Act2120 UniversityOfManitobaActuarialClub–2016/2017

InterestTheoryMidterm1–Time:70min1)Moneyaccumulatesinafundataneffectiveannualinterestrateofiduringthefirst5years,andatan

effectiveinterestrateof2ithereafter.Adepositof1ismadeintothefundattime0.Itaccumulatesto

3.09attheendof10yearsandto13.62attheendof20years.Whatisthevalueofthedepositatthe

endof7years?

2)Twofunds,AandB,startwiththesameamount.FundAgrowsatanannualinterestrateofi>0forn

years,andatanannualinterestrateofj>0forthenextnyears.FundBgrowsasanannualinterestrate

ofk>0for2nyears.FundAequals1.5timesfundBafternyears.Theamountinthetwofundsare

equalafter2nyears.Whichofthefollowingaretrue?

i)j<k<i

ii)k<(i+j)/2

iii)j=k(2/3)1/n

3)Whichofthefollowingarecorrectexpressionsford(360)?

i)1–v1/360

ii)360–360(1+i(12)/12)-30

iii)1/(1/360+1/i(360))

4)Atthebeginningofeachyearfor10years,$100isdepositedintoasavingsaccount.

Atasimpleannualinterestrateofi,thetotalamountintheaccountis$1,275attheendoftenyears.

Tothenearest$5,whatwouldbethetotalamountintheaccountattheendoftenyearsifinteresthad

beencompoundedataneffectiveannualinterestrateofi%?

5)Devinborrowed$10,000frombankXatanannualeffectiverateof8%.Heagreedtorepaythebank

withfivelevelannualinstallmentsattheendofeachyear.

Atthesametime,healsoborrowed$15,000frombankYatanannualeffectiverateof7.5%.Heagreed

torepaythisloanwithfivelevelannualinstallmentsattheendofeachyear.

Helent$25,000toCedricimmediatelyinexchangeforfourannuallevelrepaymentsattheendofeach

year,atanannualeffectiverateof8.5%.

Devincanonlyreinvesttheproceedsatanannualrateof6%.

Immediatelyafterrepayingtheloanstothebanksinfull,determinehowmuchDevinhasleft.

6)Sergedeposits$100attheendofeachyearfor20yearsintoafundearninganannualeffective

interestrateof7%.

Gabbymakes20depositsintoafundattheendofeachyearfor20years.Thefirst10depositsare$100

each,whilethelast10depositsare$100+$Xeach.Thefundearnsanannualeffectiveinterestrateof

8%duringthefirst10yearsand6%annualeffectiveinterestthereafter.

Attheendof20years,theamountinSerge’sfundequalstheamountinGabby’sfund.

CalculateX.

Act2120 UniversityOfManitobaActuarialClub–2016/2017

7)Thepresentvaluesofthefollowingthreeannuitiesareequal:

i)Perpetuity-immediatepaying$1eachyear,calculatedatanannualeffectiverateof7.25%

ii)50-yearannuity-immediatepaying$1eachyear,calculatedatanannualeffectiveinterestrateofj%

iii)n-yearannuity-immediatepaying$1eachyear,calculatedatanannualeffectiveinterestrateofj-1%

Calculaten.

Note:Certainquestionsaskingforthecalculationofaneffectiveinterestrateoftenrequiretheuseofa

financialcalculator.ThesequestionswillnotbetestedintheInterestTheorycourse,butaremorethan

likelytomakeanappearanceonExamFM.Itisstillrecommendedthatstudentsreviewthisquestionfor

thepurposeofunderstandingitssolutionprocess.

8)Youaregivenδt=2/(10+t),t>0.

Calculatethepriceofanannuity-immediatepaying$1attheendofeveryyearfor4years.

9)Anannuity-duepaying$1forn+2yearsishasapresentvalueof$13.987.Anannuity-duepaying$1

fornyearshasanaccumulatedvalueof$51.632.Calculatetheeffectiveannualrateofinterest.

Disclaimer:thisexamisnotonethathasactuallybeenpreviouslytestedinInterestTheory.ThiswasanexamcreatedbyUMACinordertoprovidestudentswithamorerecentrealisticrepresentationofwhatonemayexpectonbothInterestTheoryExams,andExamFM.ThereisastrongpossibilitythatthisexamisofeasierdifficultythanwhatwillbetestedinInterestTheory.

- SergiuBuda