Annuities and loan. repayments. Syllabus reference Financial mathematics 5 Annuities and loan. repayments

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1 8 8A Futue value of a auity 8B Peset value of a auity 8C Futue ad peset value tables 8D Loa epaymets Auities ad loa epaymets Syllabus efeece Fiacial mathematics 5 Auities ad loa epaymets Supeauatio (othewise kow as supe is a way of savig moey. It is a type of auity a egula paymet ito a ivestmet. This chapte will focus o auities, make calculatios about thei futue woth ad ivestigate applicatios of auities.

2 ARE YOU READY? Ty the questios below. If you have difficulty with ay of them, exta help ca be obtaied by completig the matchig SkillSHEET. Eithe click o the SkillSHEET ico ext to the questio o the Maths Quest HSC Couse ebookplus o ask you teache fo a copy. plus SkillSHEETSHEET 8. doc-379 Fidig values of ad i fiacial fomulas Fidig values of ad i fiacial fomulas Fid the value of ad i fo each of the followig ivestmets. a Iteest of 8% p.a. fo 5 yeas, with iteest calculated aually b Iteest of 6% p.a. fo 4 yeas, with iteest calculated six-mothly c Iteest of 7.6% p.a. fo 3 yeas, with iteest calculated quately d Iteest of 9.6% p.a. fo 0 yeas, with iteest calculated mothly e Iteest of 24% p.a. fo Novembe, with iteest calculated daily plus SkillSHEETSHEET 8.2 doc-380 Calculatig simple iteest Calculatig simple iteest 2 Fid the simple iteest o each of the followig ivestmets. a $ ivested at 5% p.a. fo 4 yeas b $5 500 ivested at 8.2% p.a. fo 6 yeas c $ ivested at 9.4% p.a. fo 8 moths plus SkillSHEETSHEET 8.3 doc-38 Calculatig compoud itees Calculatig compoud iteest 3 Fid the compoud iteest eaed o each of the followig ivestmets. a $2 000 ivested at 6% p.a. fo 3 yeas, with iteest compouded aually b $ ivested at 8% p.a. fo 5 yeas, with iteest compouded six-mothly c $ ivested at 7.2% p.a. fo 4 yeas, with iteest compouded quately ebookplus SkillSHEETSHEET 8.4 doc-383 Readig fiacial tables Readig fiacial tables 4 The table below shows the amout to which $ will gow ude compoud iteest. Iteest ate pe peiod Peiods 6% 7% 8% 9% Use the table to fid the futue value of each of the followig ivestmets. a $8000 at 6% fo 2 yeas, with iteest compouded aually b $2 500 at 8% p.a. fo 3 yeas, with iteest compouded aually c $8 000 at 2% p.a. fo 2 yeas, with iteest compouded six-mothly 248 Maths Quest Geeal Maths HSC Couse

3 8A Futue value of a auity A auity is a fom of ivestmet ivolvig egula peiodic cotibutios to a accout. O such a ivestmet, iteest compouds at the ed of each peiod ad the ext cotibutio to the accout is the made. Supeauatio is a commo example of a auity. Hee, people ivest i a fud o a egula basis, the iteest o the ivestmet compouds, while the picipal is added to fo each peiod. The auity is usually set aside fo a peso s etie wokig life ad is used to fud etiemet. It may also be used to fud a log-tem goal, such as a tip i 0 yeas time. To udestad the gowth of a auity, we eed to evise compoud iteest. The compoud iteest fomula is: A = P( whee A is the fial balace, P is the iitial quatity, is the iteest ate pe compoudig peiod ad is the umbe of compoudig peiods. WORKED EXAMPLE Calculate the value of a $5000 ivestmet made at 8% p.a. fo 4 yeas. THINK WRITE Wite the values of P, ad. P = $5000, = 0.08, = 4 2 Wite the fomula. A = P( 3 Substitute values fo P, ad. = $5000 ( Calculate the value of A. = $ A auity takes the fom of a sum of compoud iteest ivestmets. Coside the case of a peso who ivests $000 at 0% p.a. at the ed of each yea fo five yeas. To calculate this, we would eed to calculate the value of the fist $000 that is ivested fo fou yeas, the secod $000 that is ivested fo thee yeas, the thid $000 that is ivested fo two yeas, the fouth $000 that is ivested fo oe yea ad the last $000 that is added to the ivestmet. Chapte 8 Auities ad loa epaymets 249

4 WORKED EXAMPLE 2 Calculate the value of a auity i which $000 is ivested at the ed of each yea at 0% p.a. fo 5 yeas. THINK WRITE ebookplus Tutoial it-2429 Woked example 2 Use the compoud iteest fomula to calculate the amout to which the fist $000 will gow. 2 Use the compoud iteest fomula to calculate the amout to which the secod $000 will gow. 3 Use the compoud iteest fomula to calculate the amout to which the thid $000 will gow. 4 Use the compoud iteest fomula to calculate the amout to which the fouth $000 will gow. 5 Fid the total of the sepaate $000 ivestmets, emembeig to add the fial $000. A = P( = $ = $464.0 A = P( = $ = $33.00 A = P( = $ = $20.00 A = P( = $000. = $00.00 Total value = $464.0 $33.00 $20.00 $00.00 $000 = $605.0 I most cases it is moe pactical to calculate the total value of a auity usig a fomula. The amout to which a auity gows is called the futue value of a auity ad ca be calculated usig the fomula: A= M ( whee M is the cotibutio pe peiod paid at the ed of the peiod, is the iteest ate pe peiod expessed as a decimal, ad is the umbe of deposits. Fo the above example: A= M ( = $ = $ WORKED EXAMPLE 3 Beie ivests $2000 i a etiemet fud at 5% p.a. iteest compouded aually at the ed of each yea fo 20 yeas. Calculate the futue value of this auity at etiemet. THINK WRITE Wite the values of M, ad. M = $2000, = 0.05, = 20 2 Wite the fomula. A= M ( 3 Substitute values fo M, ad. = $ Calculate. = $ Maths Quest Geeal Maths HSC Couse

5 I some examples, calculatios will eed to be made whe cotibutios ae made moe ofte tha oce a yea ad whe iteest compouds moe ofte tha oce a yea. WORKED EXAMPLE 4 Chistia ivests $500 i a fud evey 6 moths at 9% p.a. iteest, compoudig six-mothly fo 0 yeas. Calculate the futue value of the auity afte 0 yeas. THINK WRITE Wite the values of M, ad by cosideig the iteest ate as 4.5% pe iteest peiod ad 20 iteest 9% p.a. = 4.5% fo 6 moths So, = ad = 20. peiods. 2 Wite the fomula. A= M ( 3 Substitute fo M, ad.. = $ Calculate. = $ WORKED EXAMPLE 5 If we eaage the fomula fo a auity to make M (the cotibutio pe peiod the subject of the fomula, we have: A M = ( This fomula would be used whe we kow the fial amout to be saved ad wish to calculate the amout of each egula deposit. Vikki has the goal of savig $0 000 i the ext five yeas. The best iteest ate that she ca obtai is 8% p.a., with iteest compouded aually. Calculate the amout of each aual cotibutio that Vikki must make. THINK WRITE Wite the values of A, ad. A = $0 000, = 0.08, = 5 2 Wite the fomula. A M = ( 3 Substitute fo A, ad. Hit: iset backets whe usig you calculato. = ( ( Calculate the value of M. = $ REMEMBER. 2. The compoud iteest fomula is: A = P( whee A is the fial balace, is the iteest ate pe peiod expessed as a decimal ad is the umbe of compoudig peiods. A auity is a fom of ivestmet whee peiodical equal cotibutios ae made to a accout, with iteest compoudig at the ed of each peiod. Chapte 8 Auities ad loa epaymets 25

6 3. The value of a auity is calculated by addig the value of each amout cotibuted as a sepaate compoud iteest ivestmet. 4. We ca calculate the value of a auity by usig the fomula: A= M ( whee M is the cotibutio pe peiod, paid at the ed of the peiod, is the iteest ate pe peiod expessed as a decimal ad is the umbe of deposits. 5. The amout of each cotibutio to auity to each a cetai goal ca be calculated usig the fomula: A M = ( EXERCISE 8A ebookplus SkillSHEET 8. doc-379 Fidig values of ad i fiacial fomulas ebookplus SkillSHEET 8.2 doc-380 Calculatig simple iteest ebookplus SkillSHEET 8.3 doc-38 Calculatig compoud iteest Futue value of a auity WE Calculate the value afte 5 yeas of a ivestmet of $4000 at 2% p.a., with iteest compouded aually. 2 Calculate the value to which each of the followig compoud iteest ivestmets will gow. a $5000 at 6% p.a. fo 5 yeas, with iteest calculated aually b $2 000 at 2% p.a. fo 3 yeas, with iteest calculated aually c $4500 at 8% p.a. fo 4 yeas, with iteest compouded six-mothly d $3000 at 9.6% p.a. fo 3 yeas, with iteest compouded six-mothly e $5 000 at 8.4% p.a. fo 2 yeas, with iteest compouded quately f $2950 at 6% p.a. fo 3 yeas, with iteest compouded mothly 3 WE 2 At the ed of each yea fo fou yeas Rodey ivests $000 i a ivestmet fud that pays 7.5% p.a. iteest, compouded aually. By calculatig each ivestmet of $000 sepaately, use the compoud iteest fomula to calculate the futue value of Rodey s ivestmet afte fou yeas. 4 Caitli is savig fo a holiday i two yeas ad so evey six moths she ivests $2000 i a accout that pays 7% p.a. iteest, with the iteest compoudig evey six moths. a Use the compoud iteest fomula to calculate the amout to which the: i fist ivestmet of $2000 will gow ii secod ivestmet of $2000 will gow iii thid ivestmet of $2000 will gow iv fouth ivestmet of $2000 will gow. b If Caitli the adds a fial deposit of $2000 to he accout immediately befoe he holiday, what is the total value of he auity? 5 WE 3 Use the fomula A= M ( to fid the futue value of a auity i which $000 is ivested each yea fo 25 yeas at a iteest ate of 8% p.a. 6 Whe baby Shao was bo, he gadpaets deposited $500 i a accout that pays 6% p.a. iteest, compouded aually. They added $500 to the accout each bithday, makig the last deposit o Shao s 2st bithday. a How may deposits of $500 wee made? b The ivestmet was give to Shao as a 2st bithday peset. What was the total value of the ivestmet at this poit? (Hit: Use the aswe to pat a. c Shao s gadpaets advised Shao to keep addig $500 to the ivestmet each bithday so that she had a etiemet fud at age 60. If Shao follows this advice, what will the ivestmet be woth at age 60? (Assume Shao makes the last deposit o he 60th bithday. 252 Maths Quest Geeal Maths HSC Couse

7 7 Calculate the futue value of each of the followig auities. a $2000 ivested at the ed of each yea fo 0 yeas, at a iteest ate of 5% p.a. b $5000 ivested at the ed of each yea fo 5 yeas, at a iteest ate of 8% p.a. c $0 000 ivested at the ed of each yea fo 20 yeas, at a iteest ate of 7.5% p.a. d $500 ivested at the ed of each yea fo 30 yeas, at a iteest ate of 5% p.a. e $ ivested at the ed of each yea fo 4 yeas, at a iteest ate of 9.2% p.a. 8 Dalee is savig fo a deposit o a uit. She hopes to buy oe i fou yeas ad eeds a $ deposit, so she ivests $5000 pe yea i a auity at 7.5% p.a. statig o Jauay a Afte the last deposit is made o Jauay 20, how may deposits has Dalee made? b Use the auity fomula to calculate if Dalee would have saved eough fo he deposit. c How much iteest was paid to Dalee o this auity? 9 WE4 At the ed of evey six moths Jaso ivests $800 i a etiemet fud which pays iteest at 6% p.a., with iteest compouded six-mothly. Jaso does this fo 25 yeas. Calculate the futue value of Jaso s auity afte 25 yeas. 0 Calculate the futue value of each of the followig auities o matuity. a $400 ivested at the ed of evey six moths fo 2 yeas at 2% p.a., with iteest compouded six-mothly b $000 ivested at the ed of evey quate fo 5 yeas at 8% p.a., with iteest compouded evey quate c $2500 ivested at the ed of each quate at 7.2% p.a. fo 4 yeas, with iteest compouded quately d $000 ivested at the ed of evey moth fo 5 yeas at 6% p.a., with iteest compouded mothly MC The iteest eaed o $0 000 ivested at 8% p.a. fo 0 yeas, with iteest compouded aually, is: A $ B $ C $ D $ MC Tacey ivests $500 i a fud at the ed of each yea fo 20 yeas. The fud pays 2% p.a. iteest, compouded aually. The total amout of iteest that Tacey eas o this fud ivestmet is: A $ B $ C $ D $ WE5 Thomas has the goal of savig $ fo his etiemet i 25 yeas. If the best iteest ate that Thomas ca obtai is 0% p.a., with iteest compouded aually, calculate the amout of each aual cotibutio that Thomas will eed to make. 4 Calculate the amout of each aual cotibutio eeded to obtai each of the followig amouts. a $ i 5 yeas at 5% p.a., with iteest compouded aually b $ i 0 yeas at 7.5% p.a., with iteest compouded aually c $ i 40 yeas at 8% p.a., with iteest compouded aually 5 Leae is 24 yeas old ad ivests $30 pe week i he supeauatio fud. Leae s employe matches this amout. a If Leae plas to etie at 60, calculate the total that Leae will cotibute to the fud at this ate. b Calculate the total cotibutios that will be made to the fud at this ate. c If the fud etus a aveage 4% p.a. iteest, compouded aually, calculate the futue value of Leae s supeauatio. Chapte 8 Auities ad loa epaymets 253

8 6 a Fid the amout of each aual cotibutio eeded to achieve a futue value of $ if the ivestmet is made fo 0 yeas at a iteest ate of % p.a., with iteest compoudig aually. b Fid the amout of each quately cotibutio eeded to save $5 000 i five yeas at 2% p.a., with iteest compoudig quately. c Fid the amout of each six-mothly cotibutio to a auity if the savigs goal is $ i 5 yeas ad the iteest ate is 8% p.a., with iteest compoudig six-mothly. Futhe developmet 7 Simo ivests $2000 pe yea ito a auity that pays 8% p.a. ove a 0-yea peiod. a Calculate the futue value of Simo s auity give that iteest is compouded aually. b Calculate the futue value of the auity if Simo wee to make six-mothly deposits of $000 ad iteest wee to be compouded six-mothly. c Calculate the pecetage icease i the futue value of the auity if paymets ae made ad iteest compouded six-mothly. Give you aswe coect to decimal place. 8 Stephaie ivests $200 pe yea ito a auity at 6% p.a. with iteest compouded aually. a If Stephaie plas to ivest each yea fo 20 yeas, fid the futue value of the auity. b Afte 0 yeas Stephaie fids that she is able to icease he aual paymet to $500. By cosideig the exta $300 as a sepaate auity fo the secod 0 yeas, fid the total value of the auity afte 20 yeas. 9 Achie ivests $2000 pe yea ito a auity. The auity pays 6.5% p.a. iteest with iteest compouded aually. a Calculate the futue value of the auity afte 0 yeas. b At the coclusio of 0 yeas Achie is able to ivest at the highe ate of 7.2% p.a. with iteest compouded aually. Achie takes his auity ad ivests the etie sum fo a futhe 0 yeas at the highe ate. Calculate the value of this ivestmet at matuity. c At the same time Achie cotiues to put $2000 pe yea ito the auity at the highe ate. Calculate the futue value of this auity afte a futhe 0 yeas. d Calculate the total value of Achie s auity at the coclusio of 20 yeas. 20 Jodie ivests $200 pe moth ito he supeauatio fud. The iteest ate is 6% p.a. with iteest compouded mothly. a Calculate the futue value of Jodie s ivestmet fo he wokig life of 40 yeas. b Calculate the futue value of the ivestmet if he employe matches Jodie s cotibutio dolla fo dolla. 2 Jeff eas a aual salay of $ Jeff cotibutes $57 pe week ito his supeauatio fud ad his employe makes a cotibutio of 3% of his salay. a Calculate the total weekly cotibutio to Jeff s supeauatio fud. b If the supeauatio fud eas a aveage 7.8% p.a. iteest compouded weekly, calculate the value of Jeff s ivestmet at etiemet afte 45 yeas of wokig. 22 Toy ivests $000 pe yea ito a auity payig 9% p.a. iteest compouded aually. a Calculate the futue value of Toy s auity afte 5 yeas. b Fo the secod paymet Toy iceases his paymet to $200. By cosideig the exta $200 as a sepaate auity that us fo 4 yeas, calculate the amout by which the futue value will icease. c Toy iceases each paymet by $200 fo the full five yeas. Calculate the value of his ivestmet afte 5 yeas. 254 Maths Quest Geeal Maths HSC Couse

9 ebookplus EXCEL Speadsheet doc-382 Auity calculato Compute Applicatio : Auity calculato Access the speadsheet Auity calculato fom the Maths Quest Geeal Mathematics HSC Couse ebookplus. The speadsheet will show you the gowth of a auity i which $000 is ivested at the ed of each yea fo 20 yeas at a ate of 8% p.a. iteest, compoudig aually. 8B. The speadsheet shows that afte 20 yeas the value of this ivestmet is $ Below is the gowth of the auity afte each deposit is made. This will allow you to see the gowth fo up to 30 deposits. Fom the Edit meu, use the Fill Dow fuctios o the speadsheet to see futhe. 2. Click o the tab, Chat. This is a lie gaph that shows the gowth of the auity fo up to 30 deposits. 3. Chage the size of the deposit to $500 ad the compoudig peiods to 2. This will show how much beefit ca be achieved by educig the compoudig peiod. 4. Check you aswes to the pevious execise by usig the speadsheet. Peset value of a auity To compae a auity with a sigle sum ivestmet, we eed to use the peset value of the auity. The peset value of a auity is the sigle sum of moey that, ivested o the same tems as the auity, will poduce the same fiacial esult. Chapte 8 Auities ad loa epaymets 255

10 To calculate the peset value of a auity, N, we ca use the fomula: A N = ( whee A is the futue value of the auity is the pecetage iteest ate pe compoudig peiod, expessed as a decimal is the umbe of deposits to be made i the auity. WORKED EXAMPLE 6 Asha has a auity that has a futue value of $ o his etiemet i 23 yeas. The auity is ivested at 8% p.a., with iteest compouded aually. Calculate the peset value of Asha s auity. THINK WRITE Wite the values of A, ad. A = $ , =.08, = 23 2 Wite the fomula. A N = ( 3 Substitute fo A, ad = Calculate. = $ I may cases you will ot kow the futue value of the auity whe calculatig the peset value. You will kow oly the amout of each cotibutio, M. We kow that: A N = ( Usig the fomula A= M ( to substitute fo A gives: N = M ( ( This fomula allows us to calculate the sigle sum eeded to be ivested to give the same fiacial esult as a auity whee we ae give the size of each cotibutio. WORKED EXAMPLE 7 Jey has a auity to which she cotibutes $000 pe yea at 6% p.a. iteest, compouded aually. The auity will matue i 25 yeas. Calculate the peset value of the auity. THINK WRITE Wite the values of M, ad. M = $000, = 0.06, = 25 2 Wite the fomula. N = M ( ( 3 Substitute fo M, ad = Calculate. = $ Maths Quest Geeal Maths HSC Couse

11 This peset value fomula ca be used to compae ivestmets of diffeet types. The ivestmet with the geate peset value will poduce the geate fiacial outcome ove time. WORKED EXAMPLE 8 Which of the followig ivestmets would give the geate fiacial etu? Ivestmet A: a auity of $00 deposited pe moth fo 20 yeas at 2% p.a. iteest, compoudig six-mothly Ivestmet B: a sigle deposit of $0 000 ivested fo 20 yeas at 2% p.a., with iteest compoudig six-mothly ebookplus Tutoial it-2430 Woked example 8 THINK WRITE The ivestmets ca be compaed by calculatig the peset value of the auity. 2 Coside the deposits of $00 pe moth to be $600 evey six moths. 3 Wite the values of M, ad. M = $600, = 0.06, = 40 4 Wite the fomula. 5 Substitute fo M, ad. N = M ( ( 6 Calculate. = $ = $ Make a coclusio. The auity has a lowe peset value tha the sigle ivestmet. Theefoe, the ivestmet of $0 000 will poduce a geate outcome ove 20 yeas. REMEMBER. The peset value of a auity is the sigle sum that ca be ivested ude the same tems as a auity ad will poduce the same fiacial outcome. 2. The peset value of a auity ca be calculated usig the fomula: A N = ( whe we kow the futue value of the auity. 3. If we kow the amout of each cotibutio of the auity, we ca calculate the peset value usig the fomula N = M ( ( whee M is the cotibutio pe peiod, paid at the ed of the peiod is the pecetage iteest ate pe compoudig peiod (expessed as a decimal is the umbe of iteest peiods. 4. Ivestmets ca be compaed usig the peset value fomula. The ivestmet with the geate peset value will poduce the geate fiacial outcome ove time. Chapte 8 Auities ad loa epaymets 257

12 EXERCISE 8B Peset value of a auity WE6 Calculate the peset value of a ivestmet that is eeded to have a futue value of $ i 30 yeas time if it is ivested at 9% p.a., with iteest compouded aually. 2 Calculate the peset value of a ivestmet equied to geeate a futue value of: a $ i 5 yeas time at 0% p.a., with iteest compouded aually b $5000 i 4 yeas time at 7.2% p.a., with iteest compouded aually c $ i 20 yeas time at 5% p.a., with iteest compouded aually. 3 Calculate the peset value of a ivestmet at 7.2% p.a., with iteest compouded quately, if it is to have a futue value of $ i 0 yeas time. 4 Calculate the peset value of the ivestmet equied to poduce a futue value of $ i 30 yeas time at 9% p.a., with iteest compouded: a aually b six-mothly c quately d mothly. 5 WE 7 Caig is payig ito a auity a amout of $500 pe yea. The auity is to u fo 0 yeas ad iteest is paid at 7% p.a., with iteest compouded aually. Calculate the peset value of this auity. 6 Calculate the peset values of each of the followig auities. a $000 pe yea fo 30 yeas at 8% p.a., with iteest compouded aually b $600 pe yea fo 20 yeas at 7.5% p.a., with iteest compouded aually c $4000 pe yea fo 5 yeas at % p.a., with iteest compouded aually d $200 pe moth fo 25 yeas at 8.4% p.a., with iteest compouded aually 7 Dae pays $250 pe moth ito a auity that pays 5.6% p.a. iteest, compouded quately. If the auity is to u fo 0 yeas, calculate the peset value of the auity. 8 Calculate the peset value of a 40-yea auity with iteest at 9.6% p.a., compouded mothly, if the mothly cotibutio to the auity is $50. 9 MC A auity is at 2% p.a. fo 0 yeas, with iteest compouded six-mothly, ad has a futue value of $ The peset value of the auity is: A $ B $ C $ D $ MC A auity cosists of quately deposits of $200 that ae ivested at 8% p.a., with iteest compouded quately. The auity will matue i 23 yeas. The peset value of the auity is: A $ B $ C $ D $ WE8 Which of the followig ivestmets will have the geate fiacial outcome? Ivestmet A: a auity of $400 pe yea fo 30 yeas at 6.9% p.a., with iteest compouded aually Ivestmet B: a sigle ivestmet of $5000 fo 30 yeas at 6.9% p.a., with iteest compouded aually 2 MC Which of the followig ivestmets will have the geatest fiacial outcome? A A auity of $200 pe yea fo 30 yeas at 8% p.a., with iteest compouded aually B A auity of $600 evey six moths fo 30 yeas at 7.9% p.a., with iteest compouded six-mothly C A auity of $300 evey quate fo 30 yeas at 7.8% p.a., with iteest compouded quately D A auity of $00 pe moth at 7.5% p.a., fo 30 yeas with iteest compouded mothly 258 Maths Quest Geeal Maths HSC Couse

13 3 Kylie wats to take a wold tip i 5 yeas time. She estimates that she will eed $ fo the tip. The best ivestmet that Kylie ca fid pays 9.2% p.a. iteest, compouded quately. a Calculate the peset value of the ivestmet eeded to achieve this goal. b Kylie plas to save fo the tip by depositig $00 pe week ito a auity. Calculate if this will be eough fo Kylie to achieve he savigs goal (take 3 weeks = quate. ebookplus WokSHEET 8. doc-383 Futhe developmet 4 A ivestmet is to gow to $ ove 25 yeas at a aveage iteest ate of 6% p.a. a Calculate the peset value of the auity if iteest is compouded aually. b Calculate the peset value if iteest is compouded quately. c Explai why the lowe peset value idicates that the quately ivestmet is bette. d Calculate the pecetage eductio i the peset value fo whe the iteest is compouded quately. 5 A auity of $2400 is paid evey yea ito a ivestmet payig 9% p.a. ove a 20-yea peiod. a Calculate the peset value of the auity if iteest is compouded aually. b Calculate the peset value of the auity if $200 is paid pe moth ad iteest is compouded mothly. c Calculate the pecetage chage i the peset value if iteest is compouded mothly. 6 Adew makes a ivestmet of $000 each yea ito a auity at 8% p.a. fo 0 yeas. Joh obtais 0% p.a. iteest ad claims the peset value of the auity will be 25% less. Detemie if Joh is coect. 7 Ricky makes a ivestmet of $300 pe quate ito a auity that pays 6.6% p.a. with iteest compouded quately. The auity is to u fo 20 yeas. a Fid the futue value of the auity. b Calculate the peset value of the auity. 8 Fioa ivests $800 pe yea fo 30 yeas ito a auity that pays 7.2% p.a. with iteest compouded aually. a Calculate the futue value of the auity. b Calculate the peset value of the auity. c Calculate the peset value of a equivalet auity whee cotibutios ae made, ad iteest is compouded, mothly. 9 A auity that is ivested at 6.9% p.a. ove 0 yeas with iteest compouded aually has a peset value of $ Calculate the amout of each aual cotibutio. Chapte 8 Auities ad loa epaymets 259

14 8C Futue ad peset value tables Poblems associated with auities ca be simplified by ceatig a table that will show eithe the futue value o peset value of a auity of $ ivested pe iteest peiod. Compute Applicatio 2: Futue value of $ Coside $ is ivested ito a auity each iteest peiod. The table we ae goig to costuct o a speadsheet shows the futue value of that $.. Ope a ew speadsheet. 2. Type i the followig ifomatio as show i step I cell B4 ete the fomula =((B$3^$A4-/B$3. (This is the futue value fomula fom execise 2A with the value of M omitted, as it is equal to. Fomat the cell, coect to 4 decimal places. 4. Highlight the age of cells B3 to M23. Fom the Edit meu, use Fill Dow ad Fill Right fuctios to copy the fomula to all othe cells i this age. This completes the table. The table shows the futue value of a auity of $ ivested fo up to 0 iteest peiods at up to 0% pe iteest peiod. You ca exted the speadsheet futhe fo othe iteest ates ad loge ivestmet peiods. The followig table is the set of futue values of $ ivested ito a auity. This is the table you should have obtaied i compute applicatio Maths Quest Geeal Maths HSC Couse

15 Peiod A table such as this ca be used to fid the value of a auity by multiplyig the amout of the auity by the futue value of $. Futue value of $ Iteest ate pe peiod % 2% 3% 4% 5% 6% 7% 8% 9% 0% % 2% WORKED EXAMPLE 9 Use the table to fid the futue value of a auity ito which $500 is deposited at the ed of each yea at 7% p.a. iteest, compouded aually fo 9 yeas. THINK WRITE Look up the futue value of $ at 7% p.a. fo 9 yeas. Futue value = $ Multiply this value by 500. = $7 967 Just as we have a table fo the futue value of a auity, we ca ceate a table fo the peset value of a auity. Compute Applicatio 3: Peset value table The table we ae about to make o a speadsheet shows the peset value of a auity of $ ivested pe iteest peiod.. Ope a ew speadsheet. 2. Ete the followig ifomatio. 3. I cell B4 type the fomula =((B$3^$A4/(B$3*(B$3^$A4. 4. Dag fom cell B4 to K3, ad the fom the Edit meu use the Fill Dow ad Fill Right fuctios to copy this fomula to the emaiig cells i you table. Chapte 8 Auities ad loa epaymets 26

16 The table ceated i compute applicatio 3 shows the peset value of a auity of $ pe iteest peiod fo up to 0% pe iteest peiod ad fo up to 0 iteest peiods. The table that you have geeated is show below. Peset value of $ Iteest ate pe peiod Peiod % 2% 3% 4% 5% 6% 7% 8% 9% 0% % 2% (cotiued 262 Maths Quest Geeal Maths HSC Couse

17 Peiod % 2% 3% 4% 5% 6% 7% 8% 9% 0% % 2% This table ca be used i the same way as the futue values table. WORKED EXAMPLE 0 Liam ivests $750 pe yea i a auity at 6% pe aum fo 8 yeas, with iteest compouded aually. Use the table to calculate the peset value of Liam s auity. THINK Use the table to fid the peset value of a $ auity at 6% fo 8 iteest peiods. WRITE 2 Multiply this value by 750. Peset value = $ = $ REMEMBER A table of futue values shows the futue value of a auity i which $ is ivested pe iteest peiod. A table of peset values shows the peset value of a auity i which $ is ivested pe iteest peiod. A table of peset o futue values ca be used to compae ivestmets ad detemie which will give the geate fiacial etu. EXERCISE 8C ebookplus SkillSHEET 8.4 doc-383 Readig fiacial tables Futue ad peset value tables WE9 Use the table of futue values o page 25 to detemie the futue value of a auity of $800 ivested pe yea fo 5 yeas at 9% p.a., with iteest compouded aually. 2 Use the table of futue values to detemie the futue value of each of the followig auities. a $400 ivested pe yea fo 3 yeas at 0% p.a., with iteest compouded aually b $2250 ivested pe yea fo 8 yeas at 8% p.a., with iteest compouded aually c $625 ivested pe yea fo 0 yeas at 4% p.a., with iteest compouded aually d $7500 ivested pe yea fo 7 yeas at 6% p.a., with iteest compouded aually 3 Samatha ivests $500 evey 6 moths fo 5 yeas i a auity at 8% p.a., with iteest compouded evey 6 moths. a What is the iteest ate pe iteest peiod? b How may iteest peiods ae thee i Samatha s auity? c Use the table to calculate the futue value of Samatha s auity. Chapte 8 Auities ad loa epaymets 263

18 4 Use the table to calculate the futue value of each of the followig auities. a $400 ivested evey 6 moths fo 4 yeas at 4% p.a., with iteest compouded six-mothly b $600 ivested evey 3 moths fo 2 yeas at 2% p.a., with iteest compouded quately c $00 ivested evey moth fo 5 yeas at 0% p.a., with iteest compouded six-mothly 5 Use the table of futue values to detemie whethe a auity at 5% p.a. fo 6 yeas o a auity at 6% p.a. fo 5 yeas will poduce the geatest fiacial outcome. Explai you aswe. 6 MC Use the table of futue values to detemie which of the followig auities will have the geatest fiacial outcome. A 6% p.a. fo 8 yeas, with iteest compouded aually B 8% p.a. fo 6 yeas, with iteest compouded aually C 7% p.a. fo 7 yeas, with iteest compouded aually D 0% p.a. fo 5 yeas, with iteest compouded six-mothly 7 WE 0 Use the table of peset values o page 252 to detemie the peset value of a auity of $250 pe yea fo 8 yeas ivested at 9% p.a. 8 Use the table of peset values to detemie the peset value of each of the followig auities. a $450 pe yea fo 5 yeas at 7% p.a., with iteest compouded aually b $2000 pe yea fo 0 yeas at 0% p.a., with iteest compouded aually c $850 pe yea fo 6 yeas at 4% p.a., with iteest compouded aually d $3000 pe yea fo 8 yeas at 9% p.a., with iteest compouded aually Futhe developmet 9 Babaa cuetly has $ i a ivestmet accout that is aveagig a iteest ate of 6% p.a. compouded aually. She wats to calculate the amout that she will eceive afte 20 yeas. a Use the compoud iteest fomula to fid the value of the $ ivestmet at matuity. b If she deposits $9000 each yea, use the table o page 000 to fid the futue value of this auity. c If she places he $ ito a diffeet savigs accout that ca offe 8% p.a. compouded aually ad iceases he deposits to $0 000 each yea, fid the amout available to he afte 20 yeas. d Ove the 20-yea peiod, calculate the exta amout saved by ivestig $ i a ivestmet accout ad $0 000 each yea at 8% p.a. compaed with $9000 each yea at 6% p.a. 0 Fid the fial value of the followig auity ivestmets by usig the compoud iteest fomula togethe with the table o page 26. a A iitial amout of $0 000 eaig 6% p.a. with aual cotibutios of $7000 fo the ext 20 yeas b A iitial amout of $ eaig 0% p.a. with aual cotibutios of $2 000 fo the ext 5 yeas c A iitial amout of $ eaig 8% p.a. compouded six-mothly with six-mothly cotibutios of $200 fo the ext 9 yeas d A iitial amout of $0 eaig 2% p.a. compouded quately with quately cotibutios of $200 fo the ext 5 yeas Justi is aged 42 ad is plaig to etie at 60 yeas of age. He estimates that he eeds $ to povide fo his etiemet. His cuet supeauatio fud has a balace of $ ad is deliveig 2% p.a. compouded aually. a Calculate the value of the $ whe Justi eaches etiemet age. b Use the table o page 000 to fid the aual cotibutios (coect to the eaest $0 eeded to meet the etiemet lump-sum taget. 264 Maths Quest Geeal Maths HSC Couse

19 2 Johy has a auity of $500 pe yea at 5.5% p.a. iteest compouded aually ove a 20-yea peiod. a Use the table o page 26 to estimate the futue value of a $ auity ove a 20-yea peiod. b Use you aswe to (a to fid the futue value of Johy s auity. c Use the table o pages to estimate the peset value of the auity. 3 Jodie has a auity ito which she pays $800 evey quate. The iteest ate is 0% p.a. with iteest compouded quately. The auity is to u fo 4 yeas. a Use the table to estimate the futue value of the auity. b Jodie s fiacé Paul wats to make a sigle ivestmet with a futue value equal to Jodie s auity. Fid the amout of the ivestmet that Paul must make. 4 Fid the cotibutios equied to meet the followig supeauatio goals. a A fial payout of $ with 0 yeas to go, with a supeauatio fud deliveig 6% p.a. compouded aually. Cotibutios ae to be paid oce a yea. b A fial payout of $ with 20 yeas to go. The cotibutio is paid mothly ito a fud aveagig 9% p.a. compouded aually. c A fial payout of $ millio with 20 yeas to go. The cotibutio is paid mothly ito a fud aveagig 2% p.a. compouded aually. 8D Loa epaymets Whe a loa is take out ad is epaid i equal mothly istalmets, the patte of epaymets woks simila to a auity. Each moth iteest compouds o the balace owig o the loa ad the a epaymet is made. Coside a loa whee the amout boowed is equal to the peset value of the auity, N, ad the amout paid o the loa each moth is equal to the cotibutio to the auity pe peiod, M. Use the fomula fo peset value, N = M (. To calculate the amout ( of each mothly epaymet, we eed to make M the subject of this fomula. Whe we do this the fomula becomes: M = N ( ( I this fomula, M is the amout of each epaymet, N is the amout boowed, is the iteest ate pe epaymet peiod as a decimal ad is the umbe of epaymets to be made. This fomula is ot give to you o the fomula sheet but will be give to you if it is eeded to solve a poblem i the exam. WORKED EXAMPLE Use the fomula M = N ( to calculate the mothly epaymets o a loa of $5000 to be ( epaid i mothly istalmets ove 4 yeas at a iteest ate of 2% p.a. THINK WRITE Calculate the values of ad. = 0.0 ad = 48 2 Wite the fomula. M = N ( ( Chapte 8 Auities ad loa epaymets 265

20 3 Substitute fo N, ad. = Calculate. = $3.67 Havig woked out the amout of each mothly epaymet, we ae also able to calculate the total cost of epayig a loa by multiplyig the amout of each epaymet by the umbe of epaymets. WORKED EXAMPLE 2 Calculate the total cost of epayig a $ home loa at 9% p.a. i equal mothly epaymets ove a 25-yea tem. THINK WRITE Calculate the values of ad. = , = Wite the fomula. M = N ( ( 3 Substitute fo N, ad = Calculate the amout of each mothly epaymet. = $ Calculate the total epaymets o the loa. Total epaymets = $ = $ By iceasig the amout of each epaymet, we ae able to shote the tem of the loa. Thee is o easy method to calculate the amout of time that it will take to epay a loa. To do this we use a guess ad efie method. We adjust the value of i the fomula util the amout of the epaymet is eached. WORKED EXAMPLE 3 A $ home loa is take out ove a 25-yea tem at a iteest ate of 2% p.a. educible iteest. The miimum mothly epaymet o the loa is $ How log will it take the loa to be epaid at $200 pe moth? THINK WRITE Calculate the value of. = Wite the fomula. M = N ( ( 3 Take a guess fo the value of (we will take 200 sice fo the oigial loa = 300 ad substitute. 4 Calculate the epaymet with = 200. As this is less tha $200 we eed to futhe educe the value of. ebookplus Tutoial it-243 Woked example 3 If = 200, M = = $ Substitute ito the fomula with = 50. If = 50, M = Maths Quest Geeal Maths HSC Couse

21 6 Calculate the epaymet. As the esult is geate tha $200, we eed to icease the value of. = $ Substitute ito the fomula with = 80. If = 80, M = As this is appoximately equal to $200, it will take 80 moths to epay the loa. = $ Give a witte aswe. It will take 5 yeas to epay the loa. REMEMBER. By cosideig the amout boowed i a loa as the peset value of a auity, we ca use the peset value fomula to calculate the amout of each epaymet. 2. The fomula used to calculate the amout of each mothly epaymet is: M = N ( ( whee N is the amout boowed, is the iteest ate pe peiod expessed as a decimal ad is the umbe of iteest peiods. 3. The total cost of a loa ca be calculated by multiplyig the amout of each epaymet by the umbe of epaymets to be made. 4. The legth of time that it will take to epay a loa ca be calculated by usig guess ad efie methods. EXERCISE 8D Loa epaymets Fo questios to 3 use the fomula, M = N (. ( Yiais takes out a $0 000 loa ove 5 yeas at 0% p.a. educible iteest with five equal aual epaymets to be made. Use the fomula to calculate the amout of each aual epaymet. 2 WE Use the fomula to calculate the amout of each mothly epaymet o a loa of $8000 to be epaid ove 4 yeas at 2% p.a. 3 Use the fomula to calculate the amout of each mothly epaymet o each of the followig loas. a $2000 at 2% p.a. ove 2 yeas b $5 000 at 9% p.a. ove 5 yeas c $ at 6% p.a. ove 20 yeas d $ at 9.6% p.a. ove 5 yeas e $ at 7.2% p.a. ove 25 yeas 4 Javie ad Diae take out a $ home loa. If the iteest ate o the loa is 8.4% p.a. educible ad the tem of the loa is 25 yeas, calculate the amout of each mothly epaymet. 5 Jio puchases a compute o tems. The cash pice of the compute is $3750. The tems ae a deposit of 0% with the balace paid i equal mothly istalmets at 9% p.a. educible iteest ove 3 yeas. a Calculate Jio s deposit o the compute. b What is the balace owig o the compute? c Calculate the amout of each mothly epaymet. Chapte 8 Auities ad loa epaymets 267

22 ebookplus WokSHEET 8.2 doc Jeemy ad Paticia sped $5 000 o ew fuishigs fo thei home. They pay a 5% deposit o the fuishigs with the balace paid i equal mothly istalmets at 8% p.a. iteest ove 4 yeas. Calculate the amout of each mothly epaymet. 7 Thah is puchasig a ca o tems. The cash pice of the ca is $ ad he pays a $7000 deposit. a What is the balace owig o the ca? b If the ca is to be epaid i equal weekly istalmets ove 5 yeas at a iteest ate of 0.4% p.a. educible iteest, calculate the amout of each weekly paymet. 8 WE 2 Ro boows $3 500 to puchase a ca. The loa is to be epaid i equal mothly istalmets ove a 3-yea tem at a iteest ate of 5% p.a. Calculate the total epaymets made o the loa. 9 Calculate the total epaymets o each of the followig loas. a $4000 at 8.4% p.a. educible iteest to be epaid ove 2 yeas i equal mothly epaymets b $ at 3.2% p.a. educible iteest to be epaid ove 6 yeas i equal mothly istalmets c $ at 7.2% p.a. educible iteest to be epaid ove 5 yeas i equal mothly istalmets d $ at 0.8% p.a. educible iteest to be epaid ove 20 yeas i equal mothly istalmets 0 MC A loa of $5000 is take out at 9% p.a. educible iteest ove 4 yeas. Which of the followig will give the amout of each mothly epaymet? A M = C M = B M = D M = MC A loa of $2 000 is take out at 2% p.a. educible iteest i equal mothly istalmets ove 5 yeas. The total amout of iteest paid o the loa is: A $ B $ C $7200 D $ WE 3 A loa of $ is take out ove 5 yeas at 9% p.a. educible iteest. The miimum mothly epaymet is $ Calculate how log it will take to epay the loa at $000 pe moth. 3 A $ loa is take out ove a 25-yea tem. The iteest ate is 9.6% p.a. a Calculate the miimum mothly epaymet. b Calculate the total epaymets o the loa. c Calculate the legth of time that it will take to epay the loa at $600 pe moth. d Calculate the total savig o the loa by epayig the loa at $600 pe moth. Futhe developmet 4 Lik boows $48 000, take out ove 0 yeas. The loa is to be epaid i mothly istalmets. Calculate the amout of each mothly epaymet at each of the followig iteest ates. a 6% p.a. b 7% p.a. c 9% p.a. d 2 % p.a. 5 A loa of $ ove 20 yeas has iteest chaged mothly at a ate of 9% p.a. a Calculate the mothly epaymet. b Afte 3 yeas the balace owig is $ The iteest ate the ises to 0%. What will be the ew epaymet equied to pay the loa off i the emaiig 7 yeas? Maths Quest Geeal Maths HSC Couse

23 6 Be took out of loa fo $ to buy a ew ca. The cotact equied that he epay the loa ove 5 yeas with mothly istalmets of $ Use a tial-ad-eo method togethe with the loa epaymet fomula to fid the iteest ate that Be is payig. 7 Tim has boowed $ to buy a house. He agees to epay the educig balace loa ove 5 yeas with mothly istalmets at 9.3% p.a. (adjusted mothly. Fid: a the mothly epaymet b the picipal epaid ad the iteest paid duig the fist epaymet. 8 Tey is epayig a $ loa ove 5 yeas with quately istalmets at 6.25% p.a. (adjusted quately. a Calculate the amout of each quately epaymet. b Calculate the total amout epaid o the loa. c Calculate the total iteest paid o the loa. d Show whethe the loa would have cost moe o less, ad by how much, had the epaymets bee made mothly ad iteest calculated mothly. 9 Stefaie boowed $8 000 fo a ca. The educig balace loa at 8% p.a. was fo a tem of 5 yeas ad ca be paid i aual, quately o mothly epaymets. The iteest will be calculated ad added at the same time as each epaymet. Fid the cheapest way to epay the loa. Types of loa aagemets Reseach oe example of each of the followig types of loas. A. Hie puchase ageemet This is the type of loa whee a majo item is puchased o tems. Usually a deposit is paid ad the the balace plus iteest is epaid ove a ageed peiod of time. B. Pesoal loa This is a loa take out fom a bak o othe fiacial istitutio. It ca be used fo ay pupose ad is usecued. This meas that thee is o item of popety that the bak ca claim if epaymets ae ot made. C. Home loa This is a secued loa, which meas that, if the epaymets ae ot made, the bak ca claim the popety ad sell it to eclaim the amout outstadig o the loa. Fo each of the above loas, aswe the followig questios. What is the iteest ate? Is iteest calculated at a flat o educible ate? 2 Ove what tem ca the loa be epaid? 3 How egulaly must epaymets be made? 4 Ca additioal epaymets be made to shote the tem of the loa? 5 Ca the iteest ate be alteed afte epaymets have begu to be made? 6 What othe fees ad chages apply to boowig the moey? ebookplus WORD doc-3330 Auities loas, gaphs Compute Applicatio 4: Gaphs of auities ad loas Most fiacial istitutios will povide gaphs that show the gowth of a auity ad the decliig balace of a loa. These gaphs ca be obtaied by eithe visitig the bak o by goig to the iteet site fo the elevat fiacial istitutio. Obtai a copy of a gaph showig the gowth of a ivestmet ad the decliig balace of a loa. Alteatively, develop a speadsheet that shows the gowth of a auity ad the decliig balace of a loa ad use the chatig fuctio of the speadsheet to daw the gaph. Access the Wod file Auities, Loas, Gaphs fom the Maths Quest Geeal Mathematics HSC Couse ebookplus. Chapte 8 Auities ad loa epaymets 269

24 SUMMARY Futue value of a auity A auity is whee egula equal cotibutios ae made to a ivestmet. The iteest o each cotibutio compouds as additios ae made to the auity. The futue value of a auity is the value that the auity will have at the ed of a fixed peiod of time. The futue value of a auity ca be calculated usig the fomula: A= M ( whee M is the cotibutio pe peiod paid at the ed of the peiod, is the pecetage iteest ate pe compoudig peiod (expessed as a decimal ad is the umbe of compoudig peiods. A The amout of each cotibutio pe peiod i a auity ca be foud usig the fomula M =. ( Peset value of a auity The peset value of a auity is the sigle sum that would eed to be ivested at the peset time to give the same fiacial outcome at the ed of the tem. The peset value of a auity ca be calculated usig the fomula: whee A is the futue value of the auity. A alteative fomula to use is: A N = ( N = M ( ( whee M is the cotibutio made to the auity pe iteest peiod. Use of tables A table ca be used to fid the peset o futue value of a auity. The table shows the peset o futue value of $ ude a auity. The peset o futue value of $ must be multiplied by the cotibutio pe peiod to calculate its peset o futue value. Loa epaymets The peset value of a auity fomula ca be used to calculate the amout of each peiodical epaymet i a educig balace loa. This is doe by cosideig the peset value of a auity as the amout boowed ad makig M the subject of the fomula. The fomula to be used is M = N (. ( The total amout to be epaid duig a loa is calculated by multiplyig the amout of each mothly epaymet by the umbe of epaymets to be made. 270 Maths Quest Geeal Maths HSC Couse

25 CHAPTER REVIEW MULTIPLE CHOICE Jey ivests $000 pe yea fo 20 yeas i a auity. The iteest ate is 6.5% p.a. ad iteest is compouded aually. The futue value of the auity is: A $ B $ 08.5 C $ D $ Madelie ivests $000 pe yea fo 20 yeas i a auity. The iteest ate is 6.5% p.a. ad iteest is compouded aually. The peset value of the auity is: A $ B $ 08.5 C $ D $ Which of the followig ivestmets has the geatest futue value afte 0 yeas? A A auity of $500 pe yea at 7.75% p.a., with iteest compouded aually B A auity of $250 pe six moths at 7.6% p.a., with iteest compouded six-mothly C A auity of $25 pe quate at 7.2% p.a., with iteest compouded quately D A sigle ivestmet of $3400 at 7.9% p.a., with iteest compouded aually 4 A loa of $ is take out ove a 20-yea tem at a iteest ate of 9% p.a. The mothly epaymet is $ What would the total savig be if the tem wee educed to 5 yeas? A $9.63 B $ C $ D $ SHORT ANSWER Calculate the amout to which each of the followig ivestmets will gow. a $3500 at 2% p.a. fo 3 yeas, with iteest compouded aually b $2000 at 8% p.a. fo 5 yeas, with iteest compouded six-mothly c $5 000 at 9.2% p.a. fo 8 yeas, with iteest compouded quately d $4200 at 3.2% p.a. fo 2 yeas, with iteest compouded mothly 2 $400 pe yea is ivested ito a auity at 7% p.a., with iteest compouded aually. Use the fomula A= M ( to calculate the value of the auity afte 20 yeas. 3 Use the fomula A= M ( to calculate the futue value of each of the followig auities. a $500 ivested pe yea fo 25 yeas at 2% p.a., with iteest compouded aually b $000 ivested evey 6 moths fo 0 yeas at 9% p.a., with iteest compouded six-mothly c $600 ivested evey 3 moths fo 5 yeas at 7.2% p.a., with iteest compouded quately d $250 ivested pe moth fo 20 yeas at 2% p.a., with iteest compouded mothly 4 A auity cosists of $00 deposits evey moth fo 5 yeas. The iteest ate is 9% p.a. ad iteest is compouded six-mothly. Fid the futue value of the auity. A 5 Use the fomula M = to calculate the ( amout of each aual cotibutio to a auity to achieve a savigs goal of $ i 40 yeas at a iteest ate of 8% p.a., with iteest compouded aually. 6 Calculate the amout of each cotibutio to the followig auities. a $ i 0 yeas at 6% p.a., with iteest compouded aually ad aual deposits b $ i 30 yeas at 2% p.a., with iteest compouded six-mothly ad cotibutios made evey six moths c $ i 20 yeas at 6% p.a., with iteest compoudig quately ad cotibutios made quately A 7 Use the fomula N = to calculate the ( peset value of a auity if it is to have a futue value of $ i 30 yeas time at a iteest ate of 0% p.a., with iteest compouded aually. 8 Calculate the peset value of the followig auities with a futue value of: a $0 000 afte 0 yeas at 5% p.a., with iteest compouded aually b $ afte 40 yeas at 2% p.a., with iteest compouded aually c $5000 afte 5 yeas at 9% p.a., with iteest compouded six-mothly d $ afte 8 yeas at 5% p.a., with iteest compouded quately. Chapte 8 Auities ad loa epaymets 27

26 9 Phuog wats to puchase a ca i 3 yeas. He feels that he will eed $ The best ivestmet he ca fid is at 8.5% p.a., iteest compouded quately. What is the peset value of this ivestmet? 0 Gayle ivests $400 pe yea i a auity. The ivestmet is at 6% p.a., with iteest compouded aually. Gayle plas to ivest i the auity fo 25 yeas. Use the fomula N = M ( to ( calculate the peset value of this auity. Whe Joae begis wok at 8, she ivests $00 pe moth i a etiemet fud. The ivestmet is at 9% p.a., with iteest compouded six-mothly. a If Joae is to etie at 60 yeas of age, what is the futue value of he auity? b What is the peset value of this auity? 2 Use the table of futue values of $ o page 26 to calculate the futue value of a auity of $4000 deposited pe yea at 7% p.a. fo 8 yeas, with iteest compouded aually. 3 Use the table of futue values of $ to calculate the futue value of the followig auities. a $750 ivested pe yea fo 5 yeas at 8% p.a., with iteest compouded aually b $3500 ivested evey six moths fo 4 yeas at 2% p.a., with iteest compouded six-mothly c $200 ivested evey 3 moths fo 2 yeas at 6% p.a., with iteest compouded quately d $250 ivested evey moth fo 3 yeas at 0% p.a., with iteest compouded six-mothly 4 Use the table of peset values of $ o pages to calculate the peset value of a auity of $500 ivested pe yea fo 6 yeas at 9% p.a., with iteest compouded aually. 5 Use the table of peset values to calculate the peset value of each of the followig auities. a $400 ivested pe yea fo 5 yeas at 0% p.a., with iteest compouded aually b $2000 ivested evey six moths fo 5 yeas at 4% p.a., with iteest compouded six-mothly c $500 ivested evey thee moths fo 2 yeas at 2 6% p.a., with iteest compouded quately d $300 ivested evey moth fo 4 yeas at 2% p.a., with iteest compouded half-yealy 6 Use the fomula M = N ( to calculate ( the amout of each mothly epaymet o a loa of $ to be epaid ove 6 yeas at 2% p.a. 7 Scott boows $ to puchase a ca. The loa is take out ove a 4-yea tem at a iteest ate of 9.6% p.a., with the loa to be epaid i equal mothly epaymets. a Calculate the amout of each mothly epaymet. b Calculate the total amout that is epaid o the loa. 8 Calculate the total epaymets made o a home loa of $ to be epaid i equal mothly epaymets ove 25 yeas at a iteest ate of 8.4% p.a. 9 Adam buys a ew louge suite fo $4400 ad pays fo it o his cedit cad. The iteest ate o the cedit cad is 2% p.a. Adam hopes to pay the cedit cad off i two yeas by makig equal mothly epaymets. a Calculate the amout of each mothly epaymet that Adam should make. b Calculate the total amout that Adam will make i epaymets. c Calculate the amout of iteest that Adam will pay. EXTENDED RESPONSE Lie ivests $2000 pe yea i a auity. The tem of the auity is 20 yeas ad the iteest ate is 8% p.a., with iteest compoudig aually. a Calculate the futue value of this auity. b Calculate the peset value of this auity. 272 Maths Quest Geeal Maths HSC Couse

27 c By how much will the futue value of the auity icease if Lie deposits $500 pe quate ad iteest is compouded quately? 2 Eddie has the goal of savig $ ove his wokig life, which he expects to be 40 yeas. Ove the peiod of his wokig life, Eddie expects to be able to obtai a aveage 7% p.a. i iteest with iteest compouded evey six moths. a Calculate the peset value of this auity. A b Use the fomula M = to calculate the amout of each six-mothly cotibutio to the auity. ( c Fo the fist 0 yeas of the auity Eddie makes o cotibutios, pefeig to diect all his moey ito payig off a motgage. At that time he makes a sigle cotibutio to catch up o the auity. What amout must Eddie deposit? 3 Jim ad Catheie take out a $ loa. The iteest ate o the loa is 2% p.a. ad the loa is to be epaid i equal mothly epaymets ove a 20-yea tem. a Use the fomula M = N ( to calculate the amout of each mothly epaymet. ( b Calculate the total amout of iteest that Jim ad Catheie will eed pay o this loa. c Calculate the savig that Jim ad Catheie will make by epayig the loa ove a 2-yea tem. ebookplus Test Youself doc-386 Chapte 8 Chapte 8 Auities ad loa epaymets 273

28 ebookplus ACTIVITIES Ae you eady? s (page 248 SkillSHEET 8. (doc-379: Fidig values of ad i fiacial fomulas. SkillSHEET 8.2 (doc-380: Calculatig simple iteest. SkillSHEET 8.3 (doc-38: Calculatig compoud iteest. SkillSHEET 8.4 (doc-383: Readig fiacial tables. 8A Futue value of a auity Tutoial WE 2 it-2429: Pefom a calculatio of auity. (page 250 s SkillSHEET 8. (doc-379: Fidig values of ad i fiacial fomulas. (page 252 SkillSHEET 8.2 (doc-380: Calculatig simple iteest. (page 252 SkillSHEET 8.3 (doc-38: Calculatig compoud iteest. (page 252 Speadsheet (doc-382: Auity calculato. (page 255 8B Peset value of a auity Tutoial WE 8 it-2430: Pefom a calculatio of auity. (page 257 WokSHEET 8. (doc-383: Apply you kowledge of auities. (page 259 8C Futue ad peset value tables SkillSHEET 8.4 (doc-383: Readig fiacial tables. (page 263 8D Loa epaymets Tutoial WE 3 it-243: Pefom a calculatio of loa epaymets. (page 266 s WokSHEET 8.2 (doc-385: Apply you kowledge of loa epaymets. (page 268 Wod (doc-3330: Auities, loas, gaphs. (page 269 Chapte eview Test Youself (doc-386: Take the ed-of-chapte test to test you pogess. (page 273 To access ebookplus activities, log o to Maths Quest Geeal Maths HSC Couse