The Insurance Risk in the SST and in Solvency II: … · ASTIN Colloquium 1-4 June 2009, Helsinki....
Transcript of The Insurance Risk in the SST and in Solvency II: … · ASTIN Colloquium 1-4 June 2009, Helsinki....
ASTIN Colloquium1-4 June 2009, Helsinki
Alois Gisler
The Insurance Risk in the SST and in Solvency II:
Modeling and Parameter Estimators
Introduction
SST and Solvency II– common goal: to install a risk based solvency regulation– solvency capital required (SCR) should depend on the risks a
company has on its bookSST2004: standard SST model developed and first field test 2008: all Swiss companies have to calculate the SST figures2011: SST SCR will be in forceSolvency II2007: SII Framework Directive Proposal adopted by the EU
Commission2008: 4th quantitative impact study 2012: "original" schedule to put the regulation into force
schedule under discussionSubject of this presentation: non-life insurance riskmodeling and parameter estimators
2 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
The Insurance Risk
Non-Life Insurance Risknon-life insurance risk = next years technical result
where
segmented into lines of business (lob) i=1,2,....,I ;
3
===== −
earned premium,administrative costs,total claim amount current year (CY), total claim amount previous years (PY)
CY
PY
PK
CC
CDR
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
(CDR = claims development result)
( )= − − −
⎡ ⎤⎡ ⎤ ⎡ ⎤− − − − −⎣ ⎦ ⎣ ⎦⎣ ⎦expected technical result
CY PY
CY CY CY PY
TR P K C CE P K E C C E C C
Modeling in SST: Insurance Risk
CY claim amountis split into "normal claim" amount
and "big claim amount"
analytical insurance risk modelmodeling ofdescribes adequately reality except for extraordinary situations
scenarioscomplements analytical model to take into account extraordinary situations;to take into account extraordinary situations;by means of scenrios , k=1,2,...,K, characterised by face amounts ck with occurrence probabilities pk .
4
CYC ,CY nC,CY bC
, ,( , , )CY n CY b PYC C C
kSC
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: Insurance Risk
5
Risk measure in the SST99% expected shortfall
SCR for insurance risk
[ ]= −99% .insSCR ES TR
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: normal claim amount CY
Model assumption
Conditional on ,
is compound Poisson;
is the "risk characteristics" of next year for lob i
6
( )1 2,Ti i i= Θ ΘΘ
,CY niC
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
1 2,i iΘ Θ are random factors with expected value 1 indicating how much next year's "true underlying" claim frequency and the "true underlying" expected claim severity will deviate from their a priori expected values due to things like weather conditions, change in economic environment, change in legislation, etc.
( )1 2,Ti i i= Θ ΘΘ
Modeling in SST: normal claim amount CY
, where
variance structure from model assumptions follows that
where
and where
7
, pure risk premium;CY ni iP E C⎡ ⎤= ⎣ ⎦=
,CY ni
ii
CXP
( )2,2 2
,: ,i flucti i i param
i
Xσ
σ σν
= = +Var
( ) ( )σ Θ + Θ2, 1 2 , i param i iVar Var
( )( )
( )
the coefficient of variation of the claim severities,a prori expected number of claims.
ii
i i
CoVa Yw
υ
ν λ=
= =
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
( )υσ = +2 2 ( ), 1.i fluct iCoVa Y
aggregation over lob
the variance of is calculated by assuming
a correlation matrix
=>
where
8 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: normal claim amount CY
• • •= , /CY nX C P
( ) ( )σ •
•
= ⋅ ⋅W R W22 1 ,T
CY CY CYXP
:= Var
( )σ σ σ=W …1 1 2 2, , , .T
CY I IP P P
( ) ( ) ( )= =R X X R, ( , , )TCY CY i ji j X XCorr Corr
Modeling in SST: big claim amount CY
Model Assumptionsi) for each lob i the big claim amount is compound Poisson-
distribution with (essentially) Pareto-distributed claim sizes
ii) are independent
=> is again compound Poisson with
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,CY biC
, ,
1
ICY b CY b
ii
C C•=
= ∑
λ λ λ•=
= = ∑1
,I
b bi
i 1
.bni
ibi
F Fλλ= •
= ∑
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
= …, , 1,2, ,CY biC i I
Modeling in SST: normal and big claim amount CY
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lob andstandard parametersnormal and bigclaim amount CY
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: claim amount PY
Reserve risk (claim amount PY)
note that
Model Assumptionsit is assumed that
11
31.12.,
outstanding claims liabilities at 1.1. for lob ,best estimate of per 1.1. = best estimate reserve,
= best estimate of per 31.12.,
i
i iPY PY
i i i i
L iR LR PA R L
=== +
.PYi i iC R R= −
.ii
i
RYR
=
( )2,2 2
,: i flucti i i param
i
YR
ττ τ= = +Var
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: claim amount PY
current standard parameters for PY-risks
12 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: claim amount PY
aggregation over lob
the variance of is calculated by assuming
a correlation matrix
current standard SST assumption Yi , i=1,2,...,I, are independent, i.e. RPY = identity matrix.
=>
Discussion on correlation assumptioncurrent standard SST assumption is questionable;reason: calendar year effects affecting several lob simultaneously; an obvious example of is claims inflation.
13 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
• • •= /Y R R
( )τ τ=•
= = ∑2 2 22
1
1:I
i ii
Y RR
Var
( ) ( ) ( )= =R Y R, ( , , )TPY PY i ji j Y YYCorr Corr
Modeling in SST: combined normal claim amount CY + claim amount PY
Notations
Model assumption It is assumed that is lognormal distributed with
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• • • • • •• • • • • • •
• • • •
+ += + = = = +
+ +
+ += + = = = +
+ +
,,
,,
, , ,
, , .
CY nCY n i i i i i i
i i i i i i ii i i i
CY nCY n
C R P X RYS C R Z V P RP R P R
C R P X R YS C R Z V P RP R P R
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
S•
[ ] ( )• • • •⎛ ⎞ ⎛ ⎞= + = ⋅ ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠W WRW W, .
TCY CY
PY PYE S P R SVar
Modeling in SST: combined normal claim amount CY + claim amount PY
Correlation matrices:
current standard SST assumption current year claims and previous year claims are uncorrelated, that is
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( )
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
=
R RX XR Y Y R R
R X Y
,
,
,
, ,
where ,
TCY CY PY
CY PY PY
TCY PY
Corr
Corr
=
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
( )( )σ τ• •
•
• •
=
+=
+
R ,
2 2 2 2
2
.
=>
CY PY
P RZP R
0
Var
Modeling in SST: correlation CY and PY; convolution with big claims
Discussion on correlation assumption between CY and PYcurrent standard SST assumption is questionable;reason: calendar year effects affecting the CY-year claim amount as the previous years' claim amounts of several lob simultaneously; an example of such a calendar year effect is claims inflation;
Convolution with big claim The distribution of can be calculated by convoluting the lognormal distribution of with the compound Poisson distribution of
=> distribution before scenarios
16 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
, ,CY n CY b PYT C C C• • •= + +,CY n PYC C• •+
,CY bC•
F
Modeling in SST: scenarios
Model Assumptions:Scenarios , k=1,2,...,K, are characterized by face amounts ckand occurrence probabilities pk. It is assumed that only one of the scenarios can occur within the next year (mutual exclusion of scenarios).
Remark:The "exclusion assumption" is not such a big restriction as it seems, since one is free in defining the scenarios. One can always define new scenarios combining two already existing scenarios.
Distribution after scenariosdistribution function of :
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inskSC
, ,CY n CY b PY insT C C C SC• • • •= + + +
( ) ( ) 0 00 1
, where 1 and 0.K K
k k kk k
F x p F x c p p c= =
= − = − =∑ ∑ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: Insurance risk
General to compare with SST: only one region, company is working in;SCR for non-life insurance risk is named SCRnl in solvency II (SII).
SII also considers CY-risk (named premium risk) and PY-risk called reserve risk. For CY-risk : no distinction is made between normal and big claims.
In addition: CAT-risks, mainly thought for natural peril risks. Characterized by face amounts similar to the scenario risks in the SST.
SII provides formulas how to calculate the SCR and not models. Models presented here = models leading to the formulas in SII to calculate the SCR .
18 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: Insurance risk
Notation
where
19
( ) ( )σ τ
= =
= =2 2
(loss ratio CY), ,
, ,
CYi i
i ii i
i i i i
C RX YP R
X YVar Var
premiumreserve per 1.1."a posteriori reserves" per 31.12.of .
i
i
i i
PRR L
===
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: premium (CY) risk
calculation premium risk per lob
where
Model assumption CY-risk (premium risk)Neither nor the credibility weight depend on the size of the company=> model assumption:
Model assumption PY-risk (reserve risk)model assumption:
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( )2 2, ,1 ,i i i ind i i Mσ α σ α σ= ⋅ + − ⋅
,
2 2,
1 1
credibility weight, standard "market" parameter,
1 ( ) with . 1
i i
i
i Mn n
ij iji ind ij i i ij
j ji i i
P PX X X X
n P P
ασ
σ= =• •
==
= − =− ∑ ∑
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
2,i Mσ
( ) 2.i iX σ=Var
( ) τ= 2.i iyVar
Modeling in SII: premium + reserve risk
premium + reserve risk per lob
correlation and aggregation
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( )1 , where .i i i i i i i ii
Z P X RY V P RV
= + = +
( ) ,Assumption: , 50%i i CY PYX Y ρ= =Corr
( ) ( ) ( )2 2,
2
2: .i i CY PY i i i i i i
i ii
P P R RZ
Vσ ρ σ τ τ
ϕ+ +
=> = =Var
( ) ρ ρ=assumption: , , given standard parametersi j ij ijZ ZCorr
( )22
1 , 1
=> , I I
i j i jii ij
i i j
VVVZ Z ZV V
ϕ ϕϕ ρ• •
= =• •
= = =∑ ∑Var
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: premium + reserve risk
implications and discussion of correlation assumptions
must hold for any company
=>– correlation between lob result from calendar year effects affecting
several lob simultaneously. To assume the same correlation matrix for X and for Y is questionable, since the calendar year effect for CY- and PY-risks might not be the same or might have a different impact.
– depend on the volumes and difficult to interpret
22 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
( ) ( ) ,, , , 50%.i j ij i i CY PYZ Z X Yρ ρ= = =Corr Corr
( ) ( ) ( ), , , ,i j i j i j ijX X Y Y Z Z ρ= = =Corr Corr Corr
( ), for i jX Y i j≠Corr
Modeling in SII: formula to calculate SCR
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lob and parameters
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: premium + reserve risk
formula for SCR premium + reserve risk
where
24
( )( )
( )
ϕ
ϕ
−
+ •
•
⎛ ⎞Φ ⋅ +⎜ ⎟= −⎜ ⎟+⎜ ⎟⎝ ⎠
= Ψ
1 2
2
0.995
exp 0.995 log( 1)1
1pr res
mean
SCR V
V VaR
( )Φ = standard normal distribution.x
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
[ ] ( ) ϕΨ = Ψ = Ψ = 2logormal distributed r.v. with 1 and ,E Var( ) ( )( )Ψ = Ψ − Ψ0.995 0.995 .meanVaR VaR E
Modeling in SII: premium + reserve risk
model assumption behind this formulahas the same distribution as where
has a lognormal distribution with
remarks and discussion
but contrary to the SST:
=> is modeled by a lognormal distribution with mean , but with a variance which is different from
25 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
[ ]S E S• •− ( )1 ,V• Ψ −Ψ [ ] ( ) ϕΨ = Ψ = 21 and .E Var
[ ] [ ]( ) ( )is aproximated by 1 .S E S V Z E Z V• • • • • •− = − Ψ −
[ ]• ≠ 1 (usually smaller than 1).E Z
S• [ ]E S•
[ ]Var S•
Modeling in SII: premium + reserve risk
Comparison of 99.5% VaR of and for
26 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
[ ]Z E Z• •− 1Ψ − [ ] 85%.E Z• =
Modeling in SII: cat risks and total insurance risk
SCR for CAT risks
total SCR for nl-insurance risk
model assumptions behind these formulas
The cat risks are independent and normally
distributed with
Same assumption for aggregating the cat risks and the other insurance risks.
27
2
1
.K
CAT kk
SCR c=
= ∑
2 2 .nl CY PY CATSCR SCR SCR+= +
ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
= …, 1,2, ,kCAT k I
( ) =0.995 .k kVaR CAT c
Modeling : Summary
STT and SII "parametrized" models;SII: factor model; STT distribution based model;
risk measure: STT 99% expected shortfall, SII 99.5% VaR
variance assumptions CY- und PY-risks (for r.v. X and Y):STT: parameter risk and random fluctuation risk, where the latter is inversely proportional to the weight (size of the company);SII: CY- and PY-risks not dependent on the size of the company
CY risk: STT distinguishes between "normal claims" and "big claims". No such distinction in SII.
Correlation Assumptions (current state):SST: no correlations between lob for the reserve risks and no correlations between CY-und PY-risks;SII: same correlation between lob for CY- and PY-risks;SST as well as SII assumptions not fully satisfactory.
28 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling : Summary
SST: scenarios for extraordinary situationscan be taken into account in a natural way in the distribution calculation;
SII: CAT-risks modeled similar to scenarios in the SST; however aggregation of cat-risks and with CY/PY-risks questionable
SST: final product is a distribution, from which the SCR is calculated;SII: final product is one figure, the SCR.
Results (AXA-Winterthur)with current standard parameters: SCRins higher in SII than in SST;split between CY- und PY-risks:SII: ca 25% CY and 75% PYSST: ca 27% CY and 73% PY
29 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Parameter Estimators: SII parameters
straightforward estimators
Remarks:can overestimate the risk in case of "strong" business cycles in
the observation period;often underestimates the reserve risks because of "smoothing"
effects in the reserves
30 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
2 2
1
1ˆ ( ) ,1
inij
i ij iji i
PX X
n Pσ
= •
= −− ∑
2 2
1
1ˆ ( ) ,1
inij
i ij iji i
RY Y
n Rτ
= •
= −− ∑
2ˆ iσ
2iτ
Parameter Estimators: SST parameters
Random fluctuation risk CY
in long-tail lob: above estimator underestimates the CoVa in recent accident years
31 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
( )( )νσ = +2 2, 1.i fluct iCoVa Y
( )( )21
11
2
ij
ij
NiijN
ij
i
Y YCoVa
Y
νυ=− −∑
=
Parameter Estimators: SST parameters
parameter risk CYspecific lob; each year j characterized by ;
r.v. belonging to different years are independent and are i.i.d.
=>
fulfill the assumptions of the Bü-Straub credibility model
=> estimator
where
32 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
( )1 2,T
j j j= Θ ΘΘ
1, 1, , J…Θ Θ Θ
( )2 2
2 2 ˆ1, ,fluct fluctj j param param
j j
E X XP
σ σσ σν
⎡ ⎤ = = + +⎣ ⎦ Var
( )222
1
ˆˆ ,1
Jj fluct
param jj
w JJc X XJ w n
σσ
= • •
⎧ ⎫= ⋅ − −⎨ ⎬−⎩ ⎭
∑
( )( )1
22
1
1 1 , ˆ 1 ,
observed number of claims.
Ii i
flucti
w wIc CoVa YI w w
n
υσ−
• •
= •• ••
•
⎧ ⎫⎛ ⎞− ⎪ ⎪= − = +⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
=
∑
Parameter Estimators: SST parameters
parameter risk CY (continued)since
one can, alternatively to the estimator given before, estimate the two components separately based on the observed claim frequencies and the observed claim sizes.
Here again one can use a credibility procedure.
more details: see paper
33 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
( ) ( )21 2paramσ Θ + ΘVar Var
Parameter Estimators: SST parameters
Estimation of the Pareto parameters for big claim CYML-estimator (adjusted for unbiasedness)
with
Number of observed big claims often rather small; combine individual estimate with market wide estimate; ML-estimators fulfill Bü-Straub cred. assumptions
=> credibility estimatorwhere
Example: => give a credibility weight of 32% to yourindividual estimate
34 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
1
1
1ˆ ln1
bn Yn c
ν
ν
ϑ−
=
⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠
∑
( ) 1ˆ ˆ, .2
E CoVan
ϑ ϑ ϑ⎡ ⎤ = =⎣ ⎦ −
0ˆ ˆ (1 )credϑ αϑ α ϑ= + −
( ) 20
2 , standard value from the SST, .1
n CoVan
ϑ κκ
−−= = Θ
− +
( ) 25%, n=16 CoVa Θ =
Parameter Estimators: SST parameters
reserve riskreserve risk should be valuated with reserving techniques; well known: Mack's mse of the ultimate for chain ladder reserving method;
for solvency purposes one needs the one-year reserve risk;the formula can be found in Bühlmann and alias (2009);
In Solvency we are interested in the one in a century adverse reserve events. What scenarios come to our mind: for instance a hyper-inflation or a big change in legislation. These are "calendar-year" events not observed in the triangles and not captured by standard reserving methods.
=> the reserve risk resulting from standard reserving methods are not sufficient for solvency purposes and should be supplemented by reserve scenarios.
35 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Parameter Estimators: SST parameters
reserve risk (continued)For small and medium sized companies the observed figures in a development triangle might fluctuate a lot. It would be helpful if one could combine industry wide patterns with the one evaluated with the data of the individual company.
For chain ladder a credibility method was developed of how one could combine the information gained from the two sources: individual data and industry wide information. The idea is to estimate the age-to-age factors by credibility techniques.
For more information see Gisler-Wüthrich (2008).
36 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
References
Bühlmann, H., De Felice M., Gisler, A., Moriconi F., Wüthrich, M.V. (2009). Recursive Credibility Formula for Chain Ladder Factors and the Claim Development Result. Forthcoming in the ASTIN Bulletin.Gisler, A., Wüthrich , M.V. (2008).Credibility for the Chain Ladder Reserving Method. ASTIN Bulletin 38/2, 565-600.Gisler, A. (2009). The Insurance Risk in the SST and in Solvency II: Modelling and Parameter Estimation. ASTIN Colloquium in Helsinki. Merz, M., Wüthrich M.V. (2008). Modelling the claims development result for solvency purposes. CAS Forum, Fall 2008, 542-568.
37 ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators