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. vii Part 1 The CARD direct reimbursement scheme for TPML insurance