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Credibility for the Linear Stochastic Reserving Methods
Sebastian Happ ∗ Rene Dahms ∗∗
∗Department of Business Administration, University of Hamburg, [email protected]
∗∗CH-4055 Basel, [email protected]
International Congress of Actuaries 2014Washington D.C.
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Overview
1 Class of linear stochastic reserving methods (LSRMs)
2 Classification of classical claims reserving methods
3 Bayesian LSRM and credibility theory
4 Remarks and outlook
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LSRM Notation
Smi ,k - m-th incremental claim property in accident year i ∈ 0, . . . , I,
development year k ∈ 0, . . . , J
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LSRM Notation
Smi ,k - m-th incremental claim property in accident year i ∈ 0, . . . , I,
development year k ∈ 0, . . . , J
For 0 ≤ m ≤ M, Smi ,k may contain:
Incremental claims payments (chain ladder (CL) method) incurred losses information (extended complementary loss ratio (ECLR)
method) prior information (estimates), for example for ultimate claim estimates
(Bornhutter-Ferguson (BF) method) insured volumes, for example salaries for workers incapacitation
(complementary loss ratio (CLR) method)
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LSRM Notation
Smi ,k - m-th incremental claim property in accident year i ∈ 0, . . . , I,
development year k ∈ 0, . . . , J
For 0 ≤ m ≤ M, Smi ,k may contain:
Incremental claims payments (chain ladder (CL) method) incurred losses information (extended complementary loss ratio (ECLR)
method) prior information (estimates), for example for ultimate claim estimates
(Bornhutter-Ferguson (BF) method) insured volumes, for example salaries for workers incapacitation
(complementary loss ratio (CLR) method)
I ≥ J
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LSRM Notation
Smi ,k - m-th incremental claim property in accident year i ∈ 0, . . . , I,
development year k ∈ 0, . . . , J
For 0 ≤ m ≤ M, Smi ,k may contain:
Incremental claims payments (chain ladder (CL) method) incurred losses information (extended complementary loss ratio (ECLR)
method) prior information (estimates), for example for ultimate claim estimates
(Bornhutter-Ferguson (BF) method) insured volumes, for example salaries for workers incapacitation
(complementary loss ratio (CLR) method)
I ≥ J
We assume that there is no tail development of claims paymentsbeyond development year J
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Information channels in LSRMs
Ln :=
M∑
m=0
I∑
i=0
(n−i)∧J∑
j=0
xmi ,jSmi ,j : xmi ,j ∈ R
Lk :=
M∑
m=0
I∑
i=0
k∑
j=0
xmi ,jSmi ,j : xmi ,j ∈ R
Lnk :=
M∑
m=0
I∑
i=0
((n−i)∧J)∨k∑
j=0
xmi ,jSmi ,j : xmi ,j ∈ R
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σ-fields of LSRMs
Bi ,k := σ(Smi ,j : 0 ≤ m ≤ M, 0 ≤ j ≤ k
)
Dn := σ (Ln) = σ
(I⋃
i=0
Bi ,(n−i)∧J
)
Dk := σ (Lk) = σ
(I⋃
i=0
Bi ,k
)
Dnk := σ (Lnk) = σ
(I⋃
i=0
Bi ,((n−i)∧J)∨k
)
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σ-fields of LSRMs
n
k
I
accidentyear
0 Jdevelopment year
I
accountingyear
Dk
Dn
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Model assumptions LSRM
A stochastic model for Smi ,k is called a LSRM :⇐⇒ For all i , m1, m2, m
and k , there are factors f mk ∈ R and σm1,m2
k ∈ R with:
i) ∃ Rmi ,k ∈ L
i+k ∩ Lk such that
E[Smi ,k+1
∣∣Di+kk
]= f mk Rm
i ,k ∈ Li+k ∩ Lk .
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Model assumptions LSRM
A stochastic model for Smi ,k is called a LSRM :⇐⇒ For all i , m1, m2, m
and k , there are factors f mk ∈ R and σm1,m2
k ∈ R with:
i) ∃ Rmi ,k ∈ L
i+k ∩ Lk such that
E[Smi ,k+1
∣∣Di+kk
]= f mk Rm
i ,k ∈ Li+k ∩ Lk .
ii) ∃ Rm1,m2
i ,k ∈ Li+k ∩ Lk such that
Cov[Sm1i ,k+1, S
m2i ,k+1
∣∣∣Di+kk
]= σ
m1,m2
k Rm1,m2
i ,k ∈ Li+k ∩ Lk .
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Is the CL model a LSRM?
Cumulated claims payments in CL model:
Ci ,k :=k∑
j=0
S0i ,j .
CL assumptions are:
i)CL E[Ci ,k+1|Bi ,k ]= gkCi ,k
ii)CL Var[Ci ,k+1|Bi ,k ]= σ2kCi ,k
iii)CL claims payments in different accident years are independent.
The independence assumption implies:
E[Ci ,k+1|D
Ik
]= E[Ci ,k+1|Bi ,k ]= gkCi ,k
Var[Ci ,k+1|D
Ik
]= Var[Ci ,k+1|Bi ,k ]= σ
2kCi ,k
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CL method
With Ci ,k ∈ Lk ∩ Li+k and
E[S0i ,k+1
∣∣Bi ,k
]= (gk − 1)Ci ,k
Var[S0i ,k+1
∣∣Bi ,k
]= σ
2kCi ,k
follows that the CL model is a LSRM.
Other well-known models like the BF method and the (extended)complementary loss ratio method are also LSRMs, see Dahms [1].
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Estimators
In Dahms [1] are derived:
BLUE estimators for the model parameter f mk and σm1,m2
k
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Estimators
In Dahms [1] are derived:
BLUE estimators for the model parameter f mk and σm1,m2
k
unbiased estimators Smi ,k+1 for E
[Smi ,k+1
∣∣∣DI]with i + k ≥ I
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Estimators
In Dahms [1] are derived:
BLUE estimators for the model parameter f mk and σm1,m2
k
unbiased estimators Smi ,k+1 for E
[Smi ,k+1
∣∣∣DI]with i + k ≥ I
estimates for the prediction uncertainty
mse
[∑
m∈M
J−1∑
k=I−i
αmi S
mi,k+1
]:= E
(∑
m∈M
J−1∑
k=I−i
αmi
(Smi,k+1 − Sm
i,k+1
))2∣∣∣∣∣∣DI
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Estimators
In Dahms [1] are derived:
BLUE estimators for the model parameter f mk and σm1,m2
k
unbiased estimators Smi ,k+1 for E
[Smi ,k+1
∣∣∣DI]with i + k ≥ I
estimates for the prediction uncertainty
mse
[∑
m∈M
J−1∑
k=I−i
αmi S
mi,k+1
]:= E
(∑
m∈M
J−1∑
k=I−i
αmi
(Smi,k+1 − Sm
i,k+1
))2∣∣∣∣∣∣DI
estimates for the one year claims development result (CDR)
mse[CDRM,I+1
i
]:= E
(∑
m∈M
αmi
J−1∑
k=I−i
(Sm,I+1i,k+1 − S
m,Ii,k+1
)− 0
)2∣∣∣∣∣∣DI
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Conclusions LSRM :
many classical distribution-free claims reserving methods belong tothe class of LSRMs, see Dahms [1]
the often stated assumption on independent accident years is notrequired in LSRMs
LSRMs do not allow for calendar year effects like inflation
prior information for f mk can not be incorporated in the classicalLSRM framework
⇒ idea:
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Conclusions LSRM :
many classical distribution-free claims reserving methods belong tothe class of LSRMs, see Dahms [1]
the often stated assumption on independent accident years is notrequired in LSRMs
LSRMs do not allow for calendar year effects like inflation
prior information for f mk can not be incorporated in the classicalLSRM framework
⇒ idea:
Bayesian LSRMs, where additional can be included via the first twomoments of the prior distributions
the unknown factors f mk are modeled as random variables
Fk : = (F 0k , . . . ,F
Mk )′ ∈ R
M+1
F : = (Fmj )0≤m≤M
0≤j≤J−1
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Model Assumptions - Bayesian LSRM
For all m1,m2,m ∈ 0, . . . ,M, i and k , there exist Rmi ,k ∈ L
i+k∩ Lk and
Rm1,m2
i ,k ∈ Li+k∩ Lk such that
i)
E[Smi ,k+1
∣∣Di+kk ,F
]= Fm
k Rmi ,k
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Model Assumptions - Bayesian LSRM
For all m1,m2,m ∈ 0, . . . ,M, i and k , there exist Rmi ,k ∈ L
i+k∩ Lk and
Rm1,m2
i ,k ∈ Li+k∩ Lk such that
i)
E[Smi ,k+1
∣∣Di+kk ,F
]= Fm
k Rmi ,k
ii)
Cov[Sm1i ,k+1, S
m2i ,k+1
∣∣∣Di+kk ,F
]= σ
m1,m2
k (F)Rm1,m2
i ,k
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Model Assumptions - Bayesian LSRM
For all m1,m2,m ∈ 0, . . . ,M, i and k , there exist Rmi ,k ∈ L
i+k∩ Lk and
Rm1,m2
i ,k ∈ Li+k∩ Lk such that
i)
E[Smi ,k+1
∣∣Di+kk ,F
]= Fm
k Rmi ,k
ii)
Cov[Sm1i ,k+1, S
m2i ,k+1
∣∣∣Di+kk ,F
]= σ
m1,m2
k (F)Rm1,m2
i ,k
iii) for all n ∈ I , . . . , I + J − 1, j ≤ J − 1 and 0 ≤ k0 < k1 < . . . < kj ≤ J − 1
gilt
E
[j∏
i=0
Ωki
∣∣∣∣∣Dn
]=
j∏
i=0
E[Ωki |Dn],
with Ωk ∈Fmk , σ
m1,m2
k (F),Fm1k Fm2
k
∣∣ 0 ≤ m,m1,m2 ≤ M.
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Bayesian LSRM
E[Smi ,k+1
∣∣∣DI]requires E
[Fmk |DI
]
For the calculation of E[Fmk |DI
]we need the prior distribution of F
and the conditional distribution Smi ,k |F .
⇒ We derive only so called credibility predictor, i.e. “best” (L2 norm)linear predictor. Therefore, only the first two moments of F are necessary:
FI ,Credk = Afk + (I− A)µk ,
fk - unbiased estimator for fk in the classical LSRM, µk - estimate for theexpected value of Fk , A is a matrix of weigths
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Bayesian LSRM
In Dahms and Happ [2] the following estimators are presented:
Sm,I ,Credi ,k+1 for i + k ≥ I
mse[∑
m∈M
∑J−1k=I−i α
mi S
m,I ,Credi ,k+1
]:=
E
[(∑m∈M
∑J−1k=I−i α
mi
(Smi ,k+1 − S
m,I ,Credi ,k+1
))2∣∣∣∣DI
]
mse[CDRM,I+1
i
]:=
E
[(∑m∈M αm
i
∑J−1k=I−i
(Sm,I+1,Credi ,k+1 − S
m,I ,Credi ,k+1
)− 0)2∣∣∣∣DI
]
where αmi DI -measurable weights.
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Conclusions and outlook
Bayesian LSRMs is a canonical generalization of the concept CLmethod - Bayes CL method in Gisler-Wuthrich [3] on the whole classof LSRMs
Bayesian LSRMs can cope with prior information in the large class ofLSRMs
In this way we get new methods, like the Bayesian BF method,Bayesian complementary loss ratio method etc.
Extension of (Bayesian) LSRMs to capture calendar year effect likeinflation having impact on the diagonals is possible, if independencebetween claims and inflation processes is assumed
Bayesian LSRMs can be applied for pricing purposes.
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References:
[1] Dahms, Rene (2012). Linear Stochastic Reserving Methods. ASTINBulletin, Vol 42, issue 1, 1-34.
[2] Dahms, R., Happ, S. (2013). Credibility for the Linear StochasticReserving Methods. Submitted to Astin Bulletin.
[3] Gisler, Alois, Wuthrich, Mario V. (2008). Credibility for the ChainLadder Reserving Method. ASTIN Bulletin, Vol. 38, no. 2, 565-600.
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