中文第八周習(xí)題

保險(xiǎn)學(xué)院

統(tǒng)計(jì)與精算系

復(fù)習(xí)(Review)利息理論第八周1例1年金在第一年末的付款為1000元，以后每年增加100元，總的付款次數(shù)為10次。如果前5年的年實(shí)際利率為5%，后5年的年實(shí)際利率為6%，這項(xiàng)年金的現(xiàn)值應(yīng)該是多少？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7一項(xiàng)年金的付款期是從第2年末至第7年末，并且在時刻t的付款率為3t-4，假設(shè)固定利息力為6％，試求此項(xiàng)年金在第7年末的終值。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：一筆1000萬元的貸款，為期10年，若年實(shí)際利率為9％，試對下面三種還款方式比較其利息總額。本金和利息在第10年末一次還清；每年產(chǎn)生利息在當(dāng)年末支付，而本金在第10年末歸還。在10年期內(nèi)，每年末償還相同的金額。問題：這些利息總額在價(jià)值上是否相等？例14某年金共有7次付款1，分別在第3期末到第9期末依次支付。求此年金的現(xiàn)值和在第12期末的積累值。16例15（1）（2）17例16

Theaccumulatedvaluejustafterthelastpaymentundera12-yearannuityof1000peryear,payinginterestattherat