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actuary Modern Actuarial Risk Theory Using R (2nd Ed).pdf

actuary Modern Actuarial Risk Th…

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简介:本文档为《actuary Modern Actuarial Risk Theory Using R (2nd Ed)pdf》,可适用于财会税务领域,主题内容包含ModernActuarialRiskTheoryJanDhaene•MichelDenuitTheoryModernActuarialRiskUs符等。

ModernActuarialRiskTheoryJanDhaene•MichelDenuitTheoryModernActuarialRiskUsingRSecondEditionRobKaas•MarcGoovaertsThisworkissubjecttocopyrightAllrightsarereserved,whetherthewholeorpartofthematerialisreproductiononmicrofilmorinanyotherway,andstorageindatabanksDuplicationofthispublicationTheuseofregisterednames,trademarks,etcinthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneralusePrintedonacidfreepaperspringercomCoverdesign:WMXDesignGmbH,HeidelbergUvAKERoetersstraatWBAmsterdamTheNetherlandsrkaasuvanlProfessorMarcGoovaertsUvAKERoetersstraatWBAmsterdamTheNetherlandsProfessorJanDhaeneAFI(Accountancy,Finance,Insurance)ResearchGroupNaamsestraatLeuvenBelgiumjandhaeneeconkuleuvenbeFaculteitEconomieenBedrijfswetenschappenProfessorMichelDenuitBelgiummicheldenuituclouvainbeISBN:ViolationsareliableforprosecutionundertheGermanCopyrightLaweISBN:InstitutdeStatistiqueKULeuvenSpringerVerlagBerlinHeidelbergconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringerVerlagLibraryofCongressControlNumber:orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember,ProfessorRobKaasmarcgoovaertseconkuleuvenbeKULeuvenUniversityofAmsterdamVoieduRomanPaysUniversitéCatholiquedeLouvainLeuvenLouvainlaNeuveBelgiumNaamsestraatUniversityofAmsterdamStudythepastifyouwoulddefinethefutureConfucius,BCBCRiskTheoryhasbeenidentifiedandrecognizedasanimportantpartofactuarialeducationthisisforexampledocumentedbytheSyllabusoftheSocietyofActuariesandbytherecommendationsoftheGroupeConsultatifHenceitisdesirabletohaveadiversityoftextbooksinthisareaThistextinrisktheoryisoriginalinseveralrespectsInthelanguageoffigureskatingorgymnastics,thetexthastwoparts,thecompulsorypartandthefreestylepartThecompulsorypartincludesChapters–,whicharecompatiblewithofficialmaterialoftheSocietyofActuariesThisfeaturemakesthetextalsousefultostudentswhopreparethemselvesfortheactuarialexamsOtherchaptersaremoreofafreestylenature,forexampleChapter(OrderingofRisks,aspecialityoftheauthors)AndIwouldliketomentionChaptersandinparticularTomyknowledge,thisisthefirsttextinrisktheorywithanintroductiontoGeneralizedLinearModelsSpecialpedagogicaleffortshavebeenmadethroughoutthebookTheclearlanguageandthenumerousexercisesareanexampleforthisThusthebookcanbehighlyrecommendedasatextbookIcongratulatetheauthorstotheirtext,andIwouldliketothankthemalsointhenameofstudentsandteachersthattheyundertooktheefforttotranslatetheirtextintoEnglishIamsurethatthetextwillbesuccessfullyusedinmanyclassroomsLausanne,HansGerbervForewordtotheFirstEditionWhenItookoffice,onlyhighenergyphysicistshadeverheardofwhatiscalledtheWorldwideWebNowevenmycathasitsownpageBillClinton,ThisbookgivesacomprehensivesurveyofnonlifeinsurancemathematicsOriginallywrittenforusewiththeactuarialscienceprogramsattheUniversitiesofAmsterdamandLeuven,itisnowinuseatmanyotheruniversities,aswellasforthenonacademicactuarialeducationprogramorganizedbytheDutchActuarialSocietyItprovidesalinktothefurthertheoreticalstudyofactuarialscienceThemethodspresentedcannotonlybeusedinnonlifeinsurance,butarealsoeffectiveinotherbranchesofactuarialscience,aswellas,ofcourse,inactuarialpracticeApartfromthestandardtheory,thistextcontainsmethodsdirectlyrelevantforactuarialpractice,forexampletheratingofautomobileinsurancepolicies,premiumprinciplesandriskmeasures,andIBNRmodelsAlso,theimportantactuarialstatisticaltooloftheGeneralizedLinearModelsisstudiedThesemodelsprovideextrapossibilitiesbeyondordinarylinearmodelsandregressionthatarethestatisticaltoolsofchoiceforeconometriciansFurthermore,ashortintroductionisgiventocredibilitytheoryAnothertopicwhichalwayshasenjoyedtheattentionofrisktheoreticiansisthestudyoforderingofrisksThebookreflectsthestateoftheartinactuarialrisktheorymanyresultspresentedwerepublishedintheactuarialliteratureonlyrecentlyInthissecondeditionofthebook,wehaveaimedtomakethetheoryevenmoredirectlyapplicablebyusingthesoftwareRItprovidesanimplementationofthelanguageS,notunlikeSPlusItisnotjustasetofstatisticalroutinesbutafullfledgedobjectorientedprogramminglanguageOthersoftwaremayprovidesimilarcapabilities,butthegreatadvantageofRisthatitisopensource,henceavailabletoeveryonefreeofchargeThisiswhywefeeljustifiedinimposingitontheusersofthisbookasadefactostandardOntheinternet,alotofdocumentationaboutRcanbefoundInanAppendix,wegivesomeexamplesofuseofRAfterageneralintroduction,explaininghowitworks,westudyaproblemfromriskmanagement,tryingtoforecastthefuturebehaviorofstockpriceswithasimplemodel,basedonstockpricesofthreerecentyearsNext,weshowhowtouseRtogeneratepseudorandomdatasetsthatresemblewhatmightbeencounteredinactuarialpracticeviiPrefacetotheSecondEditionviiiPrefaceModelsandparadigmsstudiedThetimeaspectisessentialinmanymodelsoflifeinsuranceBetweenpayingpremiumsandcollectingtheresultingpension,decadesmayelapseThistimeelementislessprominentinnonlifeinsuranceHere,however,thestatisticalmodelsaregenerallymoreinvolvedThetopicsinthefirstfivechaptersofthistextbookarebasicfornonlifeactuarialscienceTheremainingchapterscontainshortintroductionstoothertopicstraditionallyregardedasnonlifeactuarialscienceTheexpectedutilitymodelTheveryexistenceofinsurerscanbeexplainedbytheexpectedutilitymodelInthismodel,aninsuredisariskaverseandrationaldecisionmaker,whobyvirtueofJensen’sinequalityisreadytopaymorethantheexpectedvalueofhisclaimsjusttobeinasecurefinancialpositionThemechanismthroughwhichdecisionsaretakenunderuncertaintyisnotbydirectcomparisonoftheexpectedpayoffsofdecisions,butratheroftheexpectedutilitiesassociatedwiththesepayoffsTheindividualriskmodelIntheindividualriskmodel,aswellasinthecollectiveriskmodelbelow,thetotalclaimsonaportfolioofinsurancecontractsistherandomvariableofinterestWewanttocompute,forexample,theprobabilitythatacertaincapitalwillbesufficienttopaytheseclaims,orthevalueatriskatlevelassociatedwiththeportfolio,beingthequantileofitscumulativedistributionfunction(cdf)Thetotalclaimsismodeledasthesumofallclaimsonthepolicies,whichareassumedindependentSuchclaimscannotalwaysbemodeledaspurelydiscreterandomvariables,noraspurelycontinuousones,andweuseanotation,involvingStieltjesintegralsanddifferentials,encompassingboththeseasspecialcasesTheindividualmodel,thoughthemostrealisticpossible,isnotalwaysveryconvenient,becausetheavailabledatasetisnotinanywaycondensedTheobvioustechniquetouseinthismodelisconvolution,butitisgenerallyquiteawkwardUsingtransformslikethemomentgeneratingfunctionsometimeshelpsTheFastFourierTransform(FFT)techniquegivesafastwaytocomputeadistributionfromitscharacteristicfunctionItcaneasilybeimplementedinRWealsopresentapproximationsbasedonfittingmomentsofthedistributionTheCentralLimitTheorem,fittingtwomoments,isnotsufficientlyaccurateintheimportantrighthandtailofthedistributionSowealsolookatsomemethodsusingthreemoments:thetranslatedgammaandthenormalpowerapproximationCollectiveriskmodelsAmodelthatisoftenusedtoapproximatetheindividualmodelisthecollectiveriskmodelInthismodel,aninsuranceportfolioisregardedasaprocessthatproducesclaimsovertimeThesizesoftheseclaimsaretakentobeindependent,identicallydistributedrandomvariables,independentalsoofthenumberofclaimsgeneratedThismakesthetotalclaimsthesumofarandomnumberofiidindividualclaimamountsUsuallyoneassumesadditionallythatthenumberofclaimsisaPoissonvariatewiththerightmean,orallowsforsomeoverdispersionbytakinganegativePrefaceixbinomialclaimnumberForthecdfoftheindividualclaims,onetakesanaverageallytractablemodelSeveraltechniques,includingPanjer’srecursionformula,tocomputethecdfofthetotalclaimsmodeledthiswayarepresentedForsomepurposesitisconvenienttoreplacetheobservedclaimseveritydistributionbyaparametriclossdistributionFamiliesthatmaybeconsideredareforexamplethegammaandthelognormaldistributionsWepresentanumberofsuchdistributions,andalsodemonstratehowtoestimatetheparametersfromdataFurther,weshowhowtogeneratepseudorandomsamplesfromthesedistributions,TheruinmodelTheruinmodeldescribesthestabilityofaninsurerStartingfromcapitaluattimet=,hiscapitalisassumedtoincreaselinearlyintimebyfixedannualpremiums,butitdecreaseswithajumpwheneveraclaimoccursRuinoccurswhenthecapitalisnegativeatsomepointintimeTheprobabilitythatthiseverhappens,undertheassumptionthattheannualpremiumaswellastheclaimgeneratingprocessremainunchanged,isagoodindicationofwhethertheinsurer’sassetsmatchhisliabilitiessufficientlyIfnot,onemaytakeoutmorereinsurance,raisethepremiumsorincreasetheinitialcapitalAnalyticalmethodstocomputeruinprobabilitiesexistonlyforclaimsdistributionsthataremixturesandcombinationsofexponentialdistributionsAlgorithmsexistfordiscretedistributionswithnottoomanymasspointsAlso,tightupperandlowerboundscanbederivedInsteadoflookingattheruinprobabilityψ(u)withinitialcapitalu,oftenonejustconsidersanupperboundeRuforit(Lundberg),wherethenumberRisthesocalledadjustmentcoefficientanddependsontheclaimsizedistributionandthesafetyloadingcontainedinthepremiumComputingaruinprobabilityassumestheportfoliotobeunchangedeternallyMoreover,itconsidersjusttheinsurancerisk,notthefinancialriskThereforenotmuchweightshouldbeattachedtoitsprecisevaluebeyond,say,thefirstrelevantdecimalThoughsomeclaimthatsurvivalprobabilitiesare‘thegoalofrisktheory’,manyactuarialpractitionersareoftheopinionthatruintheory,howevertopicalstillinacademiccircles,isofnosignificancetothemNonetheless,werecommendtostudyatleastthefirstthreesectionsofChapter,whichcontainthedescriptionofthePoissonprocessaswellassomekeyresultsAsimpleproofisprovidedforLundberg’sexponentialupperbound,aswellasaderivationoftheruinprobabilityincaseofexponentialclaimsizesPremiumprinciplesandriskmeasuresAssumingthatthecdfofariskisknown,oratleastsomecharacteristicsofitlikemeanandvariance,apremiumprincipleassignstotheriskarealnumberusedasafinancialcompensationfortheonewhotakesoverthisriskNotethatwestudyonlyriskpremiums,disregardingsurchargesforcostsincurredbytheinsurancecompanyBythelawoflargenumbers,toavoideventualruinthetotalpremiumshouldbeatleastequaltotheexpectedtotalclaims,butadditionally,therehastobealoadinginofthecdfsoftheindividualpoliciesThisleadstoaclosefittingandcomputationbeyondthestandardfacilitiesofferedbyRxPrefacethepremiumtocompensatetheinsurerformakingavailablehisriskcarryingcapacityFromthisloading,theinsurerhastobuildareservoirtodrawuponinadversetimes,soastoavoidgettinginruinWepresentanumberofpremiumprinciples,togetherwiththemostimportantpropertiesthatcharacterizepremiumprinciplesThechoiceofapremiumprincipledependsheavilyontheimportanceattachedtosuchpropertiesThereisnopremiumprinciplethatisuniformlybestRiskmeasuresalsoattacharealnumbertosomeriskysituationExamplesarepremiums,infiniteruinprobabilities,oneyearprobabilitiesofinsolvency,therequiredcapitaltobeabletopayallclaimswithaprescribedprobability,theexpectedvalueoftheshortfallofclaimsoveravailablecapital,andmoreBonusmalussystemsWithsometypesofinsurance,notablycarinsurance,chargingapremiumbasedexclusivelyonfactorsknownaprioriisinsufficientToincorporatetheeffectofriskfactorsofwhichtheuseasratingfactorsisinappropriate,suchasraceorquiteoftensexofthepolicyholder,andalsoofnonobservablefactors,suchasstateofhealth,reflexesandaccidentproneness,manycountriesapplyanexperienceratingsystemSuchsystemsontheonehandusepremiumsbasedonapriorifactorssuchastypeofcoverageandlistpriceorweightofacar,ontheotherhandtheyadjustthesepremiumsbyusingabonusmalussystem,whereonegetsmorediscountafteraclaimfreeyear,butpaysmoreafterfilingoneormoreclaimsInthisway,premiumsarechargedthatreflecttheexactdrivingcapabilitiesofthedriverbetterThesituationcanbemodeledasaMarkovchainThequalityofabonusmalussystemisdeterminedbythedegreeinwhichthepremiumpaidisinproportiontotheriskTheLoimarantaefficiencyequalstheelasticityofthemeanpremiumagainsttheexpectednumberofclaimsFindingitinvolvescomputingeigenvectorsoftheMarkovmatrixoftransitionprobabilitiesRprovidestoolstodothisOrderingofrisksItistheveryessenceoftheactuary’sprofessiontobeabletoexpresspreferencesbetweenrandomfuturegainsorlossesTherefore,stochasticorderingisavitalpartofhiseducationandofhistoolboxSometimesithappensthatfortwolossesXandY,itisknownthateverysensibledecisionmakerpreferslosingX,becauseYisinasense‘larger’thanXItmayalsohappenthatonlythesmallergroupofallriskaversedecisionmakersagreeaboutwhichrisktopreferInthiscase,riskYmaybelargerthanX,ormerelymore‘spread’,whichalsomakesarisklessattractiveWhenweinterpret‘morespread’ashavingthickertailsofthecumulativedistributionfunction,wegetamethodoforderingrisksthathasmanyappealingpropertiesForexample,thepreferredlossalsooutperformstheotheroneasregardszeroutilitypremiums,ruinprobabilities,andstoplosspremiumsforcompounddistributionswiththeserisksasindividualtermsItcanbeshownthatthecollectivemodelofChapterismorespreadthantheindividualmodelitapproximates,henceusingthecollectivemodel,inmostcases,leadstomoreconservativedecisionsregardingpremiumstobeasked,reservestobeheld,andvaluesatriskAlso,wecanprovePrefacexithatthestoplossinsurance,demonstratedtobeoptimalasregardsthevarianceoftheretainedriskinChapter,isalsopreferable,otherthingsbeingequal,intheeyesofallriskaversedecisionmakersSometimes,stoplosspremiumshavetobesetunderincompleteinformationWegiveamethodtocomputethemaximalpossiblestoplosspremiumassumingthatthemean,thevarianceandanupperboundforariskareknownIntheindividualandthecollectivemodel,aswellasinruinmodels,weassumethattheclaimsizesarestochasticallyindependentnonnegativerandomvariablesSometimesthisassumptionisnotfulfilled,forexamplethereisanobviousdependencebetweenthemortalityrisksofamarriedcouple,betweentheearthquakerisksofneighboringhouses,andbetweenconsecutivepaymentsresultingfromalifeinsurancepolicy,notonlyifthepaymentsstoporstartincaseofdeath,butalsoincaseofarandomforceofinterestWegiveashortintroductiontotheriskorderingthatappliesforthiscaseItturnsoutthatstoplosspremiumsforasumofrandomvariableswithanunknownjointdistributionbutfixedmarginalsaremaximalifthesevariablesareasdependentasthemarginaldistributionsallow,makingitimpossiblethattheoutcomeofoneis‘hedged’byanotherInfinance,frequentlyonehastodeterminethedistributionofthesumofdependentlognormalrandomvariablesWeapplythetheoryoforderingofrisksandcomonotonicitytogiveboundsforthatdistributionWealsogiveashortintroductioninthetheoryoforderingofmultivariaterisksthesamemarginalsiftheircorrelationishigherButamorerobustcriterionistorestrictthistothecasethattheirjointcdfisuniformlylargerInthatcaseitcanbeprovedthatthesumoftheserandomvariablesislargerinstoplossorderThere’sForarandompair(X,Y),thecopulaisthejointcdfoftheranksFX(X)andFY(Y)Usingthesmallestandthelargestcopula,itispossibletoconstructrandompairswitharbitraryprescribedmarginalsand(rank)correlationsCredibilitytheoryTheclaimsexperienceonapolicymayvarybytwodifferentcausesThefirstisthequalityoftherisk,expressedthroughariskparameterThisrepresentstheaverageannualclaimsinthehypotheticalsituationthatthepolicyismonitoredwithoutchangeoveraverylongperiodoftimeTheotheristhepurelyrandomgoodandbadluckofthepolicyholderthatresultsinyearlydeviationsfromtheriskparameterCredibilitytheoryassumesthattheriskqualityisadrawingfromacertainstructuredistribution,andthatconditionallygiventheriskquality,theactualclaimsexperienceisasamplefromadistributionhavingtheriskqualityasitsmeanvalueThepredictorfornextyear’sexperiencethatislinearintheclaimsexperienceandoptimalinthesenseofleastsquaresturnsouttobeaweightedaverageoftheclaimsexperienceoftheindividualcontractandtheexperienceforthewholeportfolioTheweightfactoristhecredibilityattachedtotheindividualexperience,henceitiscalledthecredibilityfactor,andtheresultingpremiumsarecalledcredibilitypreareboundsforjointscdfsdatingbacktoFrechetinthe’sandHoffdingintheOnemightsaythattworandomsvariablesaremorerelatedthananotherpairwithxiiPrefacemiumsAsaspecialcase,westudyabonusmalussystemforcarinsurancebasedonaPoissongammamixturemodelCredibilitytheoryisactuallyaBayesianinferencemethodBothcredibilityandgeneralizedlinearmodels(seebelow)areinfactspecialcasesofsocalledGeneralizedLinearMixedModels(GLMM),andtheRfunctionglmmisabletodealwithboththerandomandthefixedparametersinthesemodelsGeneralizedlinearmodelsManyproblemsinactuarialstatisticsareGeneralizedLinearModels(GLM)Insteadofassuminganormallydistributederrorterm,othertypesofrandomnessareallowe

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