interest theory midterm 1 - university of manitobaumanitoba.ca/actuary_club/academic/exams/2120 mt 1...

Act 2120 University Of Manitoba Actuarial Club – 2016/2017 Interest Theory Midterm 1 – Time: 70 min 1) Money accumulates in a fund at an effective annual interest rate of i during the first 5 years, and at an effective interest rate of 2i thereafter. A deposit of 1 is made into the fund at time 0. It accumulates to 3.09 at the end of 10 years and to 13.62 at the end of 20 years. What is the value of the deposit at the end of 7 years? 2) Two funds, A and B, start with the same amount. Fund A grows at an annual interest rate of i > 0 for n years, and at an annual interest rate of j > 0 for the next n years. Fund B grows as an annual interest rate of k > 0 for 2n years. Fund A equals 1.5 times fund B after n years. The amount in the two funds are equal after 2n years. Which of the following are true? i) j < k < i ii) k < (i + j)/2 iii) j = k(2/3) 1/n 3) Which of the following are correct expressions for d (360) ? i) 1 – v 1/360 ii) 360 – 360(1 + i (12) /12) -30 iii) 1/(1/360 + 1/i (360) ) 4) At the beginning of each year for 10 years, $100 is deposited into a savings account. At a simple annual interest rate of i, the total amount in the account is $1,275 at the end of ten years. To the nearest $5, what would be the total amount in the account at the end of ten years if interest had been compounded at an effective annual interest rate of i %? 5) Devin borrowed $10,000 from bank X at an annual effective rate of 8%. He agreed to repay the bank with five level annual installments at the end of each year. At the same time, he also borrowed $15,000 from bank Y at an annual effective rate of 7.5%. He agreed to repay this loan with five level annual installments at the end of each year. He lent $25,000 to Cedric immediately in exchange for four annual level repayments at the end of each year, at an annual effective rate of 8.5%. Devin can only reinvest the proceeds at an annual rate of 6%. Immediately after repaying the loans to the banks in full, determine how much Devin has left. 6) Serge deposits $100 at the end of each year for 20 years into a fund earning an annual effective interest rate of 7%. Gabby makes 20 deposits into a fund at the end of each year for 20 years. The first 10 deposits are $100 each, while the last 10 deposits are $100 + $X each. The fund earns an annual effective interest rate of 8% during the first 10 years and 6% annual effective interest thereafter. At the end of 20 years, the amount in Serge’s fund equals the amount in Gabby’s fund. Calculate X.

Upload: others

Post on 05-Apr-2020

12 views

Category:

Documents

0 download

Report

Download

Embed Size (px):

TRANSCRIPT

Page 1: Interest Theory Midterm 1 - University of Manitobaumanitoba.ca/actuary_club/Academic/Exams/2120 MT 1 .pdf · Interest Theory Midterm 1 – Time: 70 min 1) Money accumulates in a fund

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.

Page 2: Interest Theory Midterm 1 - University of Manitobaumanitoba.ca/actuary_club/Academic/Exams/2120 MT 1 .pdf · Interest Theory Midterm 1 – Time: 70 min 1) Money accumulates in a fund

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