利息理论习题.docx

利息理论习题.docx

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利息理论习题

1.1

1.SallyhastwoIRAs.IRA1earnsinterestat8%effectiveannuallyandIRA2earnsinterestat10%effectiveannually.ShehasnotmadeanycontributionssinceJanuary1,1985,whentheamountinIRA1wastwicetheamountinIRA2.ThesumofthetwoaccountsonJanuary1,1993was$75000.DeterminehowmuchwasinIRA2onJanuary1,1985?

（IndividualRetirementAccount）

2.Supposewearegiventhattheeffectiverateofinterestis5%inthefirstyearand6%inthesecondyear.Weinvest$1attime0.Howmuchisinthefundattheendoftwoyears?

3.Aninvestorputs100intoFundXand100intoFundY.FundYearnscompoundinterestattheannualrateofj,andFundXearnssimpleinterestattheannualrateof1.05j.Attheendof2years,theamountinFundYisequaltotheamountinFundX.CalculatetheamountinFundYattheendof5years?

4.EricdepositsXintoasavingsaccountattime0,whichpaysinterestatanominalrateofi,compoundedsemiannually.Mikedeposits2Xintoadifferentsavingsaccountattime0,whichpayssimpleinterestatanannualrateofi.EricandMikeearnthesameamountofinterestduringthelast6monthsofthe8thyear.Calculatei.

5.Johninvests1000inafundwhichearnsinterestduringthefirstyearatanominalrateofKconvertiblequarterly.Duringthe2ndyearthefundearnsinterestatanominaldiscountrateofKconvertiblequarterly.Attheendofthe2ndyear,thefundhasaccumulatedto1173.54.CalculateK.

6.AdepositofXismadeintoafundwhichpaysanannualeffectiveinterestrateof6%for10years.Atthesametime,X/2isdepositedintoanotherfundwhichpaysanannualeffectiverateofdiscountofdfor10years.Theamountsofinterestearnedoverthe10yearsareequalforbothfunds.Calculated.

7.Youaregiven:

for

Determinetheforceofinterestattime

.

8.Attime0,100isdepositedintoFundXandalsointoFundY.FundXaccumulatesataforceofinterest

.FundYaccumulatesatanannualeffectiveinterestrateofi.Attheendof9years,theaccumulatedvalueofFundXequalstheaccumulatedvalueofFundY.Determinei.

1.2

1.Ataneffectiveannualinterestrateof

eachofthefollowingtwosetsofpaymentshaspresentvalueK:

1）Apaymentof121immediatelyandanotherpaymentof121attheendofoneyear.

2）Apaymentof144attheendoftwoyearsandanotherpaymentof144attheendofthreeyears.CalculateK.

2.Youaregiven:

1）ThesumofthepresentvaluesofapaymentofXattheendof10yearsandapaymentofYattheendof20yearsisequaltothepresentvalueofapaymentofX+Yattheendof15years.

2）X+Y=100

3）

.CalculateX.

3.Acustomerisofferedaninvestmentwhereinterestiscalculatedaccordingtothefollowingforceofinterest:

Thecustomerinvests1000attime0.Whatnominalrateofinterest,compoundedquarterly,isearnedoverthefirstfour-yearperiod?

4.Paymentsof300,500and700aremadeattheendofyearsfive,sixandeight,respectively.Interestisaccumulatedatanannualeffectiverateof4%.Youaretofindthepointintimeatwhichasinglepaymentof1500isequivalenttotheaboveseriesofpayments.Youaregiven:

1）Xisthepointintimecalculatedbythemethodofequatedtime

2）Yistheexactpointintime.CalculateX+Y.

5.Jonesagreestopayanamountof2Xattheendof3yearsandanamountofXattheendof6years.Inreturnhewillreceive2000attheendof4yearsand3000attheendof8years.Atan8%effectiveannualinterestrate,whatisthesizeofJones’secondpayment?

6.Davidcanreceiveoneofthefollowingtwopaymentstreams:

1）100attime0,200attimen,and300attime2n

2）600attime10

Atanannualeffectiveinterestrateof

thepresentvalueofthetwostreamsareequal.Given

determine

7.Youaregiventwoloans,witheachloantobepaidbyasinglepaymentinthefuture.Eachpaymentincludesbothprincipalandinterest.

Thefirstloanisrepaidbya3000paymentattheendoffouryears.Theinterestisaccruedat10%perannumcompoundedsemiannually.

Thesecondloanisrepaidbya4000paymentattheendoffiveyears.Theinterestisaccruedat8%perannumcompoundedsemiannually.

Thesetwoloansaretobeconsolidated.TheconsolidatedloanistoberepaidbytwoequalinstallmentsofX,withinterestat12%perannumcompoundedsemiannually.Thefirstpaymentisdueimmediatelyandthesecondpaymentisdueoneyearfromnow.CalculateX

8.Atacertaininterestratethepresentvalueofthefollowingtwopatternsareequal:

1）200attheendof5yearsplus500attheendof10years

2）400.94attheendof5years

Atthesameinterestrate,100investednowplus120investedattheendof5yearswillaccumulatetoPattheendof10years.CalculateP

2.1

例2.1.2一项贷款，总额为1000元，年利率为9%.设有一下三种偿还方式：

（1）贷款总额以及应付利息在第10年年末一次性偿还；

（2）每年年末偿还该年度的应付利息，本金在第10年年末偿还；

（3）在10年中美年年末进行均衡偿付。

分别计算在三种偿还方式下所支付的利息额。

例2.1.5某人购房借款50000元，计划每年年末还款10000元，直到还完为止。

设利率为7%，分三种方式求该借款人还款的整数次数以及最后的还款零头。

其中零头的还款时间分别为：

（1）在时刻n与n+1之间。

（2）在时刻n。

（3）在时刻n+1。

1.SusanandJeffeachmakedepositsof100attheendofeachyearfor40years.Startingattheendofthe41styear,SusanmakesannualwithdrawalsofXfor15yearsandJeffmakesannualwithdrawalsofYfor15years.Bothfundshaveabalanceof0afterthelastwithdrawal.

Susan’sfundearnsanannualeffectiveinterestrateof8%.Jeff’sfundearnsanannualeffectiveinterestrateof10%.Calculate

2.Atanannualeffectiveinterestrateof6.3%,anannuityimmediatewith4Nlevelannualpaymentsof1000hasapresentvalueof14113.

DeterminethefractionofthetotalpresentvaluerepresentedbythefirstsetofNpaymentsandthethirdsetofNpaymentscombined.

3.Aninvestmentrequiresaninitialpaymentof10000andannualpaymentof1000attheendofeachofthefirst10years.Startingattheendofthe11thyear,theinvestmentreturns5equalannualpaymentsofX.DetermineXtoyieldanannualeffectiverateof10%overthe15-yearperiod.

4.Thepresentvalueofaseriesofpaymentsof2attheendofevery8years,forever,isequalto5.Calculatetheeffectiverateofinterest.

5.Chuckneedstopurchaseanitemin10years.Theitemcosts200today,butitspriceinflatesat4%peryear.Tofinancethepurchase,Chuckdeposits20intoanaccountatthebeginningofeachyearforfirst6years.HedepositsanadditionalXatthebeginningofyears4,5,6tomeethisgoal.Theannualeffectiveinterestrateis10%.CalculateX

6.Depositsof1000areplacedintoafundatthebeginningofeachyearfor30years.Attheendofthe40thyear,annualpaymentscommenceandcontinueforever.Interestisataneffectiveannualrateof5%.Calculatetheannualpayment.

7.ThefollowingthreeseriesofpaymentshavethesamepresentvalueofP:

1）aperpetuity-immediateof2peryearatanannualeffectiveinterestrateof

2）a20-yearannuity-immediateofXperyearatanannualeffectiveinterestrateof

3）a20-yearannuity-dueof0.96154Xperyearatanannualeffectiveinterestrateof

.CalculateP

8.Atanominalrateofinterest

convertiblesemiannually,thepresentvalueofaseriesofpaymentsof1attheendofevery2years,forever,is5.89.Calculate

2.2

例1

在20年期间，每月支付1500元，年利率为7%。

求

（1）这些付款在前3年的现值；

（2）这些付款在最后一次付款后的第5年末的积累值。

例2

（1）已知某种年支付额为10000元，分期于每月月末支付一次相等金额，支付期10年，年实际利率是5%，该年金的现值是多少?

（2）如果年名义利率是5%，每季结算一次利息，其余条件相同，此时年金的现值又是多少？

1.Jerrywillmakedepositsof450attheendofeachquarterfor10years.Attheendof15years,JerrywillusethefundtomakeannualpaymentsofYatthebeginningofeachyearfor4years,afterwhichthefundisexhausted.Theannualeffectiverateofinterestis7%.DetermineY

2.Thepresentvalueofa5-yearannuityimmediate,withpaymentsof1000each6timesperyear,is20930.Determinetheforceofinterest.

3.Harrietwishestoaccumulate60000inafundattheendof25years.Sheplanstodeposit80intothefundattheendofeachofthefirst120months.Shethenplanstodeposit80+xintothefundattheendofeachofthelast180months.Assumethefundearnsinterestatanannualeffectiverateof3.66%.Determinex

4.JimandSueareplanningtoretireonJanuary1,1995.Theirgoalistohaveenoughmoneyinsavingstobeabletowithdraw3000permonthbeginningonemonthafterretirementandcontinuingfor25yearsafterretirement.Theyearnanannualeffectiverateofinterestof10%ontheiraccount.DeterminetheminimumamountneededintheirsavingsaccountonJanuary1,1995,toaccomplishtheirgoal.

5.Johnwins1000000inalotteryandwillbepaid20equalannualinstallmentsof50000withthefirstpaymentduetoday.AbankofferstoexchangeJohn’swinningsforaperpetuityofXpermonthwiththefirstpaymentduetoday.FindthevalueofXassuminga10%effectiverateofinterest.

6.Theproceedsofa10000deathbenefitareleftondepositwithaninsurancecompanyforsevenyearsatanannualeffectiveinterestrateof5%.Thebalanceattheendofsevenyearsispaidtothebeneficiaryin120equalmonthlypaymentsofX,withthefirstpaymentmadeimmediately.Duringthepayoutperiod,interestiscreditedatanannualeffectiveinterestrateof3%.CalculateX

7.Acardealerofferstosellacarfor10000.Thecurrentmarketloanrateisanominalrateofinterestof12%perannum,compoundedmonthly.Asaninducement,thedealeroffers100%financingataneffectiveannualinterestrateof5%.Theloanistoberepaidinequalinstallmentsattheendofeachmonthovera4-yearperiod.Calculatethecosttothedealerofthisinducement.

8.Apaymentof100ismadeattheendofeachtwomonthsforaperiodof6years.Thenominalannualrateofinterestis3%,convertibl