Credibility for the Linear Stochastic Reserving Methodswe need the prior distribution of F and the conditional distribution Sm i,k| F. ⇒ We derive only so called credibility predictor,

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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[email protected]

[email protected]

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



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








I ≥ J

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LSRM Notation








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


Lnk :=

M∑

m=0

I∑

i=0

((n−i)∧J)∨k∑

j=0


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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



k

unbiased estimators Smi ,k+1 for E

[Smi ,k+1

∣∣∣DI]with i + k ≥ I

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Estimators



k


[Smi ,k+1


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



k


[Smi ,k+1


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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i+k∩ Lk and

Rm1,m2


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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i+k∩ Lk and

Rm1,m2


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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Documents

Credibility for the Linear Stochastic Reserving Methodswe need the prior distribution of F and the conditional distribution Sm i,k| F. ⇒ We derive only so called credibility predictor,