The Insurance Risk in the SST and in Solvency II: Modeling ... · ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling

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


Modeling in SST: Insurance Risk

5

Risk measure in the SST99% expected shortfall

SCR for insurance risk

[ ]= −99% .insSCR ES TR


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


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


, 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=

= =


( )υσ = +2 2 ( ), 1.i fluct iCoVa Y

aggregation over lob

the variance of is calculated by assuming

a correlation matrix

=>

where



• • •= , /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

9

,CY biC

, ,

1

ICY b CY b

ii

C C•=

= ∑

λ λ λ•=

= = ∑1

,I

b bi

i 1

.bni

ibi

F Fλλ= •

= ∑


= …, , 1,2, ,CY biC i I

Modeling in SST: normal and big claim amount CY

10

lob andstandard parametersnormal and bigclaim amount CY


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



current standard parameters for PY-risks



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.


• • •= /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

14

• • • • • •• • • • • • •

• • • •

+ += + = = = +

+ +

+ += + = = = +

+ +

,,

,,

, , ,

, , .

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


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

15

( )

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

=

R RX XR Y Y R R

R X Y

,

,

,

, ,

where ,

TCY CY PY

CY PY PY

TCY PY

Corr

Corr

=


( )( )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


, ,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 :

17

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 .


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

===


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:

20

( )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= =• •

==

= − =− ∑ ∑


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

21

( )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



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


( ) ( ) ,, , , 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

23

lob and parameters



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


[ ] ( ) ϕΨ = Ψ = Ψ = 2logormal distributed r.v. with 1 and ,E Var( ) ( )( )Ψ = Ψ − Ψ0.995 0.995 .meanVaR VaR E


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


[ ]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•


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+= +


= …, 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.


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


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 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


( )( )νσ = +2 2, 1.i fluct iCoVa Y

( )( )21

11

2

ij

ij

NiijN

ij

i

Y YCoVa

Y

νυ=− −∑

=


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= 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 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 2paramσ Θ + ΘVar 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

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 =


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.



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).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.


Documents

The Insurance Risk in the SST and in Solvency II: Modeling ... · ASTIN Colloquium, 1-4 June 2009, Helsinki / A. Gisler / The Insurance Risk in the SST and in Solvency II: Modeling