unit 8 booklet - engage explore inspire · 2018. 9. 9. · recall: a general formula for...

MCR3U–Unit8:FinancialApplications–Lesson1 Date:___________

Learninggoal:Iunderstandsimpleinterestandcancalculateanyvalueinthesimpleinterestformula.

SimpleInterestSimpleInterestisthemoneyearned(orowed)onlyontheinvestedor

borrowed.

Theoriginalamountinvestedorborrowedisreferredtoasthe,orsimplythe

_____________________.

Example1:Supposeyouinvest$1000inabankthatoffersyou5%simpleinterest.Whatistheendingbalanceafter5

years?

Let’sbeginbyconsideringONEYEAR…

5%= ________ and 1000()=_________

Therefore,theinterest,I=_______________foroneyear.

Completethechartbelowforthissimpleinterestexample.

Year StartingBalance BalancethatInterestisCalculatedOn Interest Ending

Balance

1 $1000 $1000 I=(1000)(.05)=$50 $1050

2 $1050 $1000 I=

3

4

5

Example1(continued):

a) Nowusethesimpleinterestformulatocalculatetheinterestearnedafter1year.Didyougetthesameamount

asyourinterestcolumn?

b) Nowusetheendingbalanceformula.Didyougetthesameamountastheendingbalanceinyear5inthetable?

SimpleInterestThegeneralformulaforsimpleinterestis! = !"#where:

• !–referstotheinterestearned(indollars)• !–referstotheprincipal(orinitialinvestment,indollars)• !–referstotheinterestrateperyear(asadecimal)• !–referstothelengthoftimethemoneyisinvested(inyears)

Thegeneralformulaforendingbalanceis! = !+ !"#or! = !(!+ !")where• !–referstotheamountthatasimple-interestinvestmentorloanisworth(indollars)

Example2:$500isinvestedat6.5%simpleinterest.Findtheamountiftheinvestmentmaturesin7years.

Example3:A$700investmentdoublesin9years.Findthesimpleinterestrate.

Example4:$1400isborrowedat12.4%simpleinterestfor4months.Findthefinancingcharge(interest).

Example5:Tanyainvests$4850at7.6%/asimpleinterest.Ifshewantsthemoneytoincreaseto$8000,how

longwillsheneedtoinvesthermoney?

HW:pg.49#10,12(AnsCorr12e:15)


Learninggoal:Iunderstandthedifferencebetweensimpleandcompoundinterest.Icancalculatethefuturevalueandinterestofacompoundedinvestment.

CompoundInterestCompoundInterestisthemoneyearned(orowed)ontheinvestedor

borrowedaswellasanythatwaspreviouslyearnedorowedduringtheinvestment.

Example1:Supposeyouinvest$1000inabankthatoffersyou5%interestcompoundedannually.Whatis

theendingbalanceafter5years?

Completethechartbelowforthiscompoundinterestexample.

Year StartingBalance

BalancethatInterestis

CalculatedOn

Interest! = !"#

EndingBalance

1 $1000 $1000 I=(1000)(.05)(1)=$50 $1050

2 $1050 $1050 I=

3

4

5

Whattypeofinvestmentwouldearnyoumoremoney,simpleorcompound?Explain.

SIMPLEvs.COMPOUNDINTEREST:WHAT’STHEDIFFERENCE?SimpleInterest:Completethedifferencecolumn.


BalancethatInterestis

CalculatedonInterest Ending

Balance

FirstDifferences

1 $1000 $1000 $50 $1050

2 $1050 $1000 $50 $1100

3 $1100 $1000 $50 $1150

4 $1150 $1000 $50 $1200

5 $1200 $1000 $50 $1250

Whattypeofrelationshipexistsbetween“year”and“endingbalance”?Why?

CompoundInterest:Completethedifferencecolumns.


Balancethat

InterestisCalculated

on

Interest EndingBalance

FirstDifferences Second

DifferencesRatios

1 $1000 $1000 $50 $1050

2 $1050 $1050 $52.50 $1102.50

3 $1102.50 $1102.50 $55.13 $1157.63

4 $1157.63 $1157.63 $57.88 $1215.51

5 $1215.51 $1215.51 $60.78 $1276.29

Whattypeofrelationshipexistsbetween“year”and“endingbalance”?Why?

SUMMARYForSimpleInterest,therelationshipbetweentimeandtheendingbalanceis.

ForCompoundInterest,therelationshipbetweentimeandtheendingbalanceis.

RECALL:Ageneralformulaforexponentialgrowthis! = !(!)!,where !isthegrowthfactor,and! − 1isthegrowthrate.

Example2:Daytoninvests$1,400inanaccountthatgainsinterestatarateof8%p.a.,compoundedsemi-

annually.Hisinvestmentmaturesafter12years.

a)Findtheamountofhisinvestmentatmaturity.

b)Calculatetheinteresthegained.

FutureValueofanInvestment(Compound)Thegeneralformulaforfindingthefinalamountofaninvestment(compound)is!/!" = !(!+ !)!where:• !isthefinalamount(orfuturevalue)

• !istheprincipal(initialamountinvested)

• !istheinterestrateasadecimal

• ! = !!

• !isthefrequencyofcompoundingperiodsperyear

• !isthenumberoftimesinterestispaid

CompoundingPeriods

iftheperiodis…annual ð! = !(interestpaidonceayear)iftheperiodis…semi-annualð! = ! (interestpaidtwiceayear)iftheperiodis…quarterly ð! = ! (interestpaid4timesayear)

iftheperiodis…monthly ð! = !2(interestpaid12timesayear)

iftheperiodis…bi-weekly ð! = !"(interestpaid26timesayear)

iftheperiodis…weekly ð! = !" (interestpaid52timesayear)

iftheperiodis…daily ð! = !"#(interestpaid365timesayear)

Example3:Adamspends$520onhiscreditcard.Thecreditcompanycharges18.5%p.a.,compoundedmonthly.Hepaysoffhisloaninfullafter2years.

a)Findtheamounthemustpayback.

b)Howmuchinteresthashepaid?

Example4:Margaretcanfinancethepurchaseofa$949.99refrigeratoroneoftwoways:

•PlanA:10%/asimpleinterestfor2years

•PlanB:5%/acompoundedquarterlyfor2years

Whichplanshouldshechoose?Justifyyouranswer.

HW:pg.70#1,3,4,14,15,17,21,pg.79#7


Learninggoal:Icancalculatethepresentvalue,interestrate,andnumberofcompoundingperiodsforacompoundinvestment.

PresentValueFUTUREVALUEvs.PRESENTVALUE

referstotheamountofmoneyneededtoinvest(thepresent)sothatyouwill

obtainaparticularamountinthe.

Inotherwords,ifyouknowtheamountofmoneyyouwanttohaveinthefuture,howmuchprincipalshouldyouinvest

today?

RECALL:Thecompoundinterestformula! = !(!+ !)!,where!representsthestarting(principal)amount.

Ifwerearrangethisformulatoisolate!weobtainthePresentValueformula…

PresentValueFormula! = !(!+ !)!!

• !isthefinalamount(orfuturevalue)• !istheprincipal(initialamountinvested)


• ! = !!

• !isthenumberofcompoundingperiodsperyear

• !isthenumberoftimesinterestispaid• ! = !"

• !isthenumberofyearsoftheinvestment

Example1:Melissawouldlike$10,000in3yearstopayfortuition.Shehasfoundaninvestmentthatyields

4.7%/acompoundedbiweekly.Howmuchmustshedepositnow?

Example2:Whatannualinterestratewillcauseaninvestmenttotripleinnineyearsifinterestiscompounded

weekly?Showanalgebraicsolutionandgiveyouranswerasapercenttothenearesthundredth.

HW:pg.70#5,6,10-13,23,25,pg.80#8


Learninggoal:Icanrelatethefuturevalueannuityformulatoageometricseries.Icancalculatethefuturevalueofanannuity.

AnnuitiesFutureValue

Howarethescenariosthesame? Howarethescenariosdifferent?

Anisaseriesofearningcompoundinterestandmade

atoverafixedperiodoftime.Aswithmostinvestmenttypes,annuitiesareoften

calculatedtofindfuturevalues.Inthiscourse(unlessotherwisestated)annuitiesareordinary,wherepaymentsare

madeattheendofintervals,andthecompoundingperiodscoincidewithpaymentperiods.

Example1:Joeplanstoinvest$1000attheendofeach6-monthperiodinanannuitythatearns6%/a

compoundedsemi-annuallyforthenext5years.Drawatimelinetorepresenthisinvestment.Whatwillbe

thefuturevalueofhisannuity?

Thefuturevalueofanannuityisthefinancialequivalentoftheannuityatmaturity.

TIMELINE

Scenario1:Georgeinvests$300intoanaccountthatpays

2%/ainterestcompoundedmonthly.

Scenario2:Georgedeposits$300eachmonthintoan

accountthatpays2%/ainterestcompounded

monthly.

Year

Payment$1000 $1000 $1000 $1000 $1000

5 4 3 2 1

$1000

0

Usingtheformulaforageometricseries,here’stheformulaforcalculatingthefuturevalueofordinaryannuities:

Example2:Jonplanstoinvest$1000attheendofeach6-monthperiodinanannuitythatearns4.8%/acompounded

semi-annuallyforthenext20years.Whatwillbethefuturevalueofhisannuity?

Example3:Ms.Marshwantstoretirein25yearswith$1,000,000.Shehasfoundaninvestmentthatyields

9.6%/acompoundedmonthly(WOW!).Howmuchshouldshedepositeachmonth?

HW:pg.152#6,8,10-13,18-20(AnsCorr8d:$82,826.66)

OrdinaryAnnuitiesFutureValue

!/!" = ![(! + !)! − !]!

• !isthefinalamount(orfuturevalue)• !isthepayment,theregularamountinvestedateachinterval


• ! = !!


• !isthenumberofregularpayments

• ! = ! ∗ !"#$%& !" !"#$%&! !"!"#$%& !"# !"#$• !isthenumberofyearsoftheinvestment


Learninggoal:Icanrelatethepresentvalueannuityformulatoageometricseries.Icancalculatethepresentvalueofanannuity.

AnnuitiesPresentValueRECALL:Anordinaryannuityisaseriesofequalpaymentsearningcompoundinterestandmadeatregularintervalsoverafixedperiodoftime.

Thepresentvalueofanannuityrepresentstheinitialamountthatmustbedepositedsothatconstant

paymentsmaybetakenoutoveranintervaloftime.AloanisanexampleofPresentvalue.

Example1:Jenmakes$1000paymentseveryyeartopaybackaloanat5%/acompoundedannually.Ittakes

her5yearstopaybacktheloan.Howmuchwastheloanfor?

TIMELINE

Year

Payment$1000 $1000 $1000 $1000 $1000

5 4 3 2 1

$1000

0

Usingtheformulaforageometricseries,here’stheformulaforcalculatingthepresentvalueofordinaryannuities:

Example2:Shirleyhastakenaloantopayforherfirstcar.Torepaytheloan,herbankischargingher$327.94permonthfor1yearwithinterestat9%peryear,compoundedmonthly.Whatistheactualcostofthecar

whenShirleypurchasedit?

Example3:Lenborrowed$200000fromthebanktopurchaseayacht.Ifthebankcharges6.6%/a

compoundedmonthly,hewilltake20yearstopayofftheloan.

a) Howmuchwilleachmonthlypaymentbe?

b)Howmuchinterestwillhehavepaidoverthetermoftheloan?

HW:pg.163#4,6-11,13,18

OrdinaryAnnuitiesPresentValue

!" = ![! − (!+ !)!!]!

• !"isthefinalpresentvalueneededtoinvesttoday• !isthepayment,theregularamountinvestedateachinterval


• ! = !!


• !isthenumberofregularpayments

• ! = ! ∗ !"#$%& !" !"#$%&! !"#$%&'( !"# !"#$• !isthenumberofyearsoftheinvestment


Learninggoal:Icanusetechnologytomakefinancialcalculations.

ApplicationsofTechnologyOftenthesecalculationswehavebeencomputingthroughouttheunitaredonethroughacomputer.Wewill

beusingaTVMsolveronugCloudtodosomecalculations.

TheTVMsolverdoesafewthingsdifferentlythantheconventionswehaveestablishedinourformulae.

• Nisthetotalnumberofpaymentperiods,orthenumberofconversionperiods.Thisisthe!weusedinourformulae.

• I%istheannualinterestrateasapercent,notadecimal.Thisis!fromourformulae,butasapercent.

• PVisthepresentvalue.Recallitcanrepresenttheprincipalinacompoundinterestsituation.

• PMTistheregularpaymentamount.Itisthe!fromourformulae.

• FVisthefuturevalue.Recallitcanrepresenttheamountinacompoundinterestsituation.

• P/Yisthenumberofpaymentperiodsperyear.Thisis!fromourformulae

• C/Yisthenumberofinterestconversionperiodsperyear.Forsimpleannuitiesandmostcompound

interestquestions(allthatwehaveseen),C/YisthesameasP/Y.

• Thedifferenttabsatthebottomallowyoutosolveforyoudesiredvalue.

• DONOTchangethevalueinthehighlightedcell.

Example1:$8000isinvestedfor10yearsat8%/a,compoundedmonthly.Findtheamount.

Example2:Ms.Marshwantstobuysa$550,000house(afterdownpayment)bymakingbiweeklypayments.

Hewillpayinterestat5.7%/a.Bylaw,Canadianmortgagescan’tbecompoundedmoreoftenthansemi-

annually.IfMs.Marshwouldlikea25yearmortgage,findhisbiweeklypayment.

HW:PickingtheCorrectFormulaWorksheet

unit 8 booklet - engage explore inspire · 2018. 9. 9. · recall: a general formula for...

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