• personal loans and credit cards. • hire-purchase...

119

Unit 12.2 Managing Money 2Topic 5: Consumer credit

The focus of Unit 12.2 is mathematics relating to money, and the application of mathematical skills in practical areas such as interest, credit, inflation, loans and investment. In Topic 5 we look at credit and its different forms:• Personalloansandcreditcards.• Hire-purchaseagreements.• Reducing-balanceloans.• Theannuitiesformula.

Personal loansA personal loan is money borrowed for the purchase of goods or services by the borrower.

• Personal loans can be obtained from banks or other financial institutions. To obtain a loan from one of these institutions the borrower will need to demonstrate that they can repay the loan. The amount borrowed and the rate of interest charged will depend on whether the loan is secured or unsecured – higher rates of interest are usually payable on unsecured loans. A secured loan is a loan where the borrower has other property that the lender can access if the borrower is unable to pay back the loan.

• Whenapersonalloanisgiven,theamountborrowed,theinterestrate,thetermoftheloanand the repayment amount will all be specified.

• Thefinancialinstitutionwilloftenchargetheborrowerfortakingoutaloan:usuallyaloanestablishment fee and a monthly fee.

• Loansthatareforaveryshort term will be charged simple interest and the total amount owing will be repaid at the end of the term. A bridging loan,whichcoversthetimebetweenpurchasinganewpropertyandsellinganexistingproperty,isanexampleofashort-term loan.

• Credit card loans are short-term loans where the interest payable is calculated each month using simple interest principles.

• Longer-term loanssuchashousingloans,areusuallygivenasreducing-balance loans where the amount of interest charged reduces as the balance of the loan reduces.

• Thereareotherbusinessesthatlendmoneyforpersonalusewhereevidenceoftheabilityto repay the loan is not as strict. Because these loans are at greater risk of not being repaid (calleddefaultingontheloan)asthosegivenbybanks,thelenderusuallychargesahigherrateofinterest.Theselendersbasetheirloansonasimpleinterestrate,whichisoftennotspecified,overthetermoftheloanandusuallyrequireregularrepayments.Hire-purchase agreements are examples of this type of loan.

Short-term loans involving some simple interest calculations have been covered elsewhere in this Unit – see Topics 1 and 2. Calculations of the interest paid on credit cards and hire-purchase agreements is covered below

Credit cardsCreditcanbeprovidedbyafinancialinstitutionissuingacreditcard,orbyaretailerofferinga store loyalty card.

19_SAV_GM12_78706_TXT_5pp.indd 119 31/08/13 2:50 AM

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120 Unit 12.2 Managing Money 2

© Oxford University Press www.oup.com.au

Banks and other financial institutions will offer a credit card to a customer if the customer has demonstrated that they have a stable income and that they have a history of being able to manage their money. This is called their credit history.

Credit cards usually have a limit (maximum amount that they can accumulate in credit each month) and this is different for different individuals.

The credit card owner is sent an account each month and all or part of the credit that has been accumulated can be repaid within a specified time. There is a minimum amount that is payable and any amount that is not repaid will accumulate interest. This is usually at a higher rate than the rate for other personal loans.

It is easy to make a purchase with a credit card but it is also easy to overspend and then have to paylargeamountsininterest.Interestischargedinvariouswayswithcreditcards:

• Cashisusuallychargedinterestfromthedaythecashisborrowed.

• Somecardshaveaninterest-freeperiodbutchargeanannualfeefortheuseofthecard.If the amount that is owed is not paid within the interest-free period then the cardholder is charged interest from the date of the purchase. Interest-free periods are different for different credit cards and can be up to 55 days.

• Somecardshavenointerest-freeperiodbutdonotchargeanannualfee.

• Aminimumrepaymentamount(about3%)isseteachmonth.Ifthisistheonlyamountthatisrepaidthentheamountowedcancontinuetoincreaseandinterestratesof16–20%will be charged on the balance.

Store loyalty cards are organised along the same lines as credit cards but can only be used in the store that issues the card. There are often discounts and other incentives attached to store loyalty cards that are used to encourage customers to shop only in that store. Customers cannot usually access cash on store loyalty cards.

Example AQ. Michaelhasa‘nointerest-freeperiod’creditcardthatcharges15.6%interest.He

purchasedajacketforK65.00on15AprilandtookoutacashadvanceofK50.00on20 April.Hiscreditcardstatementarrivedon25April.HowmuchinterestwouldbeincludedinMichael’smonthlystatement?

A. MichaelwilloweinterestonK65.00for25–15+1days=11daysandinterestontheK50.00for25–20+1days=6days.Note:Thenumbersofdaysincludeboththepurchase date and the end date.

Interest= + 0.4365 15.6

11365

100

50 15.66

365100

× × × × =

K0.43willbeincludedinMichael’smonthlystatement.

Unit 12.2 Activity 5A: Credit cards1. EstherwaschargedK12interestforonemonthonaK800creditcardbalance.Whatwas

the monthly interest rate?

2. Tomi obtains a K400 cash advance on his credit card on 8 April. His credit card statement isdated26Aprilandheisbeingcharged18.5%p.a.interest.Ifthecashadvanceistheonlyitemonthestatement,howmuchinterestisTomichargedonthecreditcardstatement?


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Topic 5: Consumer credit 121


3. On1JuneHelenabuysajacket,costingK148,usinghercreditcard.Hercreditcardhasa 25-day interest-free period and her statement is dated the 15th day of the month. If she doesnotrepayanyoftheamountuntil20July,andthecreditcardhasaninterestrateof18%,whatamountofinterestwillbechargedonthestatementon:

a. 15 June?

b. 15 July?

4. Malakihasa‘nointerest-freeperiod’creditcardthatcharges16.5%p.a.interest.On9 JanuaryhepurchasesaphonecostingK395.00.HisJanuarystatementisdated28 January and the phone is the only item on it.

a. How much interest is charged on the January credit card statement?

b. Malaki pays K100.00 off the amount owed on 28 January and makes no more purchases or payments before the next credit card statement arrives on 28 February. How much will this statement show that Malaki owes?

5. Paniahasa‘nointerest-freeperiod’creditcardthatcharges16.2%p.a.interest.Herstatementisdatedthe28thdayofthemonth.InthemonthofMay,Paniamakesthefollowingtransactionsonhercreditcard:

Date Transaction Cost (K)

2 May Purchaseofshoes 45

8 May Cash advance 100

24May Utilities payment 124

Whatisthetotalamountowingthatwillappearonherstatementon28May?

6. On12OctoberKenimepurchasedhouseholdfurniture,usinghiscreditcard,tothevalueofK2 856.Kenime’screditcardhasa‘nointerest-freeperiod’andcharges17.5%p.a.interest. The statement is dated the 20th day of each month. Kenime plans to pay off the minimumamount,K50,eachmonthuntilFebruarywhenheplanstopayoffallthatisowing.AssumingthatKenimimakesthepaymentofK50onthe20thofthemonth,findthetotalamountowingonthe20thof:

a. October.

b. November.

c. December.

d. January.

e. February.

Debit cardsBecausedebitcardsaremuchmorecommonthancreditcardsinPNG,let’slookbrieflyatthedifference between them.

Acreditcardisashort-termloan,whereyouborrowfromthebank,spendnow,andpayinterestlater as well as paying back the money you borrowed. A debit card is notaloan,becauseyoucanonlywithdrawmoneyifyoualreadyhavedepositedthatmoneyinasavingsaccountorchequeaccount at your bank. The advantage is that a debit card allows you to spend your own money


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without carrying cash. That is the main difference and it means that you do not pay interest and the fees charged by the bank are lower than for credit cards.

InPNGtherearedifferenttypesofdebitcardssuchasKunduCards,SaveCardsandAccessCards.Apartfromthedifferencesmentionedabove,debitcardsoperateinawaythatissimilartocreditcards.Youmustsigntocompleteatransaction,oruseEFTPOSandenteraPIN(personal identification number); and the bank sends you regular statements that show your transactions and the balance in your account. For those who do not have a credit history and for thosewhofinditdifficulttobudget,adebitcardhasadvantages.

Hire-purchase agreements (or time-payment plans)Ifapurchasercannotpaythefullpriceofanitem,theretailermayofferahire-purchaseagreement.

Ahire-purchaseagreementusuallyrequiresthepurchasertopayadeposit(eitherasetamountorapercentageofthepurchaseprice),thentheremainderofthepurchaseprice,plusinterest,ispaidinanagreednumberofequalamounts(calledinstalments).

Withahire-purchaseagreementthepurchaserishiringtheitemfromtheretaileruntilthefinalpayment is made. If the purchaser defaults (does not pay) on any of the payments then the item will be repossessed by the retailer without any return of the payments.

Theinterestpaidonahire-purchaseplanisquotedasasimpleinterestrate(flatrate)per annum.Eventhoughtherepaymentamountcanseemaffordable,sometimestheamountof interest paid is large.

Example BQ. Bernardisofferedahire-purchaseagreement,chargingaflatrateofinterestof15%,to

buyatelevisionwhichhasapurchasepriceofK1 250.IfhepaysadepositofK250andwill pay the remainder plus interest in 18 monthly payments, how much will he pay each month?

A. TheamountowingafterthedepositispaidisK1 250–K250=K1 000

Interestat15%onK1 000for18months= = 2251000 15

1812

100

× ×; interest is K225.

BernardwillpayatotalofK1 000+K225=K1 225in18equalinstalments.

Bernard will pay = K68.06K122518

each month.

Example CQ. Adaisusingahire-purchaseagreementtobuyarefrigeratorthathasapurchaseprice

ofK990.ShehasagreedtopayK190asadepositandtheremainderinsixmonthlypaymentsofK160.

a. HowmuchinterestisAdapayingwiththiscontract?

b. Whatistheflatrateofinterestbeingcharged?

A. a. Theamountowingafterthedepositispaid=K990–K190

=K800

Sixone-monthlypaymentsofK160=6×K160

=K960

AdaispayingK960–K800=K160ininterest.


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b. AdaispayingK160ininterestfor6monthsandthiswouldbe2×K160=K320for12 months(1year).

Theyearlyinterestasapercentageoftheamountowed= = 40320800

1001

× .

Adaisbeingchargedaflatrateofinterestof40%p.a.Thisisaveryhighrateofinterestwhichwasnotimmediatelyapparentfromtheamountofinterestbeingchargedortheamountofthe repayments.

Unit 12.2 Activity 5B: Hire-purchase 1. Find the interest paid on a hire-purchase contract that is used to pay off a principal of

K1 500over2yearsandischargingaflatrateof16%p.a.

2. Findtheamountandtheflatrateofinterestwhenahire-purchaseagreementismadetopayoffaprincipalofK2 000,plusinterest,with12one-monthlypaymentsofK200.

3. Calculate the monthly instalments paid for a hire-purchase agreement that pays off a principalofK720,plusinterest,chargedataflatrateof12.5%in6monthlyinstalments.

4. Tobiashasagreedtoahire-purchasecontract,chargingaflatrateofinterestof9%,tobuyacarcostingK6 300.HeispayingadepositofK1 300andhehasachoiceofthenumberof monthly payments he can make. Calculate the amount he will be paying per month if hemakes:

a. 12 payments.

b. 18 payments.

5. Jeanniehastakenoutatime-paymentplantobuyhouseholdgoodstothevalueofK2 375.SheispayingadepositofK300andthecontractrequires20monthlypaymentsofK147.Findtheflatrateofinterestcharged.

6. Ahire-purchaseplanrequiresthepurchasertopayadepositofK40andthen9monthlypaymentsofK48.80ongoodsthatcostatotalofK400.Calculatetheflatrateofinterestper annum being charged.

7. Joseph has bought a car costing K4 500. He has paid a deposit of K500 and has agreed to payK340perfortnightforthenext6monthstopayoffthebalance.

a. How many payments will he make? (There are 26 fortnights in a year.)

b. How much interest is he paying?

c. Whatistheflatrateofinterestperannum?

8. Erinhasachoiceofhire-purchasecontractstoenablehertobuyascootercostingK5 640:

OptionA:AdepositofK1640and12monthlypaymentsofK439.

OptionB:AdepositofK1640and18monthlypaymentsofK300.

Findtheflatrateofinterestineachcaseanddecidewhichisthebestfinancialoption.

Reducing-balance loansWehavealreadylookedattheeffectofcompoundinterestonmoneyborrowedormoneyinvested.Whenwecalculatedtheamountowingfortheseloansitwasassumedthattheinterestwould be paid at the end of the loan period.


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Inrealitymostloansrequireacommitmenttoregularrepaymentsthroughoutthetermoftheloan. These types of loans are called reducing-balance loans because each repayment pays off partoftheprincipalowingandso,whenthisreduceswithtime,theamountofinterestowedalso reduces.

Reducing-balance loans are used for any long-term loan such as a loan to buy a house.

In the process of setting up a reducing-balance loan there are five variables that need to be considered:

• Theamountborrowed.

• Thecompoundinterestrate.

• Thetermoftheloan.

• Thefrequencyofpayments.

• Therepaymentamount.

Whenpeoplearenegotiatingaloantobuyahouse,forexample,allofthesevariablesareconsidered to find a loan that will fit their circumstances.

Step-by-step calculation of the amount owing on a reducing-balance loanThe amount owing on a loan after each repayment can be found by working through the addition of interest and repayment for each time period. This is a process done by computers today but the process is worth observing to understand the theory behind reducing-balance loans.

Example DQ. AloanofK10 000isbeingchargedinterestattherateof7%p.a.andregularrepayments

ofK2 000aremadeattheendofeachyear,aftertheinterestiscredited.

a. Usestep-by-stepcalculationstofindtheamountstillowingafter5years.

b. Howlongdoestheloanrunbeforeitistotallyrepaid?(Thisisthetermoftheloan.)

c. Whatistheamountofthefinalpayment?

d. Howmuchinterestispaidforthetermoftheloan?

A. Atablecanbesetuptoshowthecalculations:

Year Interest charged (K) Balance after interest (K)

Repayment (K) Amount owing at end of year (K)

0 10 000.00

1 7%of10 000.00=700.00 10 700.00 2 000.00 8 700.00

2 7%of8 700.00=609.00 9 309.00 2 000.00 7 309.00

3 7%of7309.00=511.63 7 820.63 2 000.00 5 820.63

4 7%of5820.63=407.44 6 228.07 2 000.00 4 228.07

5 7%of4228.07=295.97 4 524.04 2 000.00 2 524.04


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Year Interest charged (K) Balance after interest (K)

Repayment (K) Amount owing at end of year (K)

6 7%of2524.04=176.68 2 700.72 2 000.00 700.72

7 7%of700.72=49.05 749.77 749.77 0.00

Total K2 749.77 K12 749.77

Note:Theamountowedattheendofeachyear(thebalance)isreducingbecauseofthepayments and so the amount of interest paid also reduces.

a. Theamountstillowingafter5yearsisK2 524.04.

b. Theloanrunsfor7yearsbeforeitistotallyrepaid.

c. ThefinalrepaymentisK749.77.

d. K2 749.77ininterestispaidoverthetermoftheloan.

Graphs of reducing-balance loansThe two graphs below relate to Example D above.

Graph 1 shows the balance (amount owing) of the loan after each of the years. It can be seen that it is not a straight line. A larger amount is paid off the balance as the years go by.

Graph 2 shows the interest paid at the end of each year and shows that this is a decreasing amount as the years go by. The graph is also not a straight line.

Graph 1 Graph 2

Year1 2 3 4 5 6 7

Balance (K)

1 0002 0003 0004 0005 0006 0007 0008 0009 000

10 000

End of year

1 2 3 4 5 6 7

Inte

rest

pai

d (

K)

100

200

300

400

500

600

700

Example EQ. NancyhasnegotiatedaloanofK4 000,charging6.6%p.a.interestcalculatedonthe

reducingbalance,withmonthlyrepaymentsofK250.

a. Usestep-by-stepcalculationtofindtheamountstillowingafter5payments.

b. Findtheinterestthathasbeenpaidoverthefivepaymentperiods.

c. Comparetheinterestpaidinthereducing-balancecalculation(b.)withthesituationwherealltheinterestispaidattheendofthefivemonths.


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A. a. Ifinterestis6.6%p.a.thenthisisequivalentto6.6/12=0.55%permonth.

Settingupatableforthecalculations:

Month Interest charged (K) Balance after interest (K)

Repayment (K)

Amount owing at end of month. (K)

0 4 000.00

1 0.55%of4 000.00=22.00 4 022.00 250.00 3 772.00

2 0.55%of3 772.00=20.75 3 792.75 250.00 3 542.75

3 0.55%of3 542.75=19.49 3 562.23 250.00 3 312.23

4 0.55%of3 312.23=18.22 3 330.45 250.00 3 080.45

5 0.55%of3 080.45=16.94 3 097.39 250.00 2 847.39

Total=K97.40

Theamountowingafter5repaymentsisK2847.39

b. Theamountofinterestpaid=250×5–(amountpaidoff)

=1 250–(4 000–2 847.39)

=K97.39

Note:Thedifferenceof1toeaisduetothefactthatfiguresareroundedintheinterestcharged column.

c. Ifalltheinterestispaidattheendofthefivemonthsthenthisamountwillbe:

K(4 000×1.00555–4 000)=K111.22

Itshouldbenotedthatalthoughtheinterestismoreifitispaidattheendofthe5 months,Nancywouldhavehaduseofherrepaymentmoneyinthattime.

Unit 12.2 Activity 5C: Reducing-balance loans1. K5 000hasbeenborrowedat9%p.a.interestandregularrepaymentsofK1 200aremade

attheendofeachyear,aftertheinterestiscredited.Showstep-by-stepcalculationstofind:

a. The amount owing after 4 years.

b. The amount of interest paid in 4 years.

2. AloanofK15 000istoberepaidinquarterlyrepaymentsofK1 000.Ifinterestischargedat8.4%p.a.onthereducing-balance:

a. Whatisthequarterlyrateofinterest?

b. Show step-by-step calculations to find the amount still owing after the first year.

c. Find the interest charged in the first year.

3. Jessie has bought a K780 sound system on her credit card. The credit company charges 18%p.a.interestonthebalanceoutstandingeachmonthandJessieonlyrepaystheminimumrequired,K50,foreachofthefirstthreemonths.

a. Show step-by-step calculations to find the amount owing after three months.


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b. How much interest has she paid in three months?

c. How much more would she owe at the end of three months if she made no repayments?

4. Completethefollowingtablesshowingstep-by-stepcalculations:

a. AloanofK7 500paying9.6%p.a.interestchargedonthereducingbalancewithrepaymentsofK200madequarterly.

Quarter Interest charged (K) Balance after interest (K)

Repayment (K)

Amount owing at end of quarter.(K)

0 7 500.00

1 ….%of7 500.00=180.00 7 680.00 200.00 7 480.00

2 200.00

3 200.00

4 200.00

b. AloanofK200 000paying6.9%p.a.interestchargedonthereducingbalance,withrepayments of K2 400 made monthly.

Month Interest charged (K) Balance after interest (K)

Repayment (K)

Amount owing at end of month.(K)

0 200 000.00

1 0.575%of…….=…… 2 400.00 198 750.00

2 0.575%of19 8750=… 2 400.00

3 2 400.00

4 2 400.00

5 2 400.00

5. PeterhasboughtanannuitywithK300 000ofhissuperannuationpayout,toprovidehimwithincomeeachmonth.Ifinterestispaidat8.5%p.a.ontheremainingbalance,beforeapaymentofK2 400ispaidtoPeter,showstep-by-stepcalculationstofindthebalanceofthe initial investment remaining after 4 months.

The annuities formulaAnnuity technically means ‘annual payment’ but in the context of ‘the annuities formula’ it means a regular payment rather than an annual payment.

Finding the amount owing using the annuities formulaTheannuitiesformulaisusedtofindtheamountstillowingonaloan,whichisbeingchargedinterestonareducing-balancebasis,whereregularpaymentsarebeingmadeeachtimeperiod.


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Theformulareplacesthestep-by-stepcalculationoftheamountstillowing,andcanbeestablished from first principles involving the sum of a geometric sequence. This will not be shown here.

Theannuitiesformulais:

A PR Q Rn

Rn – –= ( 1)

–1

where:

• A is the amount owing after n time periods.

• P is the amount borrowed.

• Q is the amount of the regular repayment.

• R r=1+100

where r is the rate of interest per time period.

• n is the number of time periods.

Time periodsAtimeperiodcanbeayear,ahalf-year,aquarter(threemonths),amonth,afortnight,aweek,or a day.

Usuallyaninterestrateisquoted‘perannum’whichmeansforayear,andlong-termloanshaveatermquotedinyears.If,however,therepaymentsandtheinterestarecalculatedinashortertimeperiodthencalculationsmustreflectthisshortertimeperiod.

Example FQ. Aloanischarged9%p.a.interest,compoundingmonthlyover20years.Howmanytime

periods, n, are involved in this loan and what is the value of r and R?

A. Thenumberoftimeperiods: n=Numberofyears×numberofmonthsinayear.

=20×12

=240

The monthly interest rate, r : ryearly interest rate

number of months in a year= = % = 0.75%9

12

The value of R: R R = 1+ = 1.00750.75100

Use the following to calculate time periods:

1year=2half-years

=4quarters

=12months

=26fortnights

=52weeks

=365days


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Example GQ. Usetheannuitiesformulatofindtheamountowingafter5yearsforaloanofK10 000,

charged7%p.a.interestonthereducingbalance,whichhasrepaymentsofK2 000madeannually.(ExampleDisthestep-by-stepcalculation.)

A. Forthisexample:

P=10 000

Q=2 000

r=7%p.a.soR=1+7/100=1.07

interest is credited annually

n=5

SubstitutinginA PR Q Rn

Rn= – ( – 1)

– 1

A = 10 000 1.075 – 2000(1.075 – 1)1.07 – 1

×

=2 548.039

Theamountowingafter5yearsisK2 548.04.

Example HQ. Calculatetheamountstillowingafter3yearsonaloanofK8 000thatischarging

7.2% p.a.interestonthereducingbalanceandwherepaymentsof:

a. K200aremadeeachmonth.

b. K100arepaideachfortnight.

A. a. P=8 000

Q=200

r =7.2%peryear;7.2/12=0.6%permonth

R =1+0.6/100=1.006

n=3years=3×12=36months

Substituting A PR Q Rn

Rn= – ( – 1)

– 1

A = 8 000 1.006 – 200(1.00636 – 1)1.006 – 1

36×

=1 912.359

AftermonthlypaymentsofK200for3yearstheamountowingisK1 912.36.

b. P =8 000

Q =100

r=7.2%peryear=7.2/26%=0.2769...%perfortnight

R=1+0.2769…/100=1.002769.(Leaveonthecalculator.)

n=26×3=78fortnightsin3years

Substitutingin A PR Q Rn

Rn= – ( – 1)

– 1

A = 8 000 1.0027... 100(1.0027...78 – 1)1.0027... – 1

–78×

=1 232.757

AfterfortnightlypaymentsofK100forthreeyearstheamountowingisK1 232.76.


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Note: The differencebetweentheamountsowingfora. and b. isaboutK680.Thisisbecausethefortnightlypaymentsreducethebalancemorefrequentlyandhencelessinterestispaid.AnextraK200ispaidperyearinb.becausethereare26fortnightsinayearbutonly12 months.

Finding the repayment amountAquestionthatisoftenaskedwhenestablishingareducing-balanceloanis:‘Howmuchwillwehave to pay each month to pay off our loan in 5 years?’

In a case like this we are looking for the value of Qintheannuitiesformula,when A=0,ie.theloan is repaid. (This assumes the values of P,R and are known.)

The annuities formula can be rearranged to make Qthesubject:

A PR Q R

R– –

–= ( n 1)

1n

PR AQ Rn

R––

–=( 1)

1n

Multiply both sides by (R–1):

Q(Rn – 1) = (PRn – A)(R – 1)

Divide both sides by (Rn – 1)

Q PRn A R

Rn

– –

–= ( )( 1)

( 1)

Example IQ. TimwantstopayoffhisK20 000loan(forwhichheisbeingcharged8.4%p.a.interest

onareducing-balancebasis)in4years.

a. FindtheamountTimneedstorepayeachmonthtopayouthisloanin4years.

b. Howmuchinterest,intotal,hasTimpaidoverthetermofthisloan?

A. a. P=20 000

A=0 (Ifaloanispaid outthentheamountstillowingiszero.)

Q=?

r=8.4%p.a.whichis8.4/12=0.7%permonth

R=1+0.7/100=1.007

n=4years=48months

Substitutingin Q PRn A R

Rn= ( – )( – 1)

( – 1)

Q =(20000 1.00748 – 0)(1.007 – 1)

(1.00748 – 1)

×

=492.022

TimwouldneedtopayK492.02permonthtopayoffhisK20 000loanin4years.

Ingeneral,lessinterestispaid,andhencethetermoftheloanisshortened,ifpaymentsaremademorefrequently.


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b. TimhaspaidK492.02eachmonthfor4×12=48months.

IntotalTimhaspaidK492.02×48=K23 616.96.

HisloanwasforK20 000sohehaspaidK23 616.96–K20 000=K3 616.96ininterest.

To calculate the total interest paid for a loan:

To calculate the amount of interest paid after m payments:

Interest-only loansAninterest-onlyloanisonewheretherepaymentseachtimeperiodareequaltotheinterestdueafter that time period. No amount is paid off the principal in an interest-only loan and so there is no fixed term and the loan does not increase or decrease in value.

Interest-onlyloansareusedtomaximisetheamountofinterestbeingpaid,whichisusefuliftheinterestcanbeataxdeduction,andalsotominimisetherepaymentamount.

Ifwearetousetheannuitiesformulaforaninterest-onlyloan,thentheamountowingaftern timeperiods,A,isequaltoP,theprincipal:

P PR Q Rn

Rn= – ( – 1)

– 1

PR PQ Rn

Rn ––

–=( 1)

1

= P(Rn–1):dividebothsidesby(Rn – 1)

Q

RP

–1=

Q = P(R – 1)

Q P= 1+ –1r

100( ) Q Pr=

100Hencetherepaymentamount,Q,isequaltotheinterestcalculatedforthefirsttimeperiod;asimple-interest calculation.

Example JQ. Findthefortnightlypaymentamountforaninterest-onlyloanofK50 000,wherethe

interestrateis7.8%p.a.

A. The interest rate for fortnightly payments is = 0.3%7.826

Thepaymentamounteachfortnight= = 150; 50000 0.3

100×

K150needstobepaideachfortnighttopaytheinterestthatisowed.

Interestpaid=repaymentamount(Q)×numberofpayments(n)–loanamount(P)

Interestpaid=Q×n–P

Interestpaid=repaymentamount(Q)×m–amountpaidoffloan

=Q×m–(loanamount–amountowingaftermtimeperiods(A))

=Q ×m–(P–A)


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Unit 12.2 Activity 5D: Using the annuities formulaGive your answers correct to two decimal places where necessary.

1. Find P,Q,R and nvaluesfortheloansgiveninthetablebelow:

Loan (K) Interest rate p.a. Repayment (K) Compounded Years

a. 6 000 8.1% 500 yearly 4

b. 15 000 7.2% 250 monthly 3

c. 850 10.8% 120 quarterly 1.5

d. 200 000 8.06% 800 fortnightly 10

e. 82 000 8.32% 180 weekly 10

2. Use the formula A PR Q Rn

Rn= – ( – 1)

– 1 to calculate the amount owing (A) for each of parts

a. to e.inquestion1. above.

3. Ezekiel has an interest-only loan for an investment property. The loan is for K256 000 at aninterestrateof9.8%p.a.withfortnightlyrepayments.FindthepaymentthatEzekielmakes each fortnight.

4. Jeffrey has taken out a loan of K12 000 and he is making repayments of K200 per month. Heisbeingcharged8.4%p.a.interestcompoundedmonthlyonthereducingbalance.Use theannuitiesformulatocalculatetheamounthestillowesontheloanafter:

a. 1 year.

b. 2.5 years.

5. Use the annuities formula to find the amount still owing after 5 years on a reducing-balanceloanof:

a. K10 000,beingcharged7.75%p.a.interestwithrepaymentsofK1 500peryear.

b. K50 000,beingcharged8.8%p.a.interestwithrepaymentsofK2 500perquarter.

c. K150 000,beingcharged6.96%p.a.interestwithrepaymentsofK1 800permonth.

6. Use the formula Q PRn A R

Rn= ( – )( – 1)

( – 1)to find the repayment necessary to pay out a loan of

K24 000,in10years,thatisbeingcharged7.2%p.a.interestonthereducingbalanceiftherepaymentsaretobemade:

a. Quarterly.

b. Monthly.


Rn= ( – )( –1)

( – 1)tofindtheamountoftherepaymentrequiredtopay

outaloanof:

a. K1 000,beingcharged15%p.a.interest,paymentsbeingpaidmonthlyover6months.

b. K170 000,beingcharged7.8%p.a.interest,paymentsmadefortnightlyover20years.

c. K54 000,beingcharged7.2%p.a.paymentsmadequarterlyover10years.


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Rn= ( – )( –1)

( – 1) to find the repayment necessary to reduce a loan

fromK30 000toK20 000in4years,iftheloanisbeingcharged6%p.a.interestonthe reducing balance and the repayments are to be made monthly.

Solving for the principal, PAnothercommonlyaskedquestionisalongthelinesof:

‘How much can I borrow if I can afford to pay K1 250 per month? I want to pay out the loan in 20yearsandtheinterestbeingchargedis7.2%p.a.calculatedonthereducingbalance.’

In this case we are looking for Pintheannuitiesformula:A PR Q Rn

Rn= = ( – 1)

– 1Tomakethesubject: PR A Q Rn

Rn = + ( – 1)

–1

adding Q Rn

R

( – 1)

–1 to both sides.

PA

Q Rn

R

Rn=

+( – 1)

–1

Fortheexampleabove:

A = 0

Q = 1 250

r=7.2%p.a.whichis7.2/12=0.6%permonth

R =1+ =1.0060.6

100 n = 20 years = 240 months

Careful substitution in the formula PA

Q R

R

R=

+( n – 1)

– 1n

P =0 +

1 250(1.006240 – 1)

(1.006 – 1)

1.006240

= 158 760.54

So K159 000 (to the nearest K1 000) could be borrowed under the conditions given.

Unit 12.2 Activity 5E: Using the annuities formula1. Foreachofthefollowing:

a. Set up the annuities formula by substituting the appropriate values.

b. Solve for the unknown variable in the annuities formula.

i. Theamountowingafter5yearsonaloanofK84 000,paying8.4%p.a.interestcalculatedonthereducingbalance,withmonthlyrepaymentsofK650.

ii. TheamountthatcanbeborrowedwhenquarterlypaymentsofK5 000aremadesothattheloanispaidoutover10yearsandwhereinterestischargedat7.6%p.a.calculated on the reducing balance.


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iii. ThepaymentrequiredtoreducealoanofK100 000toK50 000over10yearswithmonthlyrepayments,whereinterestischargedat7.2%p.a.calculatedonthereducing balance.

2. Match the annuities formula substitutions A to D with the situations a. to d. below:

A Q

0 = 10 000 × 1.00548–(1.00548 – 1)

1.005 – 1

B Q

10 000 = 10 000 × 1.00548 – (1.00548 – 1)

1.005 – 1

C Q

0 = 10 000 × 1.01516 – (1.01516 – 1)

1.015 – 1

D 20 000 = P × 1.02516 – 250(1.02516 – 1)

1.025 – 1

a. ThequarterlypaymentonaloanofK10 000thatispaidoutover4yearsandwhereinterestischargedat6%p.a.

b. The amount that is borrowed over 4 years so that K20 000 is still owing and where quarterlypaymentsofK250aremade.Interestof10%p.a.isbeingchargedonthereducing balance.

c. The monthly payment on an interest-only loan of K10 000 over 4 years and where interestischargedat6% p.a.

d. The monthly payment on a loan of K10 000 that is paid out over 4 years and where interestischargedat6%p.a.

3. Greg can afford to pay K780 a month as a mortgage payment. If housing loans are being charged6.9%interestonthereducingbalance,howmuchcanGregborrow,tothenearestK1 000,iftheloanisfor:

a. 20 years.

b. 25 years.

4. Anthony and Maria have taken out a K145 000 interest-only loan for 5 years. If they are paying7.35%p.a.interest,

a. How much will they pay each fortnight?

b. How much interest will they pay over the 5 years?

5. Find the amount a couple can borrow if they can afford to repay K2 400 per month on a loanthatischarged7.15%p.a.onthereducingbalanceandtheywouldliketorepaytheloanin:

a. 15 years.

b. 20 years.

6. Match the situations (a) to (d) to the graphs I to IV of the amount owing over 5 years whereinterestischargedat7%p.a.onthechangingbalance:

a. K10 000 loan with regular payments of K200 per month.

b. K10 000 interest-only loan with regular repayments of interest.


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c. K10 000 loan with regular repayments of K50 per month.

d. K10 000 loan with regular repayments of K100 per month.

I II III

IV

7. Usetheappropriateversionoftheannuitiesformulatofind:

a. TherepaymentamountperfortnightforaloanofK35 000thatwasreducedtoK20 000over3yearsandwhereinterestof7.8%p.a.waschargedonthereducingbalance.

b. The amount of interest paid per month on an interest-only loan of K90 000 where interestischargedat6.95%p.a.

c. Theamountthatcanbeborrowed,tothenearestK1 000,ifrepaymentsofK850perfortnightcanbemadeandinterestischargedat9.1%p.a.onthereducingbalanceover 20 years.

d. The minimum amount that must be paid each month on a loan of K5 000 so that the loanamountdoesnotincrease;interestchargedat9.6%p.a.

e. Theinterestchargedinthefirst10yearsofa25-yearloanofK150 000,whereinterestischargedat7.28%p.a.onthereducingbalanceandrepaymentsofK500perfortnightare made.

Service fees and chargesBanks and other financial institutions have fees and charges for most of the services they provide. These fees and charges differ with every financial institution.

5Year

Amount (K)

10 000 --

5Year

Amount (K)

10 000 --

5

Amount (K)

Year

10 000 --

5Year

Amount (K)

10 000 --

5Year

Amount (K)

10 000 --

5Year

Amount (K)

10 000 --

5

Amount (K)

Year

10 000 --

5Year

Amount (K)

10 000 --


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Bank accounts, savings accounts, transaction accounts, cheque accountsThesearetheeverydayaccountsthatpeoplerelyonfordepositsandwithdrawals,topayregulardeductions and to accept regular payments. These accounts pay a very small amount of interest. Feesandchargesthatcanapplyinclude:

• Monthlyaccountfees(K3.00).

• Tellerservicefees(K3.00deposits;K4.00withdrawals).Someaccountshaveanumberoffree transactions per month and then a fee is charged.

• EFTPOSservicefees.

• Feesforchequesdepositedorprovided.

• Fundstransferfeestootheraccounts.

• Feesformobilephonebankingtransactions.

• Feesforprovidingastatementoftransactions.

• Chequeaccountshavefeesforprovidingchequebooks.

• Feesifthereisnotenoughmoneyintheaccounttopayachequethatiswritten(dishonouredor‘bouncing’cheque).

Credit cardsCredit cards can have a yearly fee and an interest-free period for purchases or no yearly fee and no interest-free period for purchases. Cash withdrawals from a credit card are charged interest fromthedayofthewithdrawal.Interestchargesarequitehighforcreditcardsandownersofcredit cards need to have reliable income and a good credit history.

Debit cardsDebit cards allow the owner access to their own funds in an account. Some of the fees that can applytodebitcards:

• Establishmentfees.

• Annualfees.

• Cardreplacementfees.

Term depositsThere are usually no establishment fees but prepayment fees and an interest penalty apply if a term deposit is claimed before the specified term.

Personal loansInadditiontotheinterestchargeapersonalloancanhavethefollowingcharges:

• Anapplication/processingfee.

• Earlyrepaymentfees.


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• Chargesforlaterepayments.

• ‘Bouncing’chequecharges.

• Documentationfeesfortheverificationofyourdocuments.

Home loansFees and charges from financial institutions vary significantly between lenders. The following feesandchargesmayapplytoahomeloan:

• Anapplication/setupfeethatcoversthecostofprocessingthehomeloanapplication.

• Afeetoestablishlenders’mortgageinsurance.Thisisneedediftheborrowerneedstoborrowmorethan80%ofthevalueoftheproperty.

• Avaluationfee.Thisistopayaqualifiedvaluertoassessthemarketvalueofaproperty.

• Anexitfeetocoverthecostsofpreparingthedocumentstofinishaloan.

• Anearlyexitfee.Applicableiftheloanisrepaidbeforethefixedterm.

• Regularaccountfees,usuallymonthly.

Inadditiontotheselenderfeesthereareotherchargesassociatedwithpurchasingahome:

• Governmentstampdutywhichisapercentageofthepurchaseprice.

• Legalfeesforsettingupsalecontractsandconductingenquiriesandsearchesofdocumentsrelating to the property.

Unit 12.2 Activity 5F: Service fees and chargesThis exercise can be done individually or as a group project. It is suggested that only one (or a maximumoftwo)oftheinvestigationsbedone,aseachrequiresresearchoutoftheclassroom.A well-presented report on the findings is essential.

In some instances a hypothetical situation can be used to illustrate the fees and charges; for example when investigating the cost of a loan to buy a house an imaginary property of a certain value can be used.

1. Investigate the service fees and charges for an everyday savings account with a bank. You candothisinoneofthefollowingways:

a. Ask someone you know who has a bank account for information about the fees and charges they pay.

b. Approach a local bank and gathering the information on its fees and charges.

c. Use the internet to access information on a particular bank account.

Use the list above as a guide to the possible fees and charges (there may be others that are not mentioned on the list).

2. Investigate the service fees and charges for a particular credit card issued by a bank. You should also gather information about the eligibility for owning a credit card and the limits thatareplacedontheamountofcredit.Youcandothisinoneofthefollowingways:

a. Ask someone you know who has a credit card for information about the fees and charges they pay. You will probably need to see a monthly statement.


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b. Approachalocalbankandgathertheinformationoneligibility,feesandchargesfortheir credit card.

c. Use the internet to access information on a particular credit card.

Use the list above as a guide to the possible fees and charges (there may be others that are not mentioned on the list).

3. Investigate the service fees and charges attached to a personal loan with a bank or other financial institution. You should also gather information about eligibility for a personal loan.Youcandothisinoneofthefollowingways:

a. Ask someone you know who has a personal loan for information about the fees and charges they pay.

b. Approach a local bank and gather information on its fees and charges and eligibility criteria.

c. Use the internet to access information on a personal loan from a particular bank.

Use the list above as a guide to the possible fees and charges (there may be others that are not mentioned on the list). Often the application forms attached to personal loans are useful sources of information.

4. Investigate the service fees and charges attached to a home loan from a bank or other financialinstitution.Youshouldalsogatherinformationabouttheeligibilityrequirementsforahomeloan.Youcandothisinoneofthefollowingways:

a. Ask someone you know who has a home loan for information about the fees and charges they pay.

b. Approach a local bank and gather information on its fees and charges and eligibility criteria.

c. Use the internet to access information on a home loan from a particular bank.

Use the list above as a guide to the possible fees and charges (there may be others that are not mentioned on the list). Often the application forms attached to home loans can be useful sources of information.


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• personal loans and credit cards. • hire-purchase...

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