am5am5, Studia, Semestr V, MUŻ, Bowers, Jakies zadania i rozwiązania z Bowersa - rozdziały 1-5
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TopicsinMathematicsII-ActuarialMathematics Hand-Out5:FamilyIncomeInsurance&Expense-LoadedPremiums FrankCoolen(CM206-Frank.Coolen@durham.ac.uk),March2008 Inthefinallecturesofthismodule,wediscussFamilyIncomeInsuranceasanexampleofatypeof contractforwhichwecanalsocalculaterequiredpremiums,usingsimilarmethodsasseenearlierinthe module.WealsodiscussExpense-Loadingofpremiums,consideringthreegeneraltypestoenableinsurers totakecostsofcontractsintoaccount.Thesetopicsarepartoftheexaminablematerialofthiscourse. Theexamplesinthelecturesareofcoursegoodindicationsofpossiblequestions.Suchcontracts(and quitesimilarvariations)canalsobeincludedinmoreextensivecontracts.Asanexampleexercise,we suggestsomeadditionstoExercise4-21(givenagainbelow),andweincludeafurtherexercise.Clearly, suchanalysesofcontractswithallsortsofvariations,bothintermsofbenefitsandpremiums,combineall aspectsofthecourse,andhenceareimportantexercises. 4-21.(FromHand-Out4)Considerthefollowingcontract,issuedtoalifeaged40,basedonourprovided lifetablesandwithi=0.05.Incaseofdeathbeforeage60,alumpsumof £ 100,000ispaidattheendof theyearofdeath.Else,anannuityisprovided,startingatage60,withpaymentsof £ 10,000attheendof eachfurtheryearthatthepolicyholdersurvives.Netannualpremiumsaretobepaiduntiltheageof 60,orearlierdeath.Calculate. Exercises 5-1.Calculatethenetannualpremiums(includingexpense-loadingwhereappropriate)foreachofthe following: (a)Samecontractasin4-21,butwithnetannualpremiumspaidonlyuntiltheageof50,orearlierdeath. (b)Samecontractasin4-21,butnowwitha30-yearfamilyincomeinsuranceadded,withpaymentsof £ 20,000attheendoftheyearforthisperiod(startingintheyearofdeath),iftheinsureddiesbefore reachingage70. (c)Samecontractasin(b),butnow(asin(a))withpremiumspaidonlyuntiltheageof50,orearlier death. (d)Samecontractasin4-21,butnowincludethefollowingformsandamountsofexpense-loading:Col- lectionexpenses-5%ofthepremium;Acquisitionexpenses- £ 1,000. 5-2.Considerthefollowingpolicy,issuedtoalifeaged30,andbasedonthelifeandcommutationtables provided,withi=0.05.Ifthepolicyholderdiesbeforeage65,afamilyincomeinsurancebecomesavailable consistingofanannualpaymentof20,000paidattheendofeachyear,startingtheyearinwhichthis persondiedandwiththefinalpaymentintheyeartheinsuredwouldhavereachedage65.Ifthepolicy holderreachestheageof65,thenthepolicyprovidesalifeannuitywithannualpaymentsof15,000and thefirstpaymentatthestartofthefollowingyear,untilthemomenttheinsureddies.Asafurtherbenefit ofthispolicy,thereisadeathbenefitof50,000attheendoftheyearofdeathoftheinsured,nomatter atwhatagedeathoccurs.Paymentofthepremiumsisoeredasfollows:constantnetannualpremiums aretobepaidatthestartofeachyearuntiltheageof65,ordeathifearlier. (a)Calculate. (b)Supposethattheinsuredhas20,000availablewhichhecanusetobuythepolicy,inadditiontoannual premiumsasdescribedabove.Determinehowtheannualpremiumsareaectedbytheuseofthis20,000 upfront. (c)Supposethatthispolicycanonlybeboughtatapricethatincludesexpenseloadingofthepremiums, withaone-ocostof400tocoveracquisitionexpenses,andcollectionexpensesat1.5%oftheannual premium.Answerthequestions(a)and(b)again,takingtheseexpensesintoaccount. [ Pobierz całość w formacie PDF ] |
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