Further'Mathematics'2017' Core:'RECURSION'AND'FINANCIAL'MODELLING' Loans,'investments'and'asset'values'

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1 Further'Mathematics'2017' Core:'RECURSION'AND'FINANCIAL'MODELLING' Loans,'investments'and'asset'values' Key knowledge! Amortisationofareducingbalanceloanorannuityandamortisationtables! Reducingbalanceloans,annuities,perpetuitiesandannuityinvestments.! Thedifferencebetweennominalandeffectiveinterestratesandtheuseofeffectiveinterestrates tocompareinvestmentreturnsandthecostofloanswheninterestispaidorcharged,forexample, $ daily,monthly,quarterly Key skills! Useatabletoinvestigateandanalyseonastep byastepbasistheamortisationofareducing balanceloanoranannuity,andinterpretamseatabletoinvestigateandanalyseonastep byastep basistheamortisationofareducingbalanceloanoranannuity,andinterpretamortisationtables! UsingaCAScalculator,solvepracticalproblemsassociatedwithcompoundinterestinvestments andloans,reducingbalanceloans,annuitiesandperpetuities,andannuityinvestments. $ $ $ $ Chapter$Sections Questions$to$be$completed$ 7.2$ReducingbalanceloansI 1,2,4,6,8,10,12,13,15,17 7.3$ReducingbalanceloansII 2,4,6,8,10,12,13,18 7.4$ReducingbalanceloansIII 2,4,6,8,10,12,14,15,20 7.5$Reducingbalanceandflatrateloancomparisons 2,4,5,9,10,13,17 7.6$Effectiveannualinterestrate 2,3,4,5,8,12, 7.7$Perpetuities 2,4,6,8,11,13,17 7.8$Annuityinvestments 2,4,6,8,10,11,14,16,18 Page1of35

2 Table'of'Contents$ & KEYKNOWLEDGE 1 KEYSKILLS 1 TABLE$OF$CONTENTS$ 2 7.2$REDUCING$BALANCE$LOANS$(PART$I)$ 3 INTRODUCTION$TO$ANNUITIES$ 3 ANNUITIES$ 3 WorkedExample1 3 THE$ANNUITIES$FORMULA$ 4 WorkedExample2 5 THE$FINANCIAL$SOLVER$ 5 WorkedExample3 5 WorkedExample $PERPETUITIES$ 24 WorkedExample20 24 WorkedExample21MultipleChoice 25 FINDING$V O $AND$R$ 25 WorkedExample22 25 WorkedExample $ANNUITY$INVESTMENTS$ 28 WorkedExample24 28 SUPERANNUATION$ 29 WorkedExample25 29 PLANNING$FOR$RETIREMENT$ 31 WorkedExample26 31 RETIREMENT$ 33 WorkedExample $REDUCING$BALANCE$LOANS$(PART$II)$ 8 NUMBER$OF$REPAYMENTS$ 8 WorkedExample5 8 WorkedExample6 9 WorkedExample7 9 EFFECTS$OF$CHANGING$THE$REPAYMENTS$ 10 WorkedExample8 10 INCREASING$THE$REPAYMENT$AMOUNT$ 12 WorkedExample $REDUCING$BALANCE$LOANS$(PART$III)$ 13 WorkedExample10 13 FREQUENCY$OF$REPAYMENTS$ 14 WorkedExample11 14 WorkedExample12 15 CHANGING$THE$RATE$ 16 WorkedExample13 16 INTERESTONLYLOANS $REDUCING$BALANCE$AND$FLAT$RATE$LOAN$ COMPARISONS$ 19 WorkedExample16 19 WorkedExample17 20 WorkedExample $EFFECTIVE$ANNUAL$INTEREST$RATE$ 22 Example19EffectiveInterestRate(borrowing) 22 Example:EffectiveInterestRate(Investing) 23 Page2of35

3 7.2'Reducing'Balance'Loans'(Part'I)' Introduction to Annuities Whenweinvestmoneywithafinancialinstitutionsuchasabankorcreditunion,theinstitutionpays us interest as it is using our money to lend to others. Conversely, when we borrow money, we are usingthefinancialinstitutionsmoneyandthuswearechargedinterest. Interest is usually charged monthly by financial institution and repayments are made regularly by borrowers.therepaymentsareusuallymorethaninterestchargedandthereforetheamountowing reduces.sincetheamountowingreducestheamountofinterestchargedreducesalso. Thetermsbelowareoftenusedwhentalkingaboutreducingbalanceloans: Principal,V 0 =amountborrowed($) Balance,V n =amountstillowing($) Term=lifeoftheloan=(years) Todischargealoan=topayoffaloan(whenV n =$0) Interest$ only loans exist where the repayments equal the interest added but the balance does not decrease.thisoptionisavailabletopeoplewhowishtomakethesmallestpaymentpossiblesuchas propertyinvestors. Annuities Anannuityisaninvestmentthathasregularandconstantpaymentsoverafixedperiodoftimee.g. Superannuation payments. Below is the recurrence relation that calculates the value of an annuity aftereachtimeperiod. Worked Example 1 Aloanof$100000istakenoutover15yearsatarateof7.5p.a.(interestdebitedmonthly)andisto bepaidbackmonthlywith$927instalments.completethetablebelowforthefirstfivepayments. n$$ V n $ Interest( d( V n+1 $$ 0$ $ $ 2$ 3$ 4$ 5$ Page3of35

4 WorkedExample1onCAScalculator! Enterthelabels n+1, V n, Pmt (forpayment), V n+1 Note:)You)can t)use)+)on)the)cas)so)spell)it)out)!! Nextenter1to5incolumnA,andthestartingvaluesfor V n =b1=100,000,pmt=c1=927incellsb1andc1 respectively. Incelld1inserttheequation = " Note:)where)0.625)is)r)the)interest)rate)per)period)(7.5/12)))! Incellb2enter =d1 Thisisjustusingthepreviousanswerasthestartingvalueof thenext. Nowfilldowntheequationsofcellsb2,c2andd2,downward foreachofcolumnsb,candd. The Annuities formula Theannuitiesformulacanbeusedtodeterminetheamountofmoneystillowingatanypointoftime duringthetermofareducingbalanceloan.whensomeoneborrowsmoneyfromafinancialinstitution thatpersoniscontractedtomakeregularpayments(annuities)inordertorepaytheamountborrowin theagreedtimeperiod. Note:ThatthereisacompoundingfactorRintheequationabovesoitisverysimilartocompound interestcalculations. Page4of35

5 Worked Example 2 Aloanof$50000istakenoutover20yearsatarateof6p.a.(interestdebitedmonthly)andistobe repaidwithmonthlyinstalmentsof$ findtheamountstillowingafter10years. WorkedExample2onCAScalculator Usethesolvefunction Entertheequation, - =,. / - 0 1(/- 1) / 1 when: V o =50000 d= R=1.005 N=120 Top$Tip:SavethisontheCASandjustchangethevalues.Don tknowhow?askyourteacher! The Financial Solver Note:TheFinancialSolveronCAScanbeusedinannuitiescalculationsinthesamewayitwasusedfor compoundinterestcalculations(sincethefinancialsolverworksoncompoundingcalculations). Worked Example 3 Robwantstoborrow$2800foranewsoundsystemat7.5p.a.,interestadjustedmonthly. a)! WhatwouldbeRob smonthlyrepaymentiftheloanisfullyrepaidin1½years? Page5of35

6 b)whatwouldbethetotalinterestcharged? WorkedExample3onCAScalculator UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():)= V 0 )(PV:)= ) Pmt:= V n )(FV:)= PpY:= CpY:= PlacethecursoronPmt:. PressENTERltosolve. TotalInterestpaid=totalrepayments amountborrowed Totalinterest=164.95x = =$ Worked Example 4 Joshborrows$12000forsomehomeofficeequipment.Heagreestorepaytheloanover4yearswith monthlyinstalmentsat7.8(adjustedmonthly).find: a)theinstalmentvalue. Calculatethevalueofn: n=4x12=48 UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i(:)= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt: PressENTERtosolve. Themonthlyrepaymentoverthe4yearperiodis$ Page6of35

7 b)theprincipalrepaidandinterestpaidduringthe: i)10 th repayment Tocalculatethisweneedtofindthedifferencebetweenthe9 th and10 th repayments.usingthecas financialsolver,thismeansweneedtofindtheamountowed(fv)afterthe9 th and10 th payments. UsingtheFinancial Solver Enterthefollowing: r)(i())=7.8 P)(PV):=12000 ) Pmt:= A291.82) (FV):=unknown PpY:=12 CpY:=12 With)n)(N:)=9 PlacethecursoronFV: With)n)(N:)=10 PlacethecursoronFV: PressENTERltosolve. PressENTERltosolve. Principalowingafter9 th repaymentis$ ,principalowingafter10 th repaymentis$ So,theprincipalrepaidduringthe10 th repaymentis$ a$ =$ If$291.83isthemonthlyrepaymentand$226.67istheprincipalrepaid,thentheinterestpaidis: $291.83A$226.67=$65.16 Answerinwords,Inthe10 th repayment,$226.67oftheprincipalisrepaidand$65.16interestispaid. ii)40 th repayment UsingtheFinancial Solver Enterthefollowing: r)(i()):= P)(PV):= ) Pmt:= ) (FV):= PpY:= CpY:= With)n)(N:)=39 PlacethecursoronFV: With)n)(N:)=40 PlacethecursoronFV: PressENTERltosolve. PressENTERltosolve. Principalowingafter39 th repaymentis$ ,principalowingafter40 th repaymentis$ So,theprincipalrepaidduringthe40 th repaymentis$ a$ =$ So,if$291.83isthemonthlyrepaymentand$275.30istheprincipalrepaidthen $291.83A$275.30=$16.53 Inwords,Inthe40 th repayment,$275.30oftheprincipalisrepaidand$16.53interestispaid. Thismakessense!!Atthe40 th repaymentthereislessmoneyowedsothereforethereislessinterest topay. Page7of35

8 7.3'Reducing'Balance'Loans'(Part'II)' Number of repayments Sometimesweknowhowmuchmoneyneedstobeborrowedaswellastheamountofmoneythatcan berepaideachmonth. Apersonthenwouldwanttoknowhowlongtheloanneedstobe,thatis,todeterminethenumberof repayments,n,required. Worked Example 5 Areducingbalanceloanof$60000istoberepaidwithmonthlyinstalmentsof$483.36ataninterest rateof7.5p.a.(debitedmonthly).find:a)thenumberofmonthlyrepayments(and,hence,theterm oftheloaninmoremeaningfulunits)neededtorepaytheloaninfull UsingtheFinancialSolver Enterthefollowing: n)(n:)=unknown r)(i():)= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronN,PressENTERtosolve. Answer:n=240monthsis240/12=20years, Hence,thetermoftheloanneedstobe20years b)thetotalinterestcharged :6;<;=60 = >;?7@A;:6=0 0C<D:ED?780>;?7DF$ Wemaywanttofindthetimeforonlypartoftheloanterm.Theprocedurethatisfollowedisthe sameasworkedexample5;however,v n iszeroonlyifwearecalculatingthetimetorepaytheloanin full.otherwiseweshouldconsidertheamountstillowingatthattime.thenextexampleshowsthis. Page8of35

9 Worked Example 6 Sometimeago,Petraborrowed$14000tobuyacar.Interestonthisreducingbalanceloanhasbeen chargedat9.2p.a.(adjustedmonthly)andshehasbeenpaying$446.50eachmonthtoservicethe loan.currentlyshestillowes$ howlongagodidpetraborrowthemoney? UsingtheFinancialSolver Enterthefollowing: n)(n:)=unknown r)(i():= P)(PV:)= ) Pmt*:= ) FV*:= PpY:= CpY:= PlacethecursoronN,PressENTERtosolve. *Note:$in$this$case$both$the$payment$and$the$Final$Value$are$negative$(])$because$they$are$monies$you$owe.$ Answerthequestion: Inthesituationscoveredsofar,wehaveconsideredcalculatingonlythetimefromthestartoftheloan toalaterdate(includingrepaymentinfull).itdoesnotmatterwhatperiodoftheloanisconsidered; wecanstillusethefinancialsolver.inusingcas,wecandefinev n astheamountowingattheendof thetimeperiodandv 0 astheamountowingasthestartofthetimeperiod. Worked Example 7 A loan of $ is being repaid by monthly instalments of $ with interest being charged at 11.5p.a.(debitedmonthly).Currently,theamountowingis$ Howmuchlongerwillittake to: a)reducetheamountoutstandingto$ UsingtheFinancialSolver Enterthefollowing: n)(n:)=unknown r)(i():= P*)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronN,PressENTERtosolve. *Note:InthiscasethePrincipalValueis asthisisthecurrentamount. Note:NisinmonthsbecausePpYisinmonths. Answerthequestion: Here,thetimeis18monthswhichweknowis1½years. Ifthevaluefornwas32.Thenwewoulddivideby12,giving32/12= Inthiscasewewouldhave2yearsand0.667x12months=8months.So,theanswerwouldbe2years and8months. Page9of35

10 b)repaytheloaninfull? UsingtheFinancialSolver Enterthefollowing: n)(n:)=unknown r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronN,PressENTERtosolve. Answerthequestion: Effects of changing the repayments Asmostloansaretakingoveralongtime,suchasmortgages,thefinancialsituationoftheborroweris likelytochangee.g.theymaygetapayincreaseanddecidetoincreasetheirrepaymentsortheymay havefinancialdifficultiesandseektodecreasetheirrepayments. Inthissectionwewilllookattheeffectsofchangingrepaymentsonthetermoftheloanandthetotal interestpaid. Worked Example 8 Areducingbalanceloanof$16000hasatermof5years.Itistoberepaidbymonthlyinstralmentsata rateof8.4p.a.(debitedmonthly). a)findtherepaymentvalue. Calculatethevalueofn: n=5x12=60 UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt,PressENTERtosolve. Answerthequestion: Page10of35

11 b)! Whatwillbethetermoftheloaniftherepaymentisincreasedto$393.62? UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronN,PressENTERtosolve. Answerthequestion: c)calculatethetotalinterestpaidforrepaymentsof$ d)byhowmuchdoestheinterestfigureincdifferfromthatpaidfortheoriginaloffer? Page11of35

12 Increasing the repayment amount Ifaborrowerincreasestheamountofeachrepaymentandalltheothervariablesremainthesame,the termoftheloanisreduced.converselyifadecreaseintherepaymentsoccursthetermoftheloanis increased. Worked Example 9 Bradborrowed$22000tostartabusinessandagreedtorepaytheloanover10yearswithquarterly instalments of $ and interest debited at 7.4 p.a. However, after 6 years of the loan Brad decidedtoincreasetherepaymentvalueto$ find: a)theactualtermoftheloan Calculatethevalueofn: n=6x4=24 UsingtheFinancialSolver* Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronFV,PressENTERtosolve. Nowweneedtofindthenvaluetorepaytheloanin full,inotherwordsreduce$ to$0. Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronN,PressENTERtosolve. b)thetotalinterestpaid c)theinterestsavingachievedbyincreasingtherepaymentvalue. Page12of35

13 7.4'Reducing'Balance'Loans'(Part'III)' When paying off a loan it is often wise to follow its progress through the life of the loan. The amortisation$oftheloancanbetrackedonastepabyastepbasisbyfollowingthepaymentsmade,the interest and reduction in the principal. Amortisation$ is defined as the regular decrease in value (depreciation)ofanassetorthepayingoffadebtovertimethroughregularrepayments. Worked Example 10 Sharyntakesoutaloanof$5500topayforsolarheatingforherpool.Theloanistobepaidinfullover 3yearswithquarterlypaymentsat6p.a. a)calculatethequarterlypaymentrequired. Calculatethevalueofn: n=3yearsx4quarters=12 UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt: PressENTERtosolve. b)completeanamortisationtablefortheloanwiththefollowingheadings. Payment$$ Principal$outstanding$($)$ Interest$due$($)$ Payment$($)$ Loan$outstanding$($)$ 0$ $ 2$ 3$ 4$ 5$ 6$ 7$ 8$ 9$ 10$ 11$ 12$ Page13of35

14 Frequency of repayments Here,wewillfocusontheeffectmaking$more$frequent$repaymentshasontheterm$of$the$loan,and onthetotal$amount$of$interest$charged. i.e.thefrequencyoftherepaymentswillchange,buttheactualamountofmoneypaid(theoutlay)will not e.g.a$3000quarterlyrepaymentwillbecomparedtoa$1000monthlyrepayment. Worked Example 11 Tessawantstobuyadressshop.Sheborrows$15000at8.5p.a.(debitedpriortoeachrepayment) ofthereducingbalance.shecanaffordquarterlyrepaymentsof$928.45andthiswillpaytheloanin fullinexactly5years.oneathirdofthequarterlyrepaymentsgivestheequivalentmonthlyrepayment of$ theequivalentfortnightlyrepaymentis$ find: i)thetermoftheloan UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt: PressENTERtosolve. ii)theamountstillowingpriortothelastpaymentiftessamaderepayments: a)monthly UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) b)fortnightly UsingtheFinancialSolver Enterthefollowing: FV:= ) PpY:= ) CpY:= ) n)(n:)= FV:= ) r)(i():= PpY:= ) P)(PV:)= ) CpY:= ) Pmt:= ) Whiletheoutlaywillbethesame,thetermoftheloanwillbereducedwhenrepaymentsaremade moreoften.wecandeterminethesavingsfortheloan.insuchacase,thefinal(partial)repaymentis consideredseparatelyastheinterestchargedislessincomparisontotherestoftherepayments. Page14of35

15 Worked Example 12 InWorkedexample11,Tessa s$15000loanat8.5p.a.gavethefollowingthreescenarios: 1.! quarterlyrepaymentsof$928.45for5years 2.! monthlyrepaymentsof$309.48for59monthswith$179.27stilloutstanding 3.! fortnightlyrepaymentsof$142.84for128fortnightswith$120.64stillowing. ComparethetotalinterestpaidbyTessaifsherepaidherloan: a)quarterly b)monthly c)fortnightly Savings increase when the frequency of repayments increase, this occurs as the amount owed is reducedmorefrequentlyandsotheamountofinterestchargedisslightlyless. Page15of35

16 Changing the rate Overthetermofaloan,theinterestrateislikelytochange.TheReserveBankofAustralia,themain monetaryauthorityofthefederalgovernment,istheoverallguidinginfluenceonmonetaryfactorsin theaustralianeconomy,itindirectlycontrolstheinterestrates,financialinstitutionscharge. Differentfinancialinstitutionsmayhavedifferentinterestratesandthereisinterestratevariationin aninstitutiondependingonthetypeofloan.smallerloans,suchaspersonalloans,havehigherrates comparedtohomeloans. Inthissection,theeffectofchanginginterestratesonthetermoftheloanandthetotalinterestpaid willbereviewed. Worked Example 13 Areducingbalanceloanof$18000hasbeentakenoutover5yearsat8p.a.(adjustedmonthly)with monthlyrepaymentsof$ a)whatisthetotalinterestpaid? b)if,instead,theratewas9p.a.(adjustedmonthly)andtherepaymentsremainedthesame,what wouldbe: i)thetermoftheloan UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= ii)thetotalamountofinterestpaid? UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= Page16of35

17 WorkedExample14 NatsukoandHymietookoutaloanforhomerenovations.Theloanof$42000wasduetorunfor10 years and attract interest at 7 p.a., debited quarterly on the outstanding balance. Repayments of $ weremadeeachquarter.After4yearstheratechangedto8p.a.(debitedquarterly).The repaymentvaluedidnotchange. a)findtheamountoutstandingwhentheratechanged. UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= b)findtheactualtermoftheloan. UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= c)comparethetotalinterestpaidtowhatitwouldhavebeeniftheratehadremainedat7p.a.for the10years. $ $ Page17of35

18 Interest only loans Interest only loans are loans where the borrower makes only the minimum repayment equal to the interestequaltotheinterestchargedontheloan.astheprincipalandamountowingisthesamefor theperiodofthisloan,wecanusethesimpleinterestformulaorcas.whenusingthefinancialsolver thepresentvalue(pv)andfuturevalue(fv)arethesame(withthefvnegativetoindicateitisowedto thebank. Note:Futurevalueisnegativetoindicatethemoneyisowedtothebank. This type of loan is used by investors of shares and/or property or people experiencing financial difficultiesandseekshortatermrelief. WorkedExample15 Jadewishestoborrow$40000toinvestinshares.Sheusesaninterestonlyloantominimiseher repaymentandhopestoraiseacapitalgainwhenshesellsthesharesatahighervalue.thetermof theloanis6.9p.a.compoundedmonthlywithmonthlyrepaymentsequaltotheinterestcharged. a)calculatethemonthlyinterestaonlyrepayment. b)if,in3years,shesellsthesharesfor$50000,calculatetheprofitshewouldmakeonthisinvestment strategy. Page18of35

19 7.5'Reducing'Balance'and'Flat'Rate'Loan'comparisons' Inreducingbalanceloans,interestiscalculatedonthecurrentbalance.Sincethebalance$reduces,$the$ amount$of$interest$charged$also$reduces.$incontrast,flat$rate$loans,chargeafixedamountofinterest asapercentageoftheoriginalamountborrowed.thisiscalculatedatthestartofaloanandaddedto theamountborrowed.sinceitisaflatratebasedonafixedamount,thesimpleinterestformulais usedtocalculatetheinterest: Totalsimpleinterestformula: Worked Example 16 Aloanof$12000istakenoutover5yearsat12p.a.Find: i)foraflatrateloan a)themonthlyrepayment b)thetotalamountofinterestpaid ii)forareducingbalanceloan. a)! themonthlyrepayment UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt: PressENTERtosolve. G = 0 H IJ- K.. b)thetotalamountofinterestpaidifthemoneyisborrowedon: Inthisexample,thedifferencebetweenthetwoloantypesissignificant.Forthereducingbalanceloan, each month $53.07 less is repaid and overall $ less interest is paid. Choosing a reducing balanceloanratherthanaflatrateloanresultsinasmaller$repayment$valueor$a$shorter$termandin bothcasesaninterestsaving.nowletusconsiderwhatflatrateofinterestisequivalenttotheratefor areducingbalanceloan. Page19of35

20 Worked Example 17 A reducing balance loan of $ is repaid over 8 years with monthly instalments and interest chargedat9p.a.(debitedmonthly).find: a)! therepaymentvalue UsingtheFinancialSolver Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt: PressENTERtosolve. b)thetotalamountofinterestpaid c)theequivalentflatrateofinterestforaloaninwhichallothervariablearethesame. Worked Example 18 Aloanof$76000isrepaidover20yearsbyquarterlyinstalmentsof$ andinterestischarged quarterlyat10p.a.oftheoutstandingbalance.find: a)thetotalamountofinterestpaidis: TotalInterest=PmtxN PV,sofirstcalculaten. Calculatethevalueofn: n=6x4=24 Page20of35

21 b)theamountwhichcanbeborrowedonaflatrateloaninwhichallothervariablesarethesameas above c)thedifferenceintheamountborrowedbetweenthetwotypesofloans Page21of35

22 7.6'Effective'Annual'Interest'Rate'' Wehaveconsideredpayingoffaloanatasetinterestrate,however,wehavefoundtheamountof interestpaidwouldvarywithdifferentcompoundingterms(daily,weekly,monthlyetc).theeffective$ annual$interest$rate$isusedtocomparetheannualinterestbetweenloanswithdifferingcompounding terms. L MNN = 1 + L 100 P wherelisthenominalinterestrate,andpisthenumberofcompoundingperiodsperyear. Example Aloanof$100at10p.a.compoundingquarterlyover2years,theeffectiveannualinterestrate: L MNN = 1 + (000000) 100 (0000) (0000) Thismeansthattheeffectiveannualinterestrateisactually10.38andnot10. Thecomparisonbetweenthetwocanbeshowninthefollowingtable. Amount$owing$($)$ Period$, -UK = 1 + J, K.. - usingl = K. = 2.5 V 1$ 1 + Z.[ 2$ 1 + Z.[ K.. Annual$effective$rate$calculation$($)$, -UK = 1 + J K.., - usingl = $ K = $ 1 + Z.[ K $(Year$1)$ 1 + Z.[ K..^_ K.. K $ 1 + Z.[ K.. 6$ 1 + Z.[ K $ 1 + Z.[ K $(Year$2)$ 1 + Z.[ K..^_ K.. K Example 19 Effective Interest Rate (borrowing) Jasondecidestoborrowmoneyforaholiday.Hehastwooptions i.! Apersonalloancompoundingquarterlyatanominalrateof8.75p.a. ii.! Apersonalloancompoundingfortnightlyatanominalrateof8.7p.a. calculate the effective annual rate of interest (correct to 2 decimal places) for both and determine whichisthebetterdeal.) Usethesolvefunction Entertheequation Labb = 1 + L P when:! r=8.75! n=4(quarterly) andwhen:! r=8.7! n=26(fortnightly) Note:Alternatively,youmayberequiredtofindthenominalinterestraterfromtheeffectiverate.In thiscasesolveforr(thenominalinterestrate)statingthel MNN andthenvalue. Page22of35

23 Example:Effective Interest Rate (Investing) Liana wants to invest her money, she has done some research and found the best offers from four differentfinancialinstitutions.theyare: Bank$1:8.60p.a.compoundeddaily; Bank$2:8.70p.a.compoundedfortnightly; Bank$3:8.65p.a.compoundedmonthly; Bank$4:8.75p.a.compoundedquarterly WhichbankshouldLianachooseifshewantstoearn$the$most$interest? Bank$1:$8.60$p.a.$compounded$daily$ r=8.60 n= L MNN = = 8.98 effective$interest$is:$ ^c[ 8.98$p.a.$ Bank$3:$8.65$p.a.$compounded$monthly) r=8.65 n=12 L MNN = = 9.00 effective$interest$is:$ KZ 9.00$p.a.) Bank$2:$8.70$p.a.$compounded$fortnightly$ r=8.7 n=26 L MNN = = 9.07 effective$interest$is:$ Zc 9.07$p.a.$ Bank$4:$8.75$p.a.$compounded$quarterly) r=8.75 n=4 L MNN = = 9.04 effective$interest$is:$ V 9.04$p.a.) LianashouldchooseBank$2asitpaysthe$highest$effectiveinterestrateof$9.07andwilltherefore earnmoreinterest. Page23of35

24 7.7'Perpetuities'' There a variety of ways to invest money, one is a managed$ fund, whereby, you invest an initial principalandhopethefundmanagerscaninvestthemoneytogainapositivereturn.insomeyear s moneymaybelostandthepercentagereturnscanvaryfromyeartoyear. Another option is to regularly contribute to the fund after the initial investment, thus increasing the principalandtheinterestearned.thisisanexampleofafirstaorderrecurrencerelation. Worked Example 20 Jonathaninvested$5000inamanagedfundthatwillearnanaverageof8p.a.overa2yearperiod with interested calculated monthly. If Johnathan contributes $100 at the start of the second, third, fourthandfifthmonths,completethetabletofindthevalueofhisinvestmentattheendofthefifth month. Time$period$ Principal$($)$ Interest$earned$($)$ Balance$($)$ 0$ 5000 A 1$ 2$ 3$ 4$ 5$ Aperpetuityisanannuitywhereapermanentlyinvestedsumofmoneyprovidesregularrepayments thatcontinueforever.scholarshipsorgrantsofferedtostudentsatuniversityareprovidedbyfunds knownasperpetuities.wealthypeoplewhowishtosupportaworthycausesetupperpetuities. The funds last for an indefinite period of time as the amount paid out is the same as the interest earnedontheinitiallumpsumdeposited.thebalanceoftheamountinvesteddoesnotchangeandis thesameforanindefiniteperiod. Note:Theperiodoftheregularrepaymentsmustbethesameastheperiodofthegiveninterestrate. FinanceSolvercanbeused.Astheprincipaldoesnotchange,thePV(negativecashflow)andtheFV (positivecashflow)areenteredasthesameamountbutwithoppositesigns. Page24of35

25 Worked Example 21 Multiple Choice Robertwishestousepartofhiswealthtosetupascholarshipfundtohelpyoungstudentsfromhis townfurthertheireducationatuniversity.robertinvests$200000inabondthatoffersalongaterm guaranteedinterestrateof4p.a.iftheinterestiscalculatedonceayear,thentheannualamount providedasscholarshipwillbe: A$ B$288000$ C$666.67$ D$8000$ E$4000$ Finding V o and r If) the) frequency) of) the) payments) each) year) is) not) the) same) as) the) compounding) period) of) the) given) interest)rate,)then)finance)solver)is)to)be)used)with)different(values(for(ppy(and(cpy.) Note: The principal must be known to use Finance Solver. Finance Solver gives the interest rate per annum Worked Example 22 ARotaryClubhas$100000tosetupaperpetuityasagrantforthelocaljuniorsportingclubs.Theclub investsinbondsthatreturn5.2p.a.compoundingannually. a)findtheamountoftheannualgrant. Alternatively,usetheFinanceSolver UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt:PressENTERtosolve. Page25of35

26 b) What interest rate (compounding annually) would be required if the perpetuity is to provide $6000 each year? Alternatively,usetheFinanceSolver UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronI():PressENTERtosolve. TheRotaryClubwantstoinvestigateotherpossiblearrangementsforthestructureofthegrant. c)howmuchextrawouldtheannualgrantamounttoiftheoriginalinterestratewascompoundedmonthly? UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt:PressENTERtosolve. d)whatinterestrate(compoundedmonthly)wouldberequiredtoprovide4equalpaymentsof$1500every3 months?giveyouranswercorrectto2decimalplaces. UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronI():,PressENTERtosolve. Page26of35

27 Worked Example 23 AbenefactorofacollegehasbeenapproachedtoprovideaYear7scholarshipof$1000perterm.Hecangeta financialinstitutiontoofferalongaterminterestrateof8perannum.whatistheprincipalthatneedstobe invested? Page27of35

28 7.8'Annuity'Investments'' A savings plan, like a Christmas Club account, where an initial principal is invested as well asregular depositsaremade.theinterestearnediscalculatedregularlyonthebalanceoftheinvestment,which increase with each regular deposit (annuity). This is similar to a reducing balance loan except the principalisgrowinganditisyourownmoney,notaloan. Anannuityinvestmentisaninvestmentthathasregulardepositsmadeoveraperiodoftime. Worked Example 24 Aninitialdepositof$1000wasmadeonaninvestmenttakenoutover5yearsatarateof5.04p.a. (interest calculated monthly), and an additional deposit of $100 is made each month. Complete the tablebelowforthefirstfivedepositsandcalculatehowmuchinteresthadbeenearnedoverthistime. n( $ V n( $ Interest$ d$ $ V n+1( $ 0$ $1000 1$ 2$ 3$ 4$ 5$ $ $ Page28of35

29 Superannuation InAustralia,wemustprovideenoughmoneytoliveonwhenweretire,thisiscalledSuperannuation. Whenwebeginworking,ouremployersarerequiredbylawtocontributemoneytoaSuperannuation fund, employees can also top up their superannuation account by contributing money themselves. Themoneybuildsupovermanyyearsandcanbewithdrawnwhenapersonreachesretirementage. Thefundthencanbeplacedintoanannuityorperpetuitythatpaysfortheretiree slivingexpenses andlifestyle. It is important to determine how much money you will need to retire on, to have the lifestyle you wouldlike,i.e.travel,payoffyourhouseetc.(seeingafinancialadvisorcanhelp.) ThemoneyinaSuperannuationfundisinvestedbysuperannuationfundmanagers(inshares,property etc).theperformance(valueofthefundgoesupordown)ofthefundsvariesyeartoyear. Forsimplicityindeterminingthereturnofasuperannuationfund,itwillbeassumedthattheinterest rateremainsthesameandoutsideinfluencessuchastaxationandinflation,willnotbeconsidered. Themoneythatbuildsupintheseannuitiesinvestmentscanbecalculatedusingtheannuitiesformula, excepttheamountv n growswiththeadditionofregularpaymentsd. Finance$Solver$can$also$be$used$in$a$similar$way$to$reducing$balance$loans,$with$the$difference$of$the$ cash$flows$are$reversed$(opposite$sign).$ Worked Example 25 Helen currently has $2000 in a savings account that is averaging an interest rate of 8 p.a. compounding annually. She wants to calculate the amount that she will receive in 5 years when she planstogoonanoverseastrip. a) If she deposits $6000 each year find (correct to the nearest $1000) the amount available for her overseastrip. $ Page29of35

30 Alternatively,usetheFinanceSolver UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronFV:PressENTERtosolve. b) If she places her $2000 and increases her deposits to $7000 each year into a different savings accountthatcanoffer9p.a.compoundingannually,find(correcttothenearest$1000)theamount availableforheroverseastrip. Alternatively,$use$the$Finance$Solver$ UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronFV:PressENTERtosolve. c) Calculate the extra amount saved by investing $7000 each year at 9 p.a. compared with $6000 eachyearat8p.a. Page30of35

31 Planning for retirement Whendoyouwanttoretire?Howmuchmoneydoyouneed?Areyouplanningtotravel?Asthelife expectancyforaustralian sisexpectedtobemorethan80yearsold,peoplewillneedtoplanfor20 plusyearsofretirement.acommontargetistohave60to65ofyourprearetirementincome.so,if youearn$60000p.a.now,yourretirementincomewouldbe$36000p.a.(intoday sdollars). Planning for retirement should be regularly revisited, to ensure enough money is being invested. Financialplannerscanassistinthisplanning. Worked Example 26 Andrewisaged45andisplanningtoretireat65yearsofage.Heestimatesthatheneeds$480000to provideforhisretirement.hiscurrentsuperannuationfundhasabalanceof$60000andisdelivering 7p.a.compoundedmonthly. a)findthemonthlycontributionsneededtomeettheretirementlumpsumtarget. Alternatively,usetheFinanceSolver UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt:PressENTERtosolve. b)ifinthefinaltenyearsbeforeretirement,andrewdoubleshismonthlycontributioncalculatedfrom (a),findthenewlumpsumamountavailableforretirement. Page31of35

32 Page32of35 Alternatively,usetheFinanceSolver.Weneedtosplititintothefirst10yearsandthenthelast10 years UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronFV:PressENTERtosolve. UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronFV:PressENTERtosolve. c)howmuchextracouldandrewexpectiftheinterestratefrompartb$isincreasedto9p.a.(from thefinal10years)compoundedmonthly?roundtheanswercorrecttothenearest$1000. Alternatively,usetheFinanceSolver UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronFV:PressENTERtosolve.

33 Retirement Onceapersonretires,theycanreceivealumpsumfromtheirSuperannuationfund,thislumpsumcan betransferredorrolledovertoasuitableannuity.thisannuitywillprovidearegularincometoliveon. Therearetwooptions: 1.! Perpetuities anannuitythatprovidesregularpaymentsforever.thebenefitsofthisisitwill provide income to the retiree no matter how long they live and it can be willed to relatives, whowillcollectthesameannuityindefinitely. 2.! Annuity reducing balance. The fund manager borrows the money and pays the retiree a regular income for a specified period of time. The disadvantage is if the retiree out lives the termofthereducingbalanceannuity,themoneywillrunout. Worked Example 27 Jarrod is aged 50 and is planning to retire at 55. His annual salary is $ and his employer contributionsare9ofhisgrossmonthlyincome.jarrodalsocontributesafurther$500amonthasa salary sacrifice(that is, he pays $500 from his salary to the superannuation fund. The superfund has been returning an interest rate of 7.2 p.a. compounded monthly and his current balance in the superfundis$ a)calculatejarrod stotalmonthlycontributionstothesuperannuationfund. b)calculatethelumpsumthathecanreceiveforhisplannedretirementatage55. Alternatively,usetheFinanceSolver UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronFV:PressENTERtosolve. Page33of35

34 Page34of35 Jarrodhastwooptionsforsettingupanannuitytoprovidearegularincomeafterheretiresat55. 1.! Aperpetuitythatoffersmonthlypaymentsat8p.a.compoundedmonthly. 2.! Areducingbalanceannuity,alsopaidmonthlyat8p.a.,compoundedmonthly. c)calculatethemonthlyannuityusingoptions1.expresstheannualsalaryfromthisoptionasa percentageofhiscurrentsalary. d)calculatethemonthlyannuityusingoption2ifthefundneedstolastfor25years.expressthe annualsalaryfromthisoptionasapercentageofhiscurrentsalary. Alternatively,usetheFinanceSolver UsingtheFinancialSolver,Enterthefollowing: n)(n:)= r)(i():= P)(PV:)= ) Pmt:= ) FV:= PpY:= CpY:= PlacethecursoronPmt:PressENTERtosolve.

35 7.7DifferencebetweenanAnnuityandPerpetuity Example:Johnhastwooptionstochooseforinvestinghissuperannuationfundof$100000at10pacompoundedyearlywitharegularpayment. Time' Period' (Year)' Principal' ($)' Interest'($)'! = # 1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Balance'after'payment'$15000'per'year' 100 ' ( = # +! $15000 Time' Period' (Year)' Principal' ($)' Interest'($)'! = # = $ $ $ !$15000=$ $ $ = $ $ $ !$15000=$ $ $ = $ $ $ !$15000=$ $ $ = $ $ $ !$15000=$ $ $ = $ $ $ !$15000=$ $ $ = $ $ $ !$15000=$ $ $ = $ $ $ !$15000=$ $ $ Anannuityisaninvestmentthathasregularandconstantpayments,where the principal is gradually decreased over the period of time. Therefore, the payment will eventually run out when the interest earned is less than the payment. An' annuity' investment' will' occur' when' the' regular' payment' is' more' than' the' interest'earned. Balance'after'payment'$10000'per'year' $ 100 ' ( = # +! $10000 = $10000 $ $10000 $10000=$ = $10000 $ $10000 $10000=$ = $10000 $ $10000 $10000=$ = $10000 $ $10000 $10000=$ = $10000 $ $10000 $10000=$ = $10000 $ $10000 $10000=$ = $10000 $ $10000 $10000=$ A perpetuity is an annuity where a permanently invested sum of money provides regular repayments that continue forever. The funds last for an indefinite period of time as the amount paid out is equal to the interest earnedontheinitiallumpsumdeposited(principal). The balance of the amount invested does not change and is the same for an indefiniteperiod. A' perpetuity' investment' will'occur'when'the'regular' payment' is' equal' to' the' interest'earned. DangST!RPublications Page35of35