insurance risk sst and solvency ii - aktuariat-witzel...13 dav scientific day 29.4.2009, berlin / a....

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


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


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


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


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


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



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



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


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



current standard parameters for PY-risks


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


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



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.


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



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


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


S

,

.T

CY CY

PY PY

E S P R

S

VarW WRW W


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


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


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


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 .


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



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



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



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



27

lob and parameters


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


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


,, , , 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


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


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


Comparison of 99.5% VaR of and for


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


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.


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


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


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


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 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


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 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


21 2param Var Var


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


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


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.



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).


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.


insurance risk sst and solvency ii - aktuariat-witzel...13 dav scientific day 29.4.2009, berlin / a....

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