中文第八周习题

保险学院

统计与精算系

复习(Review)利息理论第八周1例1年金在第一年末的付款为1000元，以后每年增加100元，总的付款次数为10次。如果前5年的年实际利率为5%，后5年的年实际利率为6%，这项年金的现值应该是多少？3例2Aperpetuity-immediatepays100peryear.Immediatelyafterthefifthpayment,theperpetuityisexchangedfora25-yearannuity-immediatethatwillpayXattheendofthefirstyear.Eachsubsequentannualpaymentwillbe8%greaterthantheprecedingpayment.Immediatelyafterthe10thpaymentofthe25-yearannuity,theannuitywillbeexchangedforaperpetuity-immediatepayingYperyear.Theannualeffectiverateofinterestis8%.CalculateY.4例3Aloanisbeingamortizedbymeansoflevelmonthlypaymentsatanannualeffectiveinterestrateof8%.Theamountofprincipalrepaidinthe12-thpaymentis1000andtheamountofprincipalrepaidinthet-thpaymentis3700.Calculatet.5例4Aborrows$10,000fromBandagreestorepayitwithequalquarterlyinstallmentofprincipleandinterestat8%,convertiblequarterlyoversixyears.AttheendoftwoyearsBsellstherighttoreceivefuturepaymentstoCataprice,whichproducesayieldrateof10%convertiblequarterlyforC.Findthetotalamountofinterestreceived:(1)byC,and(2)byB.6例51000isdepositedintoFundX,whichearnsanannualeffectiverateof6%.Attheendofeachyear,theinterestearnedplusanadditional100iswithdrawnfromthefund.Attheendofthetenthyear,thefundisdepleted.TheannualwithdrawalsofinterestandprincipalaredepositedintoFundY,whichearnsanannualeffectiverateof9%.DeterminetheaccumulatedvalueofFundYattheendofyear10.例6

Aperpetuitycosts77.1andmakesannualpaymentsattheendoftheyear.Thisperpetuitypays1attheendofyear2,2attheendofyear3,…,n

attheendofyear(n+1).Afteryear(n+1),thepaymentsremainconstantatn.Theannualeffectiveinterestrateis10.5%.Calculaten.例7一项年金的付款期是从第2年末至第7年末，并且在时刻t的付款率为3t-4，假设固定利息力为6％，试求此项年金在第7年末的终值。9

例8Victorinvests300intoabankaccountatthebeginningofeachyearfor20years.Theaccountpaysoutinterestattheendofeveryyearatanannualeffectiveinterestrateofi.Theinterestisreinvestedatanannualeffectiverateof(i/2).Theyieldrateontheentireinvestmentoverthe20yearperiodis8%annualeffective.Determinei.10例9Attimet=0,Sebastianinvests2000inafundearning8%convertiblequarterly,butpayableannually.Hereinvestseachinterestpaymentinindividualseparatefundseachearning9%convertiblequarterly,butpayableannually.Theinterestpaymentsfromtheseparatefundsareaccumulatedinasidefundthatguaranteesanannualeffectiverateof7%.Determinethetotalvalueofallfundsatt=10.11例10SusaninvestsZattheendofeachyearforsevenyearsatanannualeffectiveinterestrateof5%.Theinterestcreditedattheendofeachyearisreinvestedatanannualeffectiverateof6%.TheaccumulatedvalueattheendofsevenyearsisX.LoriinvestsZattheendofeachyearfor14yearsatanannualeffectiveinterestrateof2.5%.Theinterestcreditedattheendofeachyearisreinvestedatanannualeffectiverateof3%.Theaccumulatedvalueattheendof14yearsisY.CalculateY/X.12例11Maryinvests1000attheendofeachyearfor5yearsatanannualeffectiverateof9%,andreinveststheinterestatanannualeffectiverateof9%.Attheendof5years,herinvestmenthasvalueofX.Johninvests1000atthebeginningofeachyearfor5yearsatanannualeffectiverateof10%andreinveststheinterestatanannualeffectiverateof8%.Attheendof5years,hisinvestmenthasvalueY.CalculateY−X.13例12Anamountisinvestedatanannualeffectiverateofinterestiwhichisjustsufficienttopay1attheendofeachyearfornyears.Inthefirstyearthefundactuallyearnsrateiand1ispaidattheendofthefirstyear.However,inthesecondyearthefundearnsratej,wherej>i.Findtherevisedpaymentwhichcouldbemadeattheendsofyear2throughn;(1)Assumingtherateearnedrevertsbacktoiagainafterthisoneyear;(2)Assumingtherateearnedremainsjfortherestofn-yearperiod.例13例：一笔1000万元的贷款，为期10年，若年实际利率为9％，试对下面三种还款方式比较其利息总额。本金和利息在第10年末一次还清；每年产生利息在当年末支付，而本金在第10年末归还。在10年期内，每年末偿还相同的金额。问题：这些利息总额在价值上是否相等？例14某年金共有7次付款1，分别在第3期末到第9期末依次支付。求此年金的现值和在第12期末的积累值。16例15（1）（2）17例16

Theaccumulatedvaluejustafterthelastpaymentundera12-yearannuityof1000peryear,payinginterestattherat