Equity Allocation and Portfolio Selection in Insur

EquityAllocationandPortfolioSelectioninInsurance:AsimpliedPortfolioModelErikTainyJune1999,Version2000.11.07AbstractAquadraticdiscretetimeprobabilisticmodel,foroptimalportfolioselection,underriskconstraint,isintroducedinthecontextof(re-)insuranceandnance.Theportfolioiscomposedofcontractswitharbitraryunderwritingandmaturitytimes.Forpositivevaluesofun-derwritinglevels,theexpectedvalueoftheaccumulatednalresultisoptimizedunderconstraintsonitsvarianceandonannualReturnsOnEquity.ExistenceofauniquesolutionisprovedandaLagrangianfor-malismisgiven.AneectivemethodforsolvingtheEuler-Lagrangeequationsisdeveloped.Theapproximatedeterminationofthemulti-pliersisdiscussed.Thisbasicmodel,whichcanincludebothassetsandliabilities,isanimportantbuildingblockformoregeneralmod-els,withconstraintsalsoonnon-solvencyprobabilities,market-shares,short-falldistributionsandValuesatRisk.Keywords:Insurance,EquityAllocation,PortfolioSelection,Con-straintPortfolioOptimization,ValueatRiskJELClassication:C6,G11,G22,G32MathematicalSubjectClassication:90Axx,49xx,60Gxx1IntroductionOptimalequityallocationandportfolioselection,forareinsurancecompanywithseveralportfolios(orsubsidiaries)leads,inthepresenceofconstraintsonCEREMADE,UniversiteParisIX-Dauphine,PlaceduMarechal-de-Lattre-de-Tassigny,75775Paris(Cedex16),France;[e-mail]lypreparedatAXA,23,AvenueMatignon,75008Paris,France.non-solvencyprobabilities,market-sharesandROE’s1,tohighlynon-linearproblems.Thisgeneralsituationisstudiedinanaccompanyingpaper[14],whichconsiders,inadiscretetimecontext,bothassetsandliabilitiesandwhichincludesconstraintspermittingageneralformulationofdesiredrisklimitations.Togetherwithconstraintsonnon-solvabilityprobabilities,alsoconstraintsonValuesatRiskandshort-falldistributionsarecovered.Inparticular,inthismodel,constraintsonexpectedloss(meanexcessfunctionininsurance)andonhighermomentsofthelossdistributionareadmissible.Mostcommonlyusedrisk-measures(c.f.[1],[8],[15])thereforefallintotheframeworkof[14].Theportfolioweintroduceissuchthatfutureresultsofcontractswrittenatdierenttimesaredistinguishable,whicheasilyallowstoconsiderdierentmaturitytimes.Itisanextensionoftheportfoliosconsid-eredin[7](c.f.also[5])inthecaseofmultiperiodsecuritiesmarkets.Thispermitstoinclude,intheportfolio,reinsurancecontracts(andothercon-tractssuchasderivatives),witharbitraryunderwritingandmaturitytimes.Itleadstoamorecomplexgainfunction(seeformula(2.1))thaninthestandardcaseofamoneyandstockmarket,whichisaparticularcase.Theobjectivefunction(thefunctiontobeoptimized)ofthemodelissimplytheexpectednalgain(i.e.accumulatednalresult)oftheportfolio.Wenotethat,incontrastwithmanymodels(c.f.[15]),here(andin[14])thenotionofriskisnotbuiltintotheobjectivefunctionbutisintroducedthroughtheconstraintsofthemodel.Thedelicatequestionofndingthepreferredutil-ityfunctionofthecompanyisthereforetentativelycircumvented,sinceoftenriskconstraints,suchasadmissiblevaluesofnon-solvencyprobabilities,areapproximatelyknowninapplications.Incertainsituationsitispossibletoreplacethenon-solvencyprobabili-tiesbystrongerquadraticconstraints(see[14]).Thisgivesanewsimpliedquadraticstochasticoptimizationproblem,whichsolutions(ifany)respecttheconstraintsandareapproximatesolutionsoftheoriginalproblem.Thepurposeofthepresentpaperistheresolutionoftheconstraintquadraticoptimizationprobleminitsmostbasicsetting:Optimizationoftheexpectednalgain(i.e.thenalaccumulatedresult)ofonesingleportfolio,forgiveninitialequityandunderconstraintsonthevarianceofthenalgain,ontheannualROE’sandonthesignoftheunderwritinglevels,whichshouldbepos-itive.Thiscase,forwhichthealreadyintroducedportfolioisanextensionofMarkowitzportfolio[11],coversthemostbasicapplications.Itdeterminesanoptimalportfoliostrategyrespectingriskandannualprotabilityconstraints.Itisatheoreticallyimportantbuildingblockinmoregeneralsituations[14].Inparticulartheequivalence(seex2)oftheL2normoftheportfolioand1AprecisedenitionofReturnOnEquity(ROE)isgivenlaterinthisintroduction.ErikTain,CEREMADEVersion2000.11.072EquityAllocationandPortfolioSelectioninInsurance:Asimplied:::thesquarerootofthevarianceofthenalgainisatthebasisoftheproofsin[14].WeconsideraportfolioofNdierenttypesofinsurancecontractsconcludedattimes0;:::;T;whereT1:Theamountofthecontractoftypei;where1iN;beingconcludedattimet;where0tT;isdenotedi(t):Inotherwords,i(t)isthenumberofunitcontracts(e.g.theunitissettooneFF)oftypei:WesupposethattheportfoliodoesnotgenerateanynancialowsatandafteracertaintimeT+T:Let(U)(t+1;)betheresultoftheportfoliofortheperiod[t;t+1[;letthegain2of,attimet;U(t;)=P1st(U)(s;);(U(0;)=0),betheaccumulatedresultfortheperiod[0;t[;andletthenalgainoftheportfolio;U(1;)=U(T+T;)betheaccumulatedresultuntilnomorenancialowsaregenerated.(FortheprecisedenitionofUseeformula(2.1).)For1iNand0tT;i(t)and(U)(t+1;)arerandomvariables.InthecaseunderconsiderationtheequityK(t)=K(0)+U(t;);whereK(0)0istheinitialequityatt=0:LetFtbetheeventswhicharepossibleu