unit 8 booklet - engage explore inspire · 2018. 9. 9. · recall: a general formula for...
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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)
MCR3U–Unit8:FinancialApplications–Lesson2 Date:___________
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.
Year StartingBalance
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.
Year StartingBalance
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
MCR3U–Unit8:FinancialApplications–Lesson3 Date:___________
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)
• !istheinterestrateasadecimal
• ! = !!
• !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
MCR3U–Unit8:FinancialApplications–Lesson4 Date:___________
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:$82,826.66)
OrdinaryAnnuitiesFutureValue
!/!" = ![(! + !)! − !]!
• !isthefinalamount(orfuturevalue)• !isthepayment,theregularamountinvestedateachinterval
• !istheinterestrateasadecimal
• ! = !!
• !isthenumberofcompoundingperiodsperyear
• !isthenumberofregularpayments
• ! = ! ∗ !"#$%& !" !"#$%&! !"!"#$%& !"# !"#$• !isthenumberofyearsoftheinvestment
MCR3U–Unit8:FinancialApplications–Lesson5 Date:___________
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$200000fromthebanktopurchaseayacht.Ifthebankcharges6.6%/a
compoundedmonthly,hewilltake20yearstopayofftheloan.
a) Howmuchwilleachmonthlypaymentbe?
b)Howmuchinterestwillhehavepaidoverthetermoftheloan?
HW:pg.163#4,6-11,13,18
OrdinaryAnnuitiesPresentValue
!" = ![! − (!+ !)!!]!
• !"isthefinalpresentvalueneededtoinvesttoday• !isthepayment,theregularamountinvestedateachinterval
• !istheinterestrateasadecimal
• ! = !!
• !isthenumberofcompoundingperiodsperyear
• !isthenumberofregularpayments
• ! = ! ∗ !"#$%& !" !"#$%&! !"#$%&'( !"# !"#$• !isthenumberofyearsoftheinvestment
MCR3U–Unit8:FinancialApplications–Lesson6 Date:___________
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