Solvency II premium risk modeling under the direct compensation CARD system Giorgio Alfredo

147 Pages·2011·6.62 MB·English

Solvency II premium risk modeling under the
direct compensation CARD system
Giorgio Alfredo Spedicato
Via Firenze, 11
20037 Paderno Dugnano, Italy
E-mail address: spedicato [email protected]
1991 Mathematics Subject Classification. Primary 62P05; Secondary 62J12, 91B30
Professor Nino Savelli, PhD tutor
Gloria Leonardi and Stella Garnier, my actuarial supervisors at Axa Assicurazioni
& Investimenti.
iii
Abstract.
This dissertation will discuss most relevant issues and challenges for Ital-
ianTPMLpracticederivedfromtheintroductionofCARDdirectreimbursementscheme
regulation. Aspecialattentionwillbedevotedtothedevelopmentofaninternalmodel
forthenon-lifeunderwritingpremiumriskcapitalchargecoherentwiththenewbusiness
practicesthattheCARDsystemhasintroduced.
ThePhDthesisisdividedintotwopart: theexternalenvironmentdescriptionand
aninternalmodeldevelopmentprocessdetailed. Thefinalappendixshowsstatisticalre-
marks and reference with respect to the special techniques used in the thesis and the
bibliographicreferences.
TheexternalenvironmentdescriptionsummarizesthehistoricalevolutionofTPML
underwritingandactuarialpracticesduringlast20years.
AdetaileddescriptionoftheCARDdirectreimbursementstructureistherebydeveloped.
Thenatureofthecompositeclaimstructure,aspecificdescriptionofeachcomponentsof
claims,theregulatorycontextchangesarediscussedwithspecialattentiontotheactuarial
implicationonthepricingofthirdpartymotorliabilitycoverage.
Thepreviousactuarialpracticeiscontrastedwiththenewchallengesintroducedbythe
CARDscheme.
Some statistical tables coming from official industrial bureau (ANIA) and government
bodies(ISVAP)willbereportedtoshowthemarketwidefrequencyandseverityfigures
for TPML. These figures have been detailed by year, sub-line and component of claims
whetherdatahadbeenavailable.
Theexternalenvironmentdescriptionfollowswithadescriptioningeneraltermsofthe
SolvencyIIframework. Aspecialattentionhasbeengiventothenon-lifepremiumrisk
moduleoftheunderwritingrisk. QIS5andQIS4standardformulaswillbepresented,even
iftheinternalmodeldisclosedfurtherhasbeendevelopedwithrespecttoQIS4framework.
Abrieftheoreticaldescriptionofriskadjustedmeasuresofperformancehasbeenfinally
reported.
Theinternalmodeldevelopmenthasbeendevelopedintotwoparts: adescriptionof
provideddatasourcesalongwithunivariateandbivariateanalysisandthedevelopment
ofinternalmodel. Initialdatasourceanalysisshowedaggregatefiguresofexposure,fre-
quencyandseveritybycomponentofclaimsforprovidedportfolio. Theanalysishasbeen
detailed by accident year, sub - line of business and most relevant provided rate mak-
ingvariables. Theseanalyseshavebeenintroductoryforthedevelopmentoftheinternal
model.
Inordertodeveloptheinternalmodel,thepremiumriskcapitalchargehasbeendefined
usingaSolvencyIIcoherentVaRlikemeasureS˜99.5%−E(cid:104)S˜(cid:105). Acollectiverisktheory
approachhasbeenusedtodevelopthetotallossdistributionofS˜.
Original features of the internal model were: risk heterogeneity consideration by
clusteringrisks, coherencewithdirectreimbursementCARDstructure, useofGAMLSS
incapitalmodeling. 2009steadystateintermsonclaimspricelevelandtotalloss
Theheterogeneityofrisk,usuallyconsideredbyTPMLpricingactuarieswhendeveloping
losspredictivemodelingandusuallynottakenintoaccountincapitalmodeling,hasbeen
considereddividingtheportfolioinclustersdefinedbytheleveloffewrelevantratemak-
ingfactor. Atotallossdistributionhasbeendevelopedoneachclusterandtheoccurrence
ofeachspecificclusterdistributionhavebeenaggregated.
An underwriting risk modeling coherent with the CARD scheme has been imple-
mentedbytwoalternativeapproaches. Inthefirstapproach,aspecificlossdistribution
foreachcomponentofclaimshavebeendevelopedwhithineachcluster. Theiroccurrences
havebeensummedupassumingconditionaldependencebycomponentofclaims. Compen-
satingforfeitamountsforsufferedCARDclaimsdistributionsorforfeitcostsforcaused
CARDclaimsdistributionshave beenmodeledbyresampling. Inthesecondapproach
thenumberoftotalpaymentshasbeensimulatedforeachclusterofinsuredsregardless
specificcomponentofclaimsorwhethertheclaimswascausedorsuffered. Theamount
ofpaymentsdistributionhavebeenobtainedbyresamplingfromthe2009claimsdistri-
bution.
Finallythispaperisthefirstpaper(asknownbymeatDecember2010)thatshowsthe
useofGAMLSSonP&Ccapitalmodeling. GAMLSSareanextensionofgeneralizedlinear
modelsthatprovideamathematicalstructuretomodelthedependencyofmoreparame-
tersthantousualmean(asdoneinstandardGLMs)withrespecttocandidatepredictors.
GAMLSS have been used to model the frequency of each component of claims and the
severityofcomponentofclaimsdirectlyhandledbythecompany.
The two modeling frameworks have been modified therefore by testing the use of GPD
distributiontomodellargeclaims.
Atotaloffourinternalmodelshavebeendevelopedandcorrespondingcapitalcharges
havebeenestimated. Resultsshowsthatcapitalchargesseemcomparablewithstandard
formularesults(insomecaseshigherandinsomecaseslower). Mainlimitationsofthe
developedmodelsarethedifficultytoconsidertheyearlychangesofforfeitsandthetreat-
mentofclaimsdevelopmentuntilultimate(IBNR&IBNERcharge).

Contents
Preface vii
Part 1. The CARD direct reimbursement scheme for TPML
insurance 1
Chapter 1. Historical development of TPML actuarial practice in Italy 3
1. The milestones 3
2. ”‘Commissione Filippi”’ tariff 3
3. Changes since 1994 6
4. General remarks about TPML pricing 7
5. Overview of TPML ratemaking actuarial practice 7
Chapter 2. The DR scheme in Italy 11
1. The new regulatory environment 11
2. Current actuarial literature 19
3. Official risk statistic about Italian TPML 20
Chapter 3. The Solvency II framework 27
1. Synthetic overview of the Solvency II framework 27
2. SCR non-life underwriting risk 31
3. Literature on underwriting risk internal models 34
Chapter 4. Capital allocation and profit measures 37
1. Overview 37
2. Market consistent value and capital allocation for a P&C insurer 37
Part 2. A CARD - coherent premium risk internal model for a
TPML portfolio 41
Chapter 5. Preliminary data analysis 43
1. Description of data sources 43
2. Components of claim distribution 43
3. Basic univariate statistics 47
Chapter 6. CARD portfolio standard formula & internal model exemplified 51
1. Overview 51
2. The standard formula 55
3. Internal models 56
4. Discussion 68
Chapter 7. Conclusions 71
v
vi CONTENTS
1. Final remarks 71
2. Disclaimer 72
Part 3. Appendix 73
Appendix A. Review of statistics and predictive modeling 75
1. Predictive modeling 75
2. Individual and collective risk theory 82
3. Peak over threshold extreme value theory approach 84
Appendix B. One way data analysis 87
Appendix C. GAMLSS first model relativities 111
Appendix. Bibliography 137
Preface
The purpose of this PhD Thesis is to introduce the actuarial issues regarding
thenewDRschemeinforceinItalianTPMLmarketsince2007,theCARDsystem.
CARD system has been introduced in Italy along with Bersani II Law. Bersani II
law also affected TPML insurance market modifying provisions of law regarding
policyholders’ claim history recording and bonus-malus structure. Bersani II pro-
visions halted insurers’ ability to classify policyholder according to previous claims
experience.
From an actuarial point of view, the most important effect of CARD is that
both responsible and non responsible claims have to be considered in ratemaking
as long as non responsible handled claim have an impact on the total claims cost.
An historical digression on TPML insurance market in Italy will be outlined
in chapter 1. In the same paper the design of a TPML coverage used in Italy will
presented. The structure of the CARD direct reimbursement system will be ana-
lyzed in detail in chapter 2. Moreover some statistical figures regarding frequency
and severity under the card scheme will be reported in this chapter. Solvency II
framework will be presented and discussed in chapter 3, with a specific focus on
non-life premium risk.
Risk adjusted performance measures and capital allocation will be presented in
chapter 4.
The second part of the thesis will define and develop a framework to build
internalmodelstoassessSolvencyIIpremiumriskcapitalchargeonaTPMLport-
folio under the Italian CARD direct reimbursement scheme. The internal model
hasbeencalibratedonTPMLrealinsurancedata. Employeddatasetscomefroma
P&CmediumsizeinsureroperatingnationwideinItaly. Chapter5willpresentthe
dataset and descriptive risk statistics. Chapter 6 will report the standard formula
and the internal model.
Conclusions are drawn in 7.
In the appendix, chapterA reports basic remarks on most relevant statisti-
cal techniques used in this thesis. Chapter B reports one way analyses on applied
exampledataset,whilechapterCreportstheGAMLSSoutputsforthefirstmodel.
Rsoftware[R Development Core Team, 2010]hasbeenusedforallcalcu-
lations. While R script can be distributed upon request, data used will never be
given as protected by a confidentiality agreemeent with my employeer.
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Part 1
The CARD direct reimbursement
scheme for TPML insurance