Arthur CHARPENTIER, IBNR and one-year uncertainty Solvency II’ newspeak ‘one year uncertainty for IBNR’ the boostrap approach Arthur Charpentier 1 & Laurent Devineau 2 1 Universit´ Ecole Polytechnique, 2 Milliman e Rennes 1 - CREM & ´ arthur.charpentier@univ-rennes1.fr http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/ Lyon Summer School, July 2010 1
Arthur CHARPENTIER, IBNR and one-year uncertainty Agenda of the talk • Solvency II : CP 71 and the one year horizon • Solvency II : new way of looking at the ‘uncertainty’ ◦ From MSE to MSEP (MSE of prediction) ◦ From MSEP to MSEPC (conditional MSEP) ◦ CDR, claims development result • From Mack (1993) to Merz & W¨ uthrich (2009) • Updating Poisson-ODP bootstrap technique one year ultimate China ladder Merz & W¨ uthrich (2008) Mack (1993) GLM+boostrap Hacheleister & Stanard (1975) x England & Verrall (1999) 2
Arthur CHARPENTIER, IBNR and one-year uncertainty ‘ one year horizon for the reserve risk ’ 3
Arthur CHARPENTIER, IBNR and one-year uncertainty ‘ one year horizon for the reserve risk ’ 4
Arthur CHARPENTIER, IBNR and one-year uncertainty ‘ one year horizon for the reserve risk ’ 5
Arthur CHARPENTIER, IBNR and one-year uncertainty ‘ one year horizon for the reserve risk ’ 6
Arthur CHARPENTIER, IBNR and one-year uncertainty ‘ one year horizon for the reserve risk ’ 7
Arthur CHARPENTIER, IBNR and one-year uncertainty ‘ one year horizon for the reserve risk ’ 8
Arthur CHARPENTIER, IBNR and one-year uncertainty Standard models in IBNR models • Chain Ladder C i,j +1 = λ j · C i,j and Mack • Factor models Y i,j ∼ L ( A i , B j ) and GLM’s (ODP-bootstrap) • and the Bayesian approach 9
Arthur CHARPENTIER, IBNR and one-year uncertainty Standard models in IBNR models • Chain Ladder C i,j +1 = λ j · C i,j and Mack • Factor models Y i,j = ϕ ( A i , B j ) and GLM’s (ODP-bootstrap) • expert opinion and the Bayesian approach 10
Arthur CHARPENTIER, IBNR and one-year uncertainty Standard models in IBNR models • Chain Ladder C i,j +1 = λ j · C i,j and Mack • Factor models Y i,j = ϕ ( A i , B j ) and GLM’s (ODP-bootstrap) • expert opinion and the Bayesian approach 11
Arthur CHARPENTIER, IBNR and one-year uncertainty Standard models in IBNR models • Chain Ladder C i,j +1 = λ j · C i,j and Mack • Factor models Y i,j = ϕ ( A i , B j ) and GLM’s (ODP-bootstrap) • expert opinion and the Bayesian approach 12
Arthur CHARPENTIER, IBNR and one-year uncertainty Standard models in IBNR models • Chain Ladder C i,j +1 = λ j · C i,j and Mack • Factor models Y i,j = ϕ ( A i , B j ) and GLM’s (ODP-bootstrap) • expert opinion and the Bayesian approach 13
Arthur CHARPENTIER, IBNR and one-year uncertainty Standard models in IBNR models • Chain Ladder C i,j +1 = λ j · C i,j and Mack • Factor models and GLM’s (ODP-bootstrap), E ( Y i,j |F ) = ϕ ( A i , B j ) • expert opinion and the Bayesian approach 14
Arthur CHARPENTIER, IBNR and one-year uncertainty Standard models in IBNR models • Chain Ladder and Mack (1993) E ( C i,j +1 |F ) = λ j · C i,j • Factor models and GLM’s (ODP-bootstrap), E ( Y i,j |F ) = ϕ ( A i , B j ) • expert opinion and the Bayesian approach 15
Arthur CHARPENTIER, IBNR and one-year uncertainty Notations for triangle type data • X i,j denotes incremental payments, with delay j , for claims occurred year i , • C i,j denotes cumulated payments, with delay j , for claims occurred year i , C i,j = X i, 0 + X i, 1 + · · · + X i,j , 0 1 2 3 4 5 0 1 2 3 4 5 0 3209 1163 39 17 7 21 0 3209 4372 4411 4428 4435 4456 1 3367 1292 37 24 10 1 3367 4659 4696 4720 4730 et et 2 3871 1474 53 22 2 3871 5345 5398 5420 3 4239 1678 103 3 4239 5917 6020 4 4929 1865 4 4929 6794 5 5217 5 5217 • F t denotes information available at time t , available at time t , based on the first k years, only F t = { ( C i,j ) , 0 ≤ i + j ≤ t } = { ( X i,j ) , 0 ≤ i + j ≤ t } 16
Arthur CHARPENTIER, IBNR and one-year uncertainty Notations for triangle type data • X i,j denotes incremental payments, with delay j , for claims occurred year i , • C i,j denotes cumulated payments, with delay j , for claims occurred year i , C i,j = X i, 0 + X i, 1 + · · · + X i,j , 0 1 2 3 4 5 0 1 2 3 4 5 0 3209 1163 39 17 7 21 0 3209 4372 4411 4428 4435 4456 1 3367 1292 37 24 10 1 3367 4659 4696 4720 4730 et et 2 3871 1474 53 22 2 3871 5345 5398 5420 3 4239 1678 103 3 4239 5917 6020 4 4929 1865 4 4929 6794 5 5217 5 5217 • F tk denotes partial information available at time t , based on the first k years, only k = { ( C i,j ) , 0 ≤ i + j ≤ t, i ≤ k } = { ( X i,j ) , 0 ≤ i + j ≤ t, i ≤ k } F t 17
Arthur CHARPENTIER, IBNR and one-year uncertainty Chain Ladder estimation 0 1 2 3 4 5 0 1 2 3 4 5 0 3209 4372 4411 4428 4435 4456 0 3209 4372 4411 4428 4435 4456 1 3367 4659 4696 4720 4730 1 3367 4659 4696 4720 4730 4752 . 4 2 3871 5345 5398 5420 2 3871 5345 5398 5420 5430 . 1 5455 . 8 et et 3 4239 5917 6020 3 4239 5917 6020 6046 . 15 6057 . 4 6086 . 1 4 4929 6794 4 4929 6794 6871 . 7 6901 . 5 6914 . 3 6947 . 1 5 5217 5 5217 7204 . 3 7286 . 7 7318 . 3 7331 . 9 7366 . 7 with the following link ratios 0 1 2 3 4 n 1,38093 1,01143 1,00434 1,00186 1,00474 1,0000 λj One the triangle has been completed, we obtain the amount of reserves, with respectively 22, 36, 66, 153 and 2150 per accident year, i.e. the total is 2427. 18
Arthur CHARPENTIER, IBNR and one-year uncertainty How to quantify uncertainty in Solvecny II In Solvency II, uncertainty is quantified as a dispersion measure (variance or quantile) of changes in prediction, with one year of additional information. The best estimate at time t is � R t = E ( C ∞ |F t ) while it become, at time t + 1 � R t +1 = E ( C ∞ |F t +1 ). The goal is to estimate � � [ E ( C ∞ |F t +1 ) − E ( C ∞ |F t )] 2 |F t E 19
Arthur CHARPENTIER, IBNR and one-year uncertainty Quantifying uncertainty in odds/tails games In statistics, the mean squared error is a standard measure to quantify the uncertainty of an estimator, i.e. �� � 2 � mse( � � θ ) = E θ − θ In order to formalize the prediction process in claims reserving consider the following simpler case. Let { x 1 , · · · , x n } denote an i.i.d. B ( p ) sample. We want to predict S h = X n +1 + · · · + X n + h . Let n � S h = ψ ( X n +1 , · · · , X n + h ) = h · � p n denote the natural predictor for S h , at time n . Since S h is a random variable ( θ was a constant) define �� � 2 � mse( n � n � S h ) = E S h − E ( S h ) 20
Arthur CHARPENTIER, IBNR and one-year uncertainty and �� � 2 � msep( n � n � S h ) = E S h − S h Note that �� � 2 � � [ E ( S h ) − S h ] 2 � msep( n � n � E S h − E ( S h ) + E S h ) = mse( n � = S h ) + Var( S h ) where the first term is a process error and the second term a estimation error. It is also possible to calculate the information given the information available at time n , i.e. a conditional msep, �� � � 2 msepc n ( n � n � S h ) = E S h − S h |F n denoted E (msepc n ( n � S h )) = msep( n � S h ). 21
Arthur CHARPENTIER, IBNR and one-year uncertainty What are we looking for ? In Solvency II requirements, CDR n +1 = [ n � S h ] − [ x n +1 + n +1 � S h − 1 ] This defines a martingale since E ( CDR n +1 |F n ) = 0 and what is required is to estimate msepc n ( CDR n +1 ) i.e. find � msepc n ( CDR n +1 ). 22
Arthur CHARPENTIER, IBNR and one-year uncertainty What are we looking for ? 23
Arthur CHARPENTIER, IBNR and one-year uncertainty What are we looking for ? 24
Arthur CHARPENTIER, IBNR and one-year uncertainty What are we looking for ? 25
Arthur CHARPENTIER, IBNR and one-year uncertainty Having an estimator of the uncertainty Let us continue with our repeated tails/heads game. Let � p n = [ x 1 + · · · + x n ] /n , so that p n ) = p (1 − p ) Var( � n thus p n ) = h 2 p n ) = h 2 · mse( � mse( n � S h ) = mse( h · � n p (1 − p ) , or S h ) = nhp (1 − p ) + h 2 n p (1 − p ) = nh + h 2 msep( n � p (1 − p ) n i.e. S h ) = h ( n + h ) msep( n � p (1 − p ) . n 26
Arthur CHARPENTIER, IBNR and one-year uncertainty Having an estimator of the uncertainty Thus, this quantity can be estimated as S h ) = h ( n + h ) msep( n � � p n (1 − � � p n ) . n while the mse estimator was S h ) = h 2 mse( n � � n � p n (1 − � p n ) Looking at the msepc at time n , we have msepc n ( n � S h ) = Var( S |F n ) + mse( n � S h |F n ) 27
Arthur CHARPENTIER, IBNR and one-year uncertainty Having an estimator of the uncertainty Looking at the msepc at time n , we have msepc n ( n � S h ) = Var( S |F n ) + mse( n � S h |F n ) where Var( S |F n ) = Var( X n +1 + · · · + X n + h | x 1 , · · · , x n ) = Var( X n +1 + · · · + X n + h ) = hp (1 − p ) and � � 2 mse( n � E ( S h |F n ) − n � S h |F n ) = S h which can be written S h ) = hp (1 − p ) + h 2 ( p − � msepc n ( n � p n ) 2 28
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