (MLE) of the γ j s We calculate the MLE s assumng that the true µ s and σj 2 s are known and then replace the µ s by the a pror estmates ˆµ and the σj 2 s by estmates ˆσ j 2. In the Normal Model we obtan ˆγ j = I j =0 X,j I j =0 ˆµ + }{{} raw estmate ˆσ 2 j / I j =0 ˆµ J l=0 (ˆσ 2 l / I l =0 ˆµ ) ( 1 J l=0 ) I l =0 X,l I l =0 ˆµ, } {{ } correcton term where ( ˆσ j 2 = 1 I j ) I j 2 X,j =0 ˆµ X,j I j ˆµ I j =0 ˆµ, 0 j J, j I. =0 A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

21 Covarance matrx of the ˆγ j s We use the asymptotc propertes of MLE s (Fsher Informaton matrx) to estmate the covarance matrx of the ˆγ j s.: Remarks: Ĉov(ˆγ j, ˆγ k ) = ˆσ 2 j I j =0 ˆµ For the best lnear unbased estmate γ (0) j (compare wth frst summand). ( ˆσ k 2 1 / ) I k =0 ˆµ {j=k} J l=0 (ˆσ l 2/ ). I l =0 ˆµ (n the case of known µ ) we have σ 2 j Cov(γ (0) j, γ (0) k ) = 1 {j=k} I j =0 µ, Because of the sde constrant J j=0 γ j = 1 the off-dagonal correlatons must be negatve (compare wth second summand). A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

22 Predcton Uncertanty Gven all nformaton I I (.e. D I and all ˆµ ), the condtonal MSEP of the predctor R = ˆµ (1 ˆβ I ) s gven by [ ( msep R I I ( R ) = E R R ) ] 2 I I 2 J = E X,j ˆµ (1 ˆβ I ) I I j=i +1 = J j=i +1 Var(X,j ) }{{} Process Varance (PV ) ( + ) 2 ˆµ (1 ˆβ I ) µ (1 β I ) } {{ } Estmaton Error (EE ). A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

23 MSEP We estmate the condtonal MSEP gven I I as follows msep R I I ( R J ) = Var(X,j ) + Var(ˆµ )(1 ˆβ I ) 2 j=i +1 I + ˆµ 2 Var(ˆγ j ) + 2 j=0 0 j<k I Ĉov(ˆγ j, ˆγ k ), msep I =I J+1 R I I + 2 I J+1 <k I ( I ( =I J+1 R ) = I =I J+1 (1 ˆβ I )(1 ˆβ I k )Ĉov(ˆµ, ˆµ k ) ) I I k + ˆµ ˆµ k Ĉov(ˆγ j, ˆγ l ). j=0 l=0 msep R I I ( R ) A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12

24 Conclusons and Remarks Under dstrbutonal assumptons we have derved estmates for the development pattern takng all relevant nformaton nto account formulas for the smoothng from the raw estmates ˆγ (0) j to the fnal estmates ˆγ j estmates for the correlatons of these estmates. We recommend to use these formulas also n the dstrbuton-free case (currently there are no estmators avalable from whch we know that they perform better). A. Saluz, A. Gsler, M. V. Wüthrch (ETH Zurch) Pattern and MSEP for BF June, / 12