DAV Scientific Day, 30th April 2009, Berlin
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 book
SST2004: standard SST model developed and first field test with
10 companies2008: all Swiss companies have to carry out the SST2011: SST SCR will be in force
Solvency II2007: SII Framework Directive Proposal adopted by the EU
Commission2008: 4th quantitative impact study 2012: "original" schedule to put the regulation into force
? : SII SCR in force
2 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Introduction
Subject of this presentation: non-life insurance risk– modeling– parameter estimators
3 DAV Scientific Day 29.4.2009, Berlin / 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
total claim PY:CDR = claims development result where
4
CY PYTR P K C C
,PYC CDR
earned premium,administrative costs,total claim amount current year (CY), total claim amount previous years (PY).
CY
PY
PK
CC
31.12.,PY PYCDR R PA R
31.12., .
claims reserves per 1.1. (best estimate),payments for claims of previous years,claims reserves per 31.12. for claims of previous years.
PY
PY
RPA
R
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
The Insurance Risk
Technical Result can be written as
TR composed of three components:– expected technical result– deviation of CY-claim amount from expected value (premium risk)– PY risk (reserve risk)
segmented into lines of business (lob) i=1,2,....,I ;
5
expected technical result
CY CY CY PY
CY CY CY PY
TR P K E C C E C C
E P K E C C E C C
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: Insurance Risk
CY claim amountis split into "normal claim" amount
and "big claim amount"
analytical insurance risk modelmodelling ofdescribes adequately reality except for extraordinary situations
scenarioscomplements analytical model to take into account extraordinary situations;modelling of extraordinary situations;scenarios , k=1,2,...,K, characterised by face amounts ck and occurrence probability pk .
6
CYC ,CY nC,CY bC
, ,( , , )CY n CY b PYC C C
kSC
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: Insurance Risk
7
Technical Result
Risk measure in the SST99% expected shortfall
SCR for insurance risk
, , , ,
expected technical result
.
CY CY n CY n CY b CY b
PY ins
TR E P K E C C E C C E C
C SC
99%
99% ,ins
mean
SCR ES TRES TR E TR
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: normal claim amount CY
Model assumptionlob i characterized in coming year by
risk characteristics for claim frequencyrisk characteristics for claim sizes
independent with
conditional on .
is compound Poisson with where = known weight and with claim sizes having the same distribution as ,where has a distribution with
Remark:no further specified distributional assumption onwe will however make a distributional ass. on
8
1 2,T
i i i 1i2i
1 2,Ti i i
,CY niC
1 2,i i 1 2 1,i iE E
1 1, ,i i i i iw w iw( )
iY 2i iY
iY iF y .i iE Y
( )iY
,CY n PYC C C DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: normal claim amount CY
Introduce
variance structure from model assumptions follows that
where
and where
9
, pure risk premium;CY ni iP E C
,CY ni
iCX
P
2,2 2
,: ,i flucti i i param
i
X
Var
2, 1 2 1 2
1 22 2 ( ),
,1.
i param i i i i
i i
i fluct iCoVa Y
Var Var Var VarVar Var
( )
( )
the coefficient of variation of the claim severities,a prori expected number of claims.
ii
i i
CoVa Yw
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: normal claim amount CY
Variance structure IISince we can also write
Correlation and Variance of
10
i i iP
2,2 2
,: ,i flucti i i param
i
XP
Var
2 2, ,where .i fluct i i fluct
1 2 1 1 2 2, , , , , , , , , .T
T TI CY CY I IX X X P P P X R X X W Corr
,
1: .
CY n I
ii
C PX XP P
,CY nC
, .CY n TCY CY CYC W R WVar
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: normal claim amount CY
11
lob description1 motor liability2 motor hull3 property4 general liability5 workers compensation (UVG)6 corporate accident without UVG7 corporate health8 individual health9 marine10 aviation11 credit and surety12 legal protection13 others
standard parameters for CY- riskslob σpar CoVa (claim size)1 3.50% 7.002 3.50% 2.503 5.00% 5.004 3.50% 8.005 3.50% 7.506 4.75% 4.507 5.75% 2.508 5.75% 2.259 5.00% 6.50
10 5.00% 2.5011 5.00% 5.0012 4.50% 2.2513 4.50% 5.00
Correlation matrix for CY- year riskslob 1 2 3 4 5 6 7 8 9 10 11 12 13
1 1 0.5 0 0.25 0.25 0.25 0 0 0 0 0 0 02 0.5 1 0.25 0 0 0 0 0 0 0 0 0 03 0 0.25 1 0.25 0 0 0 0 0 0 0 0 04 0.25 0 0.25 1 0 0 0 0 0 0 0 0 05 0.25 0 0 0 1 0.5 0.5 0 0 0 0 0 06 0.25 0 0 0 0.5 1 0.5 0 0 0 0 0 07 0 0 0 0 0.5 0.5 1 0.25 0 0 0 0 08 0 0 0 0 0 0 0.25 1 0 0 0 0 09 0 0 0 0 0 0 0 0 1 0 0 0 0
10 0 0 0 0 0 0 0 0 0 1 0 0 011 0 0 0 0 0 0 0 0 0 0 1 0 012 0 0 0 0 0 0 0 0 0 0 0 1 013 0 0 0 0 0 0 0 0 0 0 0 0 1
lob andstandard parametersnormal claim amount CY
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
big claims CYlob Pareto parameter1 2.502 1.853 1.404 1.805 2.006 2.007 3.008 3.009 1.501011 0.751213 1.50
Modeling in SST: claim amount PY
Reserve risk (claim amount PY)
note that
Model Assumptionsit is assumed that
12
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
Var
DAV Scientific Day 29.4.2009, Berlin / 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
13 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
standard parameters for PY-riskslob τpar τfluct
1 3.5% 2.5%2 3.5% 20.0%3 3.0% 15.0%4 3.5% 4.0%5 2.0% 4.0%6 3.0% 5.0%7 3.0% 7.0%8 5.0% 0.0%9 5.0% 25.0%
10 5.0% 20.0%11 5.0% 15.0%12 5.0% 0.0%13 5.0% 50.0%
lob description1 motor liability2 motor hull3 property4 general liability5 workers compensation (UVG)6 corporate accident without UVG7 corporate health8 individual health9 marine10 aviation11 credit and surety12 legal protection13 others
Modeling in SST: claim amount PY
Correlation and Variance of Var of
then
current standard SST assumption on correlations PY risksYi , i=1,2,...,I, are independent, i.e. RPY = identity matrix
=>
14
Y
1 2 1 1 2 2, , , , , , , , ,
.
TT TI PY PY I IY Y Y R R R
RYR
CorrY R Y Y , W
22
1, : .PY T TPY PY PY PY PY PYC Y
R
Var VarW R W W R W
2 2 22
1
1 I
i ii
RR
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: claim amount PY
Remark and discussion on correlation assumptionthe assumption, that the reserve risks of different lob are independent, is questionable because of calendar year effectsaffecting several lob simultaneously; the most obvious such calendar year effect is claims inflation.
15 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: combined normal claim amount CY + claim amount PY
Notations
Correlation matrices:
16
,
,
,
,
.
CY ni i i
CY ni i i i i i
ii i i i
i i i
S C RC R P X RYZ
P R P RV P R
,
,
, ,
where , , ,
TCY PYCY PY
CY PYCY PY i ji j C C
R C C
R
Corr
Corr
,
,
,
.
TCY CY
PY PY
CY CY PY
CY PY PY
Corr C CRC C
R RR R
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: combined normal claim amount CY + claim amount PY
Variance on the total over all lob
then
Current standard assumption in the SSTCY and PY risks are independent => is a matrix with zero's
=>
17
, .CY
CY C RS C R ZP R
21,
T TCY CY CY CY
PY PY PY PYS Z
V
Var VarW W W WR RW W W W
,CY PYR
2 2 2 2
2 2 2 22, P RS P R Z
P R
Var Var
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Remark and discussion on correlation assumptionThe assumption, that CY risks and PY risks are independent is questionable because of calendar year effects affecting CY-risk and reserve risk simultaneously; the most obvious such calendar yeareffect is claims inflation having an impact on CY as well as on the reserve risk.
Model assumption It is assumed that is lognormal distributed with
18
Modeling in SST: combined normal claim amount CY + claim amount PY
S
,
.T
CY CY
PY PY
E S P R
S
VarW WRW W
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: big claim amount CY
Model AssumptionsIt is assumed that i) for each lob i the big claim amount has a compound Poisson-
dstribution, i.e.
where is Poisson and
are independent and independent from
ii) are independent
=> is again compound Poisson with
and
19
,CY biC
,
1
,biN
CY b bi iC Y
biN b
i iY F
, 1,2,..., ,b bi iY N b
iN
, : 1,2, ,CY biC i I
, ,
1
ICY b CY b
ii
C C
1
Ib b
ii
1
.bni
ibi
F F
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SST: big claim amount CY
Remarks on big claims modeling– The claim size distribution is essentially assumed to be Pareto;
upper limits and xs-reinsurance can easily be taken into account
– The distribution function of can be calculated for instance by means of the Panjer algorithm.
ConvolutionThe distribution of can be calculated by convoluting the lognormal distribution of with the compound Poisson distribution of
=> distribution before scenarios
20
,CY bC
, ,CY n CY b PYT C C C ,CY n PYC C
,CY bC
F
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
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 :
21
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
DAV Scientific Day 29.4.2009, Berlin / 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 .
22 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: formula to calculate SCR
Notation
where
calculation premium risk per lob
where
23
2 2
(loss ratio CY), ,
, ,
CYi i
i ii i
i i i i
C RX YP R
X Y
Var Var
premiumreserve per 1.1."a posteriori reserves" per 31.12.of .
i
i
i i
PRR L
2 2, ,1 ,i i i ind i i M
,
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
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: formula to calculate SCR
calculation premium + reserve risk per lob
Correlation and aggregation
24
1 , where .i i i i i i i ii
Z P X RY V P RV
,Assumption: , 50%i i CY PYX Y Corr
2 2,
2
2 : .i i CY PY i i i i i i
i ii
P P R RZ
V
Var
Assumption: , , given standard parametersi j ij ijZ Z Corr
22
1 , 1
=> , I I
i j i jii ij
i i j
VVVZ Z ZV V
Var
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: formula to calculate SCR
formula for SCR
where
25
1 2
2
0.995
exp 0.995 log( 1)1
1pr res
mean
SCR V
V VaR
2
0.995
logormal distributed r.v. with 1 and ,99.5% value at risk of ,
,standard normal distribution.
mean
EVaR E
V P Rx
Var
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: formula to calculate SCR
SCR for CAT risks
Total SCR for nl-insurance risk
26
2
1
.K
CAT kk
SCR c
2 2 .nl CY PY CATSCR SCR SCR
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: formula to calculate SCR
27
lob and parameters
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
lob1 motor, third party liability2 motor, other classes3 marine, aviation & transport (MAT)4 fire and other damage to property5 third-party liability6 credit and suretyship7 legal expenses8 assistance9 miscellaneous non-life insurance10 NP reins property11 NP reins casualty12 NP reins MAT
description
lob1 9% 12% 152 9% 7% 53 13% 10% 104 10% 10% 55 13% 15% 156 15% 15% 157 5% 10% 58 8% 10% 59 11% 10% 10
10 15% 15% 511 15% 15% 1512 15% 15% 10
standard parameters
CY-risk (premium)
σ τ
PY-risk (reserve)
max historical
yearsmn
correlation matrix for Z-variables (combined CY and PY risk)lob 1 2 3 4 5 6 7 8 9 10 11 12
1 1 0.50 0.50 0.25 0.50 0.25 0.50 0.25 0.50 0.25 0.25 0.252 0.50 1 0.25 0.25 0.25 0.25 0.50 0.50 0.50 0.25 0.25 0.503 0.50 0.25 1 0.25 0.25 0.25 0.25 0.50 0.50 0.25 0.25 0.504 0.25 0.25 0.25 1 0.25 0.25 0.25 0.50 0.50 0.50 0.25 0.505 0.50 0.25 0.25 0.25 1 0.50 0.50 0.25 0.50 0.25 0.50 0.256 0.25 0.25 0.25 0.25 0.50 1 0.50 0.25 0.50 0.25 0.25 0.257 0.50 0.50 0.25 0.25 0.50 0.50 1 0.25 0.50 0.25 0.50 0.258 0.25 0.50 0.50 0.50 0.25 0.25 0.25 1 0.50 0.50 0.25 0.259 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 1 0.25 0.25 0.50
10 0.25 0.25 0.25 0.50 0.25 0.25 0.25 0.50 0.25 1 0.25 0.2511 0.25 0.25 0.25 0.25 0.50 0.25 0.50 0.25 0.25 0.25 1 0.2512 0.25 0.50 0.50 0.50 0.25 0.25 0.25 0.25 0.50 0.25 0.25 1
credibility weights αi for σ2
historical years availablemn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5 0 0 0.64 0.72 0.79 - - - - - - - - - -
10 0 0 0 0 0.64 0.69 0.72 0.74 0.76 0.79 - - - - -15 0 0 0 0 0 0 0.64 0.67 0.69 0.71 0.73 0.75 0.76 0.78 0.79
Modeling in SII: CY and PY risk
CY risk (premium risk)Neither nor the credibility weight in the formula of depends on the size of the company
=> model assumption:
PY risk (reserve risk)
model assumption:
Remark and discussion:The assumptions, that the loss ratio or the reserve risk is independent of the size of the company is questionable.
28
2,i M i 2
,i ind
2.i iX Var
2.i iY Var
DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling in SII: CY + PY
Correlation Assumptions
Implications:since assumptions must hold for any company =>
Discussion and Remarks:– correlation between lob result from calendar year effects affecting
several lob simultaneously. To assume the same correlation matrix for X and for Y is a bit questionable, since the calendar year effect for CY- and PY-risks might not be the same or have a different impact.
– depend on the volumes and difficult to interpret
29 DAV Scientific Day 29.4.2009, Berlin / 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 Y Corr Corr
, , , ,i j i j i j ijX X Y Y Z Z Corr Corr Corr
, for i jX Y i jCorr
Modeling in SII: CY + PY
Model Assumptionhas the same distribution as where
has a lognormal distribution with
Remarks and Discussion
Contrary to the SST:
=> is modeled by a lognormal distribution with mean , but with a variance which is different from
30 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
S E S 1 ,V 2
211 and .E
V
Var
is aproximated by 1 .S E S V Z E Z V
1.E Z
S E S
Var S
Modeling in SII: CY + PY
Comparison of 99.5% VaR of and for
31 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 1 85%.E Z
φ =
1% 0.070 0.070 1.0042% 0.144 0.145 1.0073% 0.224 0.226 1.0114% 0.309 0.313 1.0155% 0.399 0.407 1.0196% 0.496 0.508 1.0237% 0.599 0.615 1.0278% 0.709 0.731 1.0329% 0.826 0.856 1.036
10% 0.951 0.989 1.04011% 1.084 1.132 1.04512% 1.225 1.285 1.04913% 1.376 1.449 1.05314% 1.536 1.625 1.05815% 1.706 1.813 1.06316% 1.887 2.014 1.06717% 2.080 2.229 1.07218% 2.284 2.459 1.07619% 2.501 2.704 1.08120% 2.732 2.966 1.086
(a)VaRα
mean(Ψ) VaRαmean(Z)
(b)ratio
(c)=(b)/(a)
Modeling in SII: Aggregation Cat Risks
32 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
2
1
,K
CAT kk
SCR c
2 2 .nl CY PY CATSCR SCR SCR
Modelassumption:The Cat-risks Catk , k=1,2,....,K are independent and normally distributed with VaR0.995 (Catk)=ck
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: risk not dependent on the size of the company
CY risk: STT distinguishes between "normal claims" and "big claims".
Correlation Assumptions (current state):SST: no correlations between lob for the reserve risks and no correlations between CY- und PY-risksSII: same correlation between lob for CY- and PY-risksboth assumptions not fully satisfactory.
33 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
Modeling : Summary
SST: scenarios for extraordinary situationsSII: CAT-risks modeled similar to scenarios in the SST
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
34 DAV Scientific Day 29.4.2009, Berlin / 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
35 DAV Scientific Day 29.4.2009, Berlin / 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 P
2 2
1
1ˆ ( ) ,1
inij
i ij iji i
RY Y
n R
2ˆi
2i
Parameter Estimators: SST parameters
Random fluctuation risk CY
in long-tail lob: above estimator underestimates the CoVa in recent accident years
36 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
2 2, 1.i fluct iCoVa Y
21
11
2
ij
ij
NiijN
ij
i
Y YCoVa
Y
Development triangle CoVa claim amounts in motor liabilityAY/DY 0 1 2 3 4 5 6 7 8 9
1998 6.1 7.1 8.1 8.5 9.1 9.7 10.1 10.2 10.2 10.91999 6.1 7.6 8.1 9.0 10.0 10.3 10.6 10.6 11.2 11.22000 5.8 6.9 8.4 8.9 9.4 9.5 9.5 10.3 10.3 10.32001 5.8 7.4 8.9 9.5 9.6 9.6 10.4 10.7 10.7 10.72002 5.7 7.8 9.0 9.0 9.2 9.9 10.3 10.6 10.6 10.62003 6.6 8.5 8.8 9.1 9.8 10.2 10.6 10.9 10.9 10.92004 6.6 8.5 9.2 10.1 10.7 11.1 11.5 11.9 11.9 11.92005 5.2 7.2 8.7 9.2 9.7 10.1 10.5 10.8 10.8 10.82006 5.4 7.5 8.5 9.0 9.5 9.9 10.3 10.6 10.6 10.62007 5.7 7.4 8.4 8.8 9.4 9.7 10.1 10.4 10.4 10.4
mean 10.8
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
37 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
1 2,T
j j j
1, 1, , J
2 2
2 2 ˆ1, ,fluct fluctj j param param
j j
E X XP
Var
222
1
ˆˆ ,1
Jj fluct
param jj
w JJc X XJ w n
1
22
1
1 1 , ˆ 1 ,
observed number of claims.
Ii i
flucti
w wIc CoVa YI w w
n
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
38 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
21 2param Var 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
39 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators
1
1
1ˆ ln1
bn Yn c
1ˆ ˆ, .2
E CoVan
0ˆ ˆ (1 )cred
20
2 , standard value from the SST, .1
n CoVan
25%, n=16 CoVa
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 Merz/Wüthrich (2008) and Bühlmann and alias (2009);
technical remark:we believe that the parameter risk can be treated in rigorous mathematical way only within a Bayesian framework. Taking an non-informative prior then leads to a slightly different formula than the classical formula of Mack. Mack's formula is then obtained by a first order Taylor approximation. The results obtained with the two formulae are mostly so close to each other, that there is no difference from a practical point of view.
In Solvency we are interested in the one in a century adverse reserve rbents. 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.
40 DAV Scientific Day 29.4.2009, Berlin / 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).
41 DAV Scientific Day 29.4.2009, Berlin / 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). The estimation error in the chain-ladder reserving method: a Bayesian approach. ASTIN Bulletin 36/2, 554-565.
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.
42 DAV Scientific Day 29.4.2009, Berlin / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling and Parameter Estimators