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1、FaircostsofguaranteedminimumdeathbenefitcontractsFranc¸oisQuittard-PinonandRivoRandrianarivonyAbstract.Theauthorsofferanewperspectiveonthedomainofguaranteedminimumdeathbenefitcontracts.Theseproductshavetheparticularfeatureofofferinginvestorsaguaranteedcapitalupondeath.Acompletemethodologybasedonth
2、egeneralisedFouriertransformisproposedtoinvestigatetheimpactsofjumpsandstochasticinterestrates.ThispaperthusextendsMilevskyandPosner(2001).Keywords:lifeinsurancecontracts,variableannuities,guaranteedminimumdeathbenefit,stochasticinterestrates,jumpdiffusionmodels,mortalitymodels1IntroductionThecon
3、tractanalysedinthisarticleisaGuaranteedMinimumDeathBenefitcontract(GMDB),whichisalifeinsurancecontractpertainingtotheclassofvariableannuities(VAs).Foranintroductiontothissubject,seeHardy[4]andBauer,KlingandRuss[2].Theprovidedguaranty,onlyineffectupondeath,ispaidbycontinuouslydeductingsmallamounts
4、fromthepolicyholder’ssubaccount.Itisshowninthischapterhowthesefeescanbeendogenouslydetermined.MilevskyandPosner[8]foundthesefeesoverpricedbyinsurancecompanieswithrespecttotheirmodelfairprice.Toanswerthisoverpricingpuzzle,theeffectsofjumpsinfinancialprices,stochasticinterestratesandmortalityarecon
5、sidered.Forthispurpose,anewmodelisproposedwhichgeneralisesMilevskyandPosner[8].2Generalframeworkandmainnotations2.1FinancialriskandmortalityFinancialriskisrelatedtomarketriskfirstlybecausethepolicyholder’saccountislinkedtoafinancialassetoranindex,andsecondlyviainterestrates.WedenotebyrM.Corazzaeta
6、l.(eds.),MathematicalandStatisticalMethodsforActuarialSciencesandFinance©Springer-VerlagItalia2010284F.Quittard-PinonandR.Randrianarivonythestochasticprocessmodellingtheinstantaneousrisk-freerate.Thediscountfactoristhusgivenby:!tδt=e−0rsds.(1)Thepolicyholder’saccountvalueismodelledbythestochasti
7、cprocessS.Inthatmodel,"standsforthefeesassociatedwiththeMortalityandExpense(M&E)riskcharge.Thefuturelifetimeofapolicyholderagedxisther.v.Tx.Foranindividualagedx,theprobabilityofdeathbeforetimet≥0is"#t$P(Tx≤t)=1−exp−λ(x+s)ds,
