CMM Subject Support Strad: FINANCE Uit 3 Loas ad Mortgages: Text m e p STRAND: FINANCE Uit 3 Loas ad Mortgages TEXT Cotets Sectio 3.1 Aual Percetage Rate (APR) 3.2 APR for Repaymet of Loas 3.3 Credit Purchases 3.4 Mortgage Repaymets
CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text 3 Loas ad Mortgages 3.1 Aual Percetage Rate (APR) Most baks ad buildig societies calculate iterest more tha oce a year - ofte o a day-to-day basis. This may ivolve usig a variety of iterest rates which ca be calculated from oe aother. There is much cofusio at this poit, however, ad it was to help cosumers that laws were itroduced several years go requirig all baks, fiace houses ad other istitutios offerig loa facilities to state i their advertisemets the true APR (aual percetage rate) beig charged by way of iterest. The APR is the iterest rate figure that idicates the total cost of borrowig, icludig ay charges. This iformatio helps the borrower to compare like with like whe cosiderig various optios. Worked Example 1 The iterest charges for borrowig o a major credit card are 1.85% per moth. What is the APR? Solutio Iterest of 1.85% gives a multiplier of 1.0185 for each moth. So the multiplier for 12 moths is 1. 0185 12 = 1.2460... Thus the APR is 24.6%. Note This is more tha 12 185. % = 222. % ad although the differece is small, it becomes more sigificat with higher iterest charges. Worked Example 2 The iterest charge for borrowig o Twego is 4.95% per moth. What is the APR? Solutio Here the mothly multiplier is 1.0495, so for 12 moths it is 12 1. 0495 1. 7856... = So the APR is 78.6% which is cosiderably more tha 12 495. % = 594. % The situatio is eve more extreme with a daily rate of iterest. Worked Example 3 The iterest charge for borrowig o Twego is 0.5% per day. (a) What is the APR? (b) How much would you owe after 2 weeks if you borrowed 100 o these terms? 1
3.1 CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text Solutio (a) (b) Note The daily multiplier is 1.0, so for a year (365 days), it is So the APR is 517.5%. 365 = 1. 0 6. 1746... To calculate how much you would owe after 2 weeks, you use 14 14 100 daily multiplier 100 1. 0 = = 107. 23 Note that such loas (sometimes referred to as 'payday loas', usually for betwee 80 ad 1000 ad desiged to keep your fiaces afloat util you receive your mothly wages), ca be helpful for short term loas of just a few days or maybe weeks, but ca lead to severe fiacial problems i the loger term. Most payday loa UK compaies charge rates of iterest ito the thousads, with a optio to roll o debt from moth to moth - a very bad idea cosiderig the amout of iterest you would be payig. New rules for payday leders, desiged to protect vulerable borrowers from spirallig costs, were itroduced i Jauary 2015 ad will be reviewed i 2017. Payday loas cotiue to be a expesive ad potetially problematic optio for obtaiig moey. Worked Example 4 What weekly iterest rate correspods to 26% APR? Solutio 52 It the yearly multiplier is 1.26, the weekly multiplier is 1. 26 = 1. 0045... So the weekly iterest rate is 0.45% per week. 1 Worked Example 5 A fiace compay charges iterest at 26% per year calculated weekly. What APR is it chargig? Solutio 26% per year is equivalet to 26 % =. % per week. 52 So the multiplier for 1 week is 1.0. The multiplier for 52 weeks is 1. 0 52 which is 1.2960. So the true APR is 29.6%. You should bear i mid that real-life calculatios are ofte ot this simple. A credit card compay, for example, may charge iterest from the date of purchase, or from the date of the statemet or from the ed of the moth, or from some other date. The compay may eve waive iterest altogether as log as the whole balace is repaid withi a certai 2
3.1 CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text period. Details will be give i the 'Coditios of Use' issued to all cardholders; it is importat that you read these very carefully. Agai, baks ad buildig societies are ormally required by law to deduct icome tax at the curret basic rate before payig iterest to idividual depositors, except i some cases whe the depositors themselves are ot liable for icome tax. For this reaso, baks ad buildig societies usually quote et iterest after allowig for the tax deducted: you should be very careful if you have to compare these rates with the gross rates (i.e., before deductio of tax) quoted by may other istitutios. This is dealt with i Uit 5. Exercises Calculate: 1. the true APR equivalet to 2% per caledar moth; 2. the true APR equivalet to 10% per week; 3. the mothly rate equivalet to a true APR of 9% p.a.; 4. the true APR equivalet to 12% p.a. calculated weekly; 5. the true APR equivalet to 14% p.a. calculated daily. 3.2 APR for Repaymet of Loas Here we cosider the reverse problem to that cosidered i the previous sectio, that is, repaymet of a sum of moey, say C, borrowed o a certai date ad repaid after time years. We eed to determie the APR charged. Note that the APR, say i, is give by A = C 1 + i where A is the repaymet sum. Worked Example 1 A loa of 2000 is repaid by a paymet of 2500 made 4 years later. What is the APR for this loa? Solutio Usig the formula above, 2500 = 2000 1 + i 4 2500 Hece ( 1 + i ) = = 2000 ad ( 1 + ) i = ( 125. ) So the APR is approximately 5.7%. 1 4 125. i = 0. 737... 4 3
3.2 CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text Worked Example 2 A borrower is loaed 4000 o 1st Jauary 2015 ad agrees to repay with a sigle paymet of 5000 o 21st March 2016. What is the APR for this loa. Solutio Here A = 5000 ad C = 4000 but the time iterval,, is eeded i years. = 1year + 3 moths = 1 + = 1 + ( 31 + 29 + 31) 366 91 366 years years = 1. 2486... years Thus the APR is give, approximately, by 5000 = 4000 1 + i 1. 2486 ( 1 + i) 1. 2486 = 5000 4000 = 125. Hece 1. 1. 2486 ( 1 + i ) = ( 125) = 1. 19568... givig i = 0. 19568... So the APR is approximately 19.6%. Exercises 1. Fid the APR for each of the followig loas. (a) 1000 repaid by sigle repaymet of 1100 made after 1 year. (b) 1000 repaid by a sigle repaymet of 1400 made after 4 years. (c) (d) 4500 repaid by sigle repaymet of 5000 made after 2 1 2 years. 25 000 repaid by a sigle repaymet of 30 000 after 10 years. 4
3.2 CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text 2. Fid the APR for each of the loas show i the table below. Amout borrowed Date of loa Amout repaid Date of repaymet (a) 1000 1/12/2016 1100 31/3/17 (b) 5000 1/1/15 6000 30/9/17 (c) 25 000 1/4/14 32 500 1/8/16 3.3 Credit Purchases I reality, may loas are repaid ot i a lump sum at the ed of the loa period but i a series of weekly, mothly or aual paymets. Very few people buyig a house ca afford to pay for it i cash, for example: istead, the usual procedure is to obtai a loa from a bak or buildig society ad to repay this by mothly istalmets (with iterest, of course) over a period of 25 years or so. The calculatios are quite complicated because the outstadig balace o which iterest is charged varies from moth to moth as repaymets are made. Oe method of fidig how log it takes to repay a load is to calculate the positio at the ed of each repaymet period. The ext worked example shows this method. Worked Example 1 A fried has agreed to led you 1000 at a rate of 5% per aum ad you have agreed to pay back 150 per year. (a) (b) How log will it take you to pay off the loa? How much do you repay i total? Solutio You ca set out the calculatios i a table: Iterest Repaymet Balace ( ) ( ) ( ) At the start 1000.00 After 1 year + 50.00 150.00 900.00 After 2 years + 45.00 150.00 795.00 After 3 years + 39.75 150.00 684.75 After 4 years + 34.24 150.00 568.99 After 5 years + 28.45 150.00 447.44 After 6 years + 22.37 150.00 319.81 After 7 years + 15.99 150.00 185.80 After 8 years + 9.30 150.00 45.10 After 9 years + 2.26 47.36 0.00 5
3.3 CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text (a) So it takes you 9 years to repay the loa. (b) You repay 1247.36 i total. This log ad repetitive calculatio is tedious whe doe by had but ca be processed almost istatly usig a calculator or spreadsheet. Exercises For each of the followig loas, calculate (i) the repaymet period ad (ii) the actual total amout paid. 1. 500 borrowed at 9% iterest ad repaid at 200 per year. 2. 1200 borrowed at 14% iterest ad repaid at 350 per year. 3. 200 borrowed at 25% iterest ad repaid at 80 per year. 4. 250 borrowed at 1.5% per moth ad repaid at 50 per moth. 3.4 Mortgage Repaymets The method of calculatio i Sectio 3.3 is far from ideal. Apart from the time take, its mai disadvatage is that it ca be used oly whe the size of the repaymet is already kow. More commoly, it is the period of the loa that is kow ad the repaymet which has to be calculated, ad for this reaso a differet techique is eeded. You will eed to have studied both geometric series ad logarithms to uderstad what follows. First, take the problem i Worked Example 1 of Sectio 3.3, ad cosider it i a slightly differet way. You borrow 1000 at 5% iterest ad pay it back at 150 per year. How log will this take? Balace, i, at... 0 years = 1000 1 year = 1000 1. 150 2 years = 1000 1. 150 1. 150 (because iterest is calculated as 5% of the previous balace ad the a paymet is deducted). 2 3 years = 1000 1. 150 1. 150 1. 150 3 2 = 1000 1. 150 1. 150 1. 150 This should suggest a fairly obvious patter: 6
3.4 CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text Balace,,, i, at 1 2 years = 1000 1. 150 1. 150 1.... 1. 1. 150 1 2 = 1000 1. 150 1. + 1. +... + 1. + 1 Now the terms i brackets form a geometric series with terms. It is coveiet to read this series backwards, so that its 'first' term is 1 ad its commo ratio 1.; the usual 11 ( 1. ) formula the gives its sum as. 1 1. 150 1 1. So balace = 1000 1. 1 1. = 1000 1 150 1 1.. 0. = 1000 1. + 3000 1 1. = 1000 1. + 3000 3000 1. = 3000 2000 1. The loa is repaid whe this balace reaches zero. 0 = 3000 2000 1. 2000 1. = 3000 1. = 15. l 1. = l 1. = l 15. l 1. = 831.... So the loa is repaid after about 8.3 years, which agrees with the aswer already obtaied. This method may seem very log ad complicated i compariso with the earlier oe, but it becomes easier with practice ad it ca be used for log periods without ay difficulty. Worked Example 1 Adrew ad Aliso borrow 45 000 from the buildig society to buy a house. (They would probably say that they get a mortgage, although from a lawyer's poit of view they actually give the buildig society a mortgage - a legal charge o the house - i exchage for a loa.) The effective iterest they will pay after allowig for tax relief is 9% per aum, calculated mothly, ad they pla to repay 380 per moth. How log will it take them to pay off the loa? 7
3.4 CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text Solutio The mothly iterest rate is 9% 12 = 075. %, so the mothly multiplier is 1.0075. Owig, i, at 0 moths = 45 000 1 moth = 45000 1. 0075 380 2 moths = 45000 1. 0075 380 1. 0075 380 ad so o... 2 = 45000 1. 0075 380 1. 0075 380 1 moths = 45000 1. 0075 380 1. 0075... 380 380 1 1. 0075 = 45000 1. 0075 1 1. 0075 = 45000 1. 0075 + 50666. 67 1 1. 0075 = 45000 1. 0075 + 50666. 67 50666. 67 1. 0075 = 50666. 67 5666. 67 1. 0075, summig the arithmetic progressio The loa is repaid whe this balace is zero, which happes whe 5666. 67 1. 0075 = 50666. 67 1. 0075 = 8. 9411... log 1. 0075 = log 8. 9411... = 293. 18... So the period of the loa is 293 moths (just uder 25 years). The real advatage of this method over the simpler step-by-step method is that it ca be used to calculate the repaymets eeded to pay off the loa withi a give period, as we will see i the ext worked example. Worked Example 2 Suppose your brother borrows 400 from you to go o a sports weeked. He agrees to pay it back (with iterest at 10% p.a. calculated mothly) by twelve mothly istalmets. How much should each istalmet be? Solutio Suppose each paymet is P. The mothly iterest rate is 10% 12 = 0. 833...%. 8
3.4 CMM Subject Support Strad: Fiace Uit 3 Loas ad Mortgages: Text Balace at 0 moths = 400 1 moth = 400 1. 00833 P 2 moths = 400 1. 00833 P 1. 00833 P ad so o... 2 = 400 1. 00833 P 1. 0083 P 12 11 12 moths = 400 1. 00833 P 1. 00833 +... + 1 12 = 400 1. 00833 = 441. 87 12. 57P But this balace is to be zero, givig P = 441. 87 12. 57 = 35. 152... He should pay back 35.15 each moth. 12 P( 1 1. 00833 ) 1 1. 00833 Exercises 1. Sam ad Sheila borrow 50 000 from a buildig society at a et iterest rate of 10% p.a., added at the begiig of each year to the balace the outstadig. Each year, Sam ad Sheila make 12 mothly repaymets of 450 each, totallig 5400. How log will it be before they have paid off the loa completely? 2. Repeat Questio 1, with the buildig society calculatig the iterest mothly at a rate equivalet to 10% p.a., ad Sam ad Sheila still repayig 450 per moth. 3. Joh ad Julie borrow 40 000 from a buildig society to buy a house. The rate of iterest (which is added to the balace at the begiig of each year) is 10.5% p.a., ad Joh ad Julie wat to pay off the mortgage over a period of 25 years. What should their mothly repaymet be? 4. Repeat Questio 1, assumig that iterest is added mothly. (Remember that 10.5% is ot the APR.) 5. The maager of a electrical shop wats to offer her customers ew credit terms. A laptop computer costig 239 is to be offered for twety-four equal mothly paymets so that the APR is 15%. How much should each paymet be? 6. A used car is advertised at a cash price of 6674, but the garage offers credit terms with a quoted APR of 17%. These terms call for a 99 deposit followed by 48 mothly paymets of 185.75. Check this calculatio ad commet o your result. 7. Agela buys a computer ad pays by istalmets. The cash price of the computer would be 550, but Agela pays 250 dow, 200 after six moths, ad 150 after twelve moths. Work out the half-yearly iterest rate ad the APR. 9