APPLICATIONS OF GEOMETRIC

Transcription

1 APPLICATIONS OF GEOMETRIC SEQUENCES AND SERIES TO FINANCIAL MATHS The mos powerful force i he world is compoud ieres (Alber Eisei) Page of 52

2 Fiacial Mahs Coes Loas ad ivesmes - erms ad examples... 3 Derivaio of Amorisaio morgages ad loas formula... 3 Amorisaio schedule... 5 Sample paper Q6 20 LCHL... 9 Pre-Leavig paper Q6 20 LCHL Quesios Se A Quesios Se A - suggesed soluios Quesios Se B Suggesed soluios - Quesios Se B Coiuous compoudig ad e Appedix - formulae for amorisaio schedule Page 2 of 52

3 Fiacial Mahs Loas ad Ivesmes - erms ad examples Loas ad ivesmes - associaed ermiology People adverisig loas ad ivesme producs wa o make heir producs seem as aracive as possible. They ofe have differe ways of calculaig he ieres, ad he producs migh ivolve differe periods of ime. This makes i difficul for cosumers o compare he producs. Because of his, govermes have rules abou wha iformaio mus be provided i adverisemes for fiacial producs ad i he agreemes ha busiesses make wih heir cusomers. Before defiig erms such as APR used for loas, for which here is a sauory regulaio, we eed o look a he cocep of prese value. Prese Value If you received 00 oday ad deposied i io a savigs accou, i would grow over ime o be worh more ha 00. This is a resul of wha is called he ime value of moey, a cocep which says ha i is more valuable o receive 00 ow raher ha say a year from ow. To pu i aoher way he prese value of receivig 00 oe year from ow is less ha 00. Assumig a 0% ieres rae per aum, he 00 I will receive i oe years ime is worh ow. Tha. is is prese value. (Ivesig 90.9 ow for oe year a 0% per aum yields 00 i oe year s 00 ime.) The prese value of 00 which I will receive i wo year s ime is (Ivesig ow for wo year a 0% per aum yields 00 i wo year s ime.) For example he owers of a piece of lad migh say ha hey will sell i o you ow for 60,000 oday or for 200,000 a he ed of wo years. Usig a prese value calculaio you ca see ha he ieres rae implici i he secod opio is.8% per aum. We will see ha prese value calculaios ca ell you such higs as: The amou of each regular payme for a give loa (give he ieres rae as a APR (see below) ad he ime i years over which he loa is o be repaid) ( See example page 8) How much moey o ives righ ow i reur for specific cash amous o be received i he fuure The size of he pesio fud required o he dae of reireme o give a fixed icome every year for a cerai umber of years (See Sample paper LCHL 20 Q6) The fair marke value of a bod (Pre Leavig NCCA paper LCHL Q6 The cash value opio available i mos US loery games ( LC HL 20 Paper Q6(d) ad Q4 page 25 ad Q6 Page 4) Prese value versus fuure value Whe regular paymes are beig used o pay off a loa, he we are usually ieresed i calculaig heir prese values (value righ ow), because his is he basis upo which he loa repaymes ad/or he APR are calculaed. Whe regular paymes are beig used for ivesme, we may isead be ieresed i heir fuure values (value a some ime i he fuure), sice his ells us how much we ca expec o have whe he ivesme maures. Page 3 of 52

4 Fiacial Mahs Loas ad Ivesmes - erms ad examples Loas ad oher forms of credi APR I he case of loas ad oher forms of credi, here is a legal obligaio o display he Aual Perceage Rae (APR) promiely. There are also clear rules laid dow i legislaio (Cosumer Credi Ac) abou how his APR is o be calculaed. Noe ha you eed o be careful if you search he web or oher resources for iformaio abou APR. The erm is o used i he same way i all couries. We are cocered here oly wih he meaig of he erm i Irelad, where is use is govered by Irish ad Europea law. (Formulae ad Tables bookle page 3- sauory formula for APR) APR is based o he idea of he prese value of a fuure payme. There are hree key feaures of APR:. All he moey ha he cusomer has o pay mus be icluded i he calculaio he loa repaymes hemselves, alog wih ay se-up charges, addiioal uavoidable fees, ec. 2. The defiiio saes ha he APR is he aual ieres rae (expressed as a perceage o a leas oe decimal place) ha makes he prese value of all of hese repaymes equal o he prese value of he loa. 3. I calculaig hese prese values, ime mus be measured i years from he dae he loa is draw dow. Noe ha he effec of his mehod of calculaio is ha he ieres rae has he same effec as if a fixed amou of moey was borrowed a his rae of aual ieres, compouded aually. APR akes accou of he possible differe compoudig periods i differe producs ad equalises hem all o he equivale rae compouded aually. Nomial rae If a credi ieres rae is o a APR, he i may be referred o as a omial rae or headlie rae. These have somewha less relevace ow, as i is o loger legal o quoe a ieres rae oher ha he APR i a adveriseme for a loa or credi agreeme. Noeheless, here is a example: if a loa or overdraf faciliy is govered by a charge of % per moh calculaed o he ousadig balace for ha moh, ha migh have bee cosidered o be omially a 2% aual rae, calculaed mohly. However, i is acually a APR of 2.68%, sice owed a he sar of a year would become (.0) 2 =.268 by he ed of he year. (See Page 32 Tables ad Formulae Bookle.) Savigs ad Ivesmes - AER, EAR, CAR For o paricular reaso, he erm APR is reserved for loas ad credi agreemes, where he cusomer is borrowig from he service provider. I he opposie case, where he cusomer is savig or ivesig moey, he comparable erm is he Equivale Aual Rae (EAR), someimes referred o as Aual Equivale Rae (AER) or Compoud Aual Reur / Compoud Aual Rae (CAR). The Fiacial Regulaor s office cosiders hese erms (EAR/AER/CAR) o be equivale. The erm CAR is approved for use i relaio o racker bods. For oher ivesme producs, he Fiacial Regulaor s office cosiders ha he erms AER ad EAR should be used. Page 4 of 52

5 Fiacial Mahs Loas ad Ivesmes - erms ad examples The regulaor s code iself uses he erm Equivale Aual Rae, implyig he acroym EAR, bu AER may be more commo ieraioally. The rules goverig heir use i adverisemes ad agreemes are o as clearly specified i law as is he case wih he erm APR, ad i is o as clear wha, if ay, fees ad charges have o be ake accou of whe calculaig EAR/AER. Also, i he case of ivesmes ha do o have a guaraeed reur, he calculaio of EAR ofe ivolves esimaes of fuure growh. Despie all his, he mehod of calculaio is he exac same as is he case wih he APR. Example Bak of Irelad offered a 9 moh fixed erm reward accou payig 2.55% o mauriy, for ew fuds from 0,000 o 500,000. (Tha is, you go your moey back i 9 mohs, alog wih 2.55% ieres.) Cofirm ha his was a EAR of 3.4%. Soluio For every euro you ivesed, you go back.0255 i 3 4 of a year s ime. A 3.4%, he prese value.0255 of his reur is which is as i should be. 3 4 Aleraively, jus cofirm ha , or ha I summary: The fuure value or fial value of he ivesme of afer 9 mohs is The fuure value or fial value of he ivesme afer year is.034. The prese value of.034 which due i year s ime usig his rae is. The prese value of.0255 due i 9 moh s ime usig his rae is. Example 2 The goverme s Naioal Solidariy Bod offers 50% gross reur afer 0 years. Calculae he EAR for he bod. Soluio For every you ives you ge back.5 i 0 years ime. F P( i).5 ( i) i (.5) 0 i i EAR = 4.4% Page 5 of 52

6 Fiacial Mahs Loas ad Ivesmes - erms ad examples Auiies A auiy is a form of ivesme ivolvig a series of periodic equal coribuios made by a idividual o a accou for a specified erm. Ieres may be compouded a he ed or begiig of each period. The erm auiy is also used for a series of regular paymes made o a idividual for a specified ime, such as i he case of a pesio. The word auiy comes from he word aual meaig yearly. Pesio fuds ivolve makig coribuios o a auiy before reireme ad receivig paymes from a auiy afer reireme. Calculaios ca be made o fid ou (i) Wha a cerai coribuio per period amous o as a fud (ii) Wha size of coribuio eeds o be made o creae of fud of a specific amou Whe receivig paymes from a auiy he prese value of he auiy is he lump sum ha mus be ivesed ow i order o provide hose regular paymes over he erm. Examples of auiies: Mohly re paymes Regular deposis i a savigs accou Social welfare beefis Aual premiums for a life isurace policy Periodic paymes o a reired perso from a pesio fud Divided paymes o socks ad shares Loa repaymes The fuure value of a auiy is he oal value of he ivesme a he ed of he specified erm. This icludes all paymes deposied as well as he ieres eared. The followig exrac is ake from Mahemaics A Pracical Odyssey, Johso Mowry The prese value of a auiy is he lump sum ha ca be deposied a he begiig of he auiy s erm, a he same ieres rae ad wih he same compoudig period, ha would yield he same amou as he auiy. This value ca help he saver o udersad his or her opios; i refers o a aleraive way of savig he same amou of moey i he same ime. I is called he prese value because i refers o he sigle acio he saver ca ake i he prese (i.e. a he begiig of he auiy s erm) ha would have he same effec as would he auiy.) (See 20 LCHL Sample Paper Q6 for a example of auiies i pracice.) Amorisaio ad amorised loas The process of accouig for a sum of moey by makig i equivale o a series of paymes over ime, such as arises whe payig off a deb over ime is called amorisaio. Accordigly, a loa ha ivolves payig back a fixed amou a regular iervals over a fixed period of ime is called a amorised loa. Term loas ad auiy morgages (as opposed o edowme morgages) are examples of amorised loas. See example 3 below. Page 6 of 52

7 Fiacial Mahs Loas ad Ivesmes - erms ad examples Bods A bod is a cerificae issued by a goverme or a public compay promisig o repay borrowed moey a a fixed rae of ieres a a specified ime. (See Q6 NCCA Pre Leavig 20) Regular paymes over ime geomeric series Arragemes ivolvig savigs ad loas ofe ivolve makig a regular payme a fixed iervals of ime. For example, a regular saver accou migh ivolve savig a cerai amou of moey every moh for a umber of years. A erm loa or a morgage migh ivolve borrowig a cerai amou of moey ad repayig i i equal isalmes over ime. Calculaios ivolvig such regular payme schedules, whe hey are cosidered i erms of he prese values of he paymes as i loas - example 3 below, (or he fuure values as i ivesmes - example 4 below) will ivolve he summaio of a geomeric series. Page 7 of 52

8 Fiacial Mahs Loas ad Ivesmes - erms ad examples Amorised loa example Whe regular paymes are beig used o pay off a loa, he we are usually ieresed i calculaig heir prese values (value righ ow) raher ha heir fuure values, because his is he basis upo which he loa repaymes ad/or he APR are calculaed. We have see ha he APR is he ieres rae for which he prese value of all he repaymes is equal o he prese value of he loa. I he case of a amorised loa, hese prese values form a cosise paer ha urs ou o be a geomeric series. Example 3 Seá borrows 0,000 a a APR of 6%. He was o repay i i five equal isalmes over five years, wih he firs repayme oe year afer he akes ou he loa. How much should each repayme be? Soluio Le each repayme equal A. The he prese value of he firs repayme is A/.06, he prese value of he secod repayme is A/.06 2, ad so o. The oal of he prese values of all he repaymes is equal o he loa amou. A A A Toal of he prese values of all he repaymes This is a geomeric series, wih 5, firs erm A a ad commo raio r..06 The sum of he firs 5 erms which is he loa amou is S 5 A A 0000 If S 5 has o equal he loa amou of 0,000, he A (i) (Sudes could fid S for a small umber of erms by addig he erms idividually firs ad he checkig heir aswer by usig he formula for S of a geomeric series.) This ype of calculaio is so commo ha i is coveie o derive a formula o shorcu he calculaio for he regular repayme A. By cosiderig he geeral case of a amorised loa wih ieres rae i, ake ou over years, for a loa amou of P, a geomeric series ca be used o derive he geeral formula: i( i) A P ( i) (.06) This formula gives he same resul as (i) above: A (.06) (The formula assumes payme a he ed of each payme period. We will derive his formula laer. (See page 4) Page 8 of 52

9 Fiacial Mahs Loas ad Ivesmes - erms ad examples Amorised loas - regular paymes a iervals oher ha yearly The calculaios ivolved are he same, excep ha oe mus be careful o rea he APR properly. Suppose, i he above case, Seá waed o make mohly repaymes isead of yearly oes. How much would each repayme he be? There are wo ways o approach his. We ca eiher keep he year as he ui of ime, or chage o mohs. If we say wih years, we keep he same rae for he APR bu he we eed o deal wih fracioal uis of ime. If we swich o mohs, we mus cover he APR o a equivale mohly compouded rae ad he we have ieger amous of ime.. Sick wih years If we keep he year as he ui of ime, he he firs repayme is made afer of a year, so is 2 A A prese value is he prese value of he secod repayme is 2, ad so o I he case of example 3 o he previous page, if Seá swiches o mohly paymes, he geomeric A A series will have 60 erms, wih a =, ad r = The sum of he prese values of all he repaymes = Loa amou. S 60 A A 5.923A = Loa amou = 0,000, givig A = (Noe: If we muliply his aswer by 2, we ge which is less ha he amou Sea repaid whe repayig yearly. How much will his save him over he five years? ) 2. Swich o mohs If we swich o mohs as he ui of ime, he we firs have o deermie wha mohly ieres rae, compouded mohly, is equivale o a APR of 6%. This ivolves geig he welfh roo of.06, which is The, we ca rea he prese values as a geomeric series agai, wih firs A erm a, commo raio r ad umber of erms = A S As above his gives A = A Page 9 of 52

10 Fiacial Mahs Loas ad Ivesmes - erms ad examples Aleraively, we ca use he amorisaio formula wih i = , P = 0,000, ad = 60 (usig mohs as he ime periods ad he correspodig i) ( ) A ( ) 60 Regular saver accous ad similar ivesmes Whe regular paymes are beig used for savigs or ivesme, we are ieresed i heir fuure values (value a some ime i he fuure), sice his ells us how much we ca expec o have whe he ivesme maures. As meioed earlier, ay regular payme over ime will give rise o a geomeric series, irrespecive of wheher is purpose is o repay a loa or o geerae savigs for he fuure. Accordigly, he aalysis is very similar. I he case of savigs ad ivesmes, we are geerally ieresed i he fuure value raher ha he prese value. Example 4 Accordig o isyourmoey.ie, oe of he baks is offerig a regular mohly savigs accou wih a AER of 4.00% o balaces up o 5,000. If I save 00 per moh, sarig oday, how much will I have i he accou i five years ime, assumig he rae says he same? Soluio: (Usig years as he ui of ime) The fuure value (value i 5 years ime) of my firs 00 is 00(.04) 5. The fuure value (value i 5 years ime) of my secod 00 is 00(.04) (59/2). The fuure value (value i 5 years ime) of my 60 h (ad las) 00 is 00(.04) (/2). The fuure value of all of hese regular paymes io he savigs accou is: (.04) 00(.04) 00(.04)...00(.04) 00(.04) This is a geomeric series wih firs erm 00(.04) 5, commo raio (.04) (-/2) ad 60 erms. S (.04) (To make calculaios easier, sore 2.04 eeded.) i he calculaor s memory ad jus recall i each ime i is Aoher opio, o avoid fracios, whe addig up he erms of he geomeric series is o reverse he order: S 00(.04) 00(.04)... 00(.04) 00(.04) 00(.04) Page 0 of 52

11 Fiacial Mahs Loas ad Ivesmes - erms ad examples 2 2 a 00(.04), r.04, 60 S (.04) 6, Coecio bewee he calculaios ivolved i a amorised loa ad hose ivolved i a regular saver accou There is clearly a coecio bewee he calculaios ivolved i a amorised loa ad hose ivolved i a regular saver accou. I may be oed ha we have a formula o page 3 of he formulae ad ables bookle for calculaig he repaymes ivolved i a amorised loa bu we do have a formula for he equivale problem of calculaig he aual or mohly savigs required o geerae a paricular ivesme fud i he fuure. However, if we wish o ake advaage of he loa formula o shorcu he cosideraio of he geomeric series ivolved i he savigs case, he we ca readily do so by observig ha a sigle jump ca ake us from he prese value o he fuure value of ay fud, or back agai. Tha is, ay give sream of paymes has a oal prese value ad a oal fuure value. We have already see how we ca jump forward usig he compoud ieres formula or backwards usig prese value formula (firs wo formulae o page 30 of he formulae ad ables book). *Thus, for example, if we wa o kow wha mohly ivesme (paid a he ed of each moh) is required o geerae a fud of 20,000 i 0 years ime a a EAR of 5%, we ca isead solve he equivale problem of fidig he mohly repaymes required o pay back a loa whose pricipal is he prese value of 20,000, ad which is o be paid off over 0 years a a APR of 5%. Thus, we ca firs fid he prese value of 20,000, ad he use he amorised loa repayme formula o fid he mohly ivesme required o geerae he 20,000 i 0 years ime. The prese value of 20,000, ivesed a a EAR of 5% for 0 years is: F P ( i) (.05) Usig he amorised loa formula o calculae he mohly ivesme required o geerae a fud of 20,000 where he mohly ieres rae is (.05) /2 = : ( ) A ( ) 20 Hece a mohly ivesme of would geerae a fud of 20,000 i 0 years ime a a EAR of 5%. ( See Q6 LCHL Sample paper 20 - Padraig eeds 358,70.84 i 40 years ime i a reireme fud - wha fixed amou should he ives every moh a a EAR of 3% o achieve his fud?) Page of 52

12 Fiacial Mahs Loas ad Ivesmes - erms ad examples Use of raio ad proporio Aoher possible approach o deermiig he mohly ivesme required o geerae a paricular fud is o use raio/proporio. Oe ca firs calculae, usig a geomeric series, he fud ha would be geeraed by a mohly savig of (or 00). By comparig his amou o he fud required, we ca scale his savig proporioally upwards or dowwards as required, sice he fial fud is i direc proporio o he mohly amou saved. ivesed a he ed of each moh for 20 mohs a a EAR of 5% would give he followig geomeric series whose sum is: S 9 8 ( ) ( )...( ) ( )... ( ( Hece if oe ivess a he ed of each moh for 0 years a a EAR of 5% he fud geeraed is To geerae a fud of 20,000 oe would eed o ives 20,000/ = Page 2 of 52

13 Fiacial Mahs Amorisaio Morgages ad loas formula Amorisaio formula (Page 3 Formulae ad Tables ) Terms associaed wih he amorisaio formula revisied: Prese Value is he value o a give dae of a fuure payme or series of fuure paymes discoued o reflec he ime value of moey ad oher facors such as ivesme risk. A auiy is a series of equal paymes or receips ha occur a evely spaced iervals. Each payme occurs a he ed of each period for a ordiary auiy. A amorised loa is a loa for which he loa amou plus ieres is paid off i a series of regular paymes. A amorised loa is a auiy whose fuure value is he same as he loa amou s fuure value, uder compoud ieres. A amorised loa s paymes are used o pay off a loa. Oher ypes of auiies paymes ca be used o geerae savigs as for example for reireme fuds. We ca hik of he siuaio i wo ways which give he same ed resul: ) The sum of he prese values of all he aual repayme amous = sum borrowed. (This priciple is eshried i Europea Law) 2) Fuure value of loa amou = Fuure value of he aual repayme amous (i.e. fuure value of he auiy) Noe: Compoudig periods oher ha aual ca be applied e.g. quarerly, mohly, daily ec. Give ha A = aual repayme amou, he prese value of oe aual repayme amou paid i years ime is A P, where i is he aual rae of ieres expressed as a decimal or fracio ( i) So if I borrow 0,000 over 5 years, whe I add up he prese values of all he aual repayme amous, his sum should equal 0,000. A A A A A 0, 000 A ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) Sudes see ha he prese value of each repayme is less ha he oe before i. Two mehods of derivig he Amorisaio morgages ad loas formula (Page 3 Formulae ad Tables book) are used below. Page 3 of 52

14 Fiacial Mahs Amorisaio Morgages ad loas formula Mehod Loa amou sum of he prese value of all he repaymes (assumig payme a he ed of each payme period) P Loa amou, A periodic repayme amou, i he umber of payme periods he ieres rae for he payme period expressed as a decimal or fracio A A A A P ( i) ( i) ( i) ( i) A P S of a geomeric series, = = umber of compoudig periods,, a r i i a( r ) P r Mehod 2 P A A ( i) ( i) ( i) A ( i) i ( ) ( i) i i A ( i) ( i) i ( i) i ( A) ( i) i( i) i( i) P ( i) The fuure value of he loa amou P sum of he fuure values of equal repaymes each of value A made a he ed of each compoudig period. 2 2 P( i) A( i) A( i)... A( i) A( i) A 2 A A( i) A( i) A P( i) S of a geomeric series ( i)... A( i) 3 ar ( ) S where a A, r i, r A(( i) ) P( i) Exercise: Adap he formula ( i) for he siuaio where he A(( i) ) repaymes are made a he P( i) begiig of each payme i period as opposed o a he ( i) i A P ed of each payme period. ( i) Page 4 of 52

15 amou i euro Fiacial Mahs Amorisaio schedule Assumig a loa is repaid i fixed aual repaymes he aual repayme is made up of wo pars oe par ieres ad he remaider is par of he capial sum borrowed, called pricipal porio i he graph below. The graphs below show ha ha eve hough he periodic payme is fixed, he par of i which is ieres is decreasig as more of he loa is paid off ad he par of i which is pricipal is icreasig. The graph below refers o a amorisaio schedule for a loa paid back mohly over 360 mohs. (See Sample quesios Se A quesio 8.),400.00, Showig pricipal ad ieres porios of he paymes Ieres porio, Pricipal porio Payme umber Amorisaio Schedule A amorisaio schedule is a lis of several periods of paymes, he pricipal ad ieres porios of hose paymes ad he ousadig pricipal (or balace) afer each of hose paymes is made. Below is he amorisaio schedule for example 3 Page 8, showig i figures he reds i he ieres ad pricipal porios of each payme for successive paymes for he loa i example 3: 0,000 loa paid back over 5 years a 6% APR ivolvig a fixed aual repayme of per year. Payme # Fixed payme Ieres porio Pricipal porio balace 0 0, rae 6.00% 2, , , years 5 2 2, , , fixed payme per year 2, , , , , ,2.82 2, , , See Appedix for he formulae for his spreadshee. Page 5 of 52

16 Fiacial Mahs Amorisaio schedule Seps i a amorisaio schedule:. Fill i he firs balace = loa amou (payme umber 0) 2. For payme, fill i he payme umber ad fixed repayme amou 3. For payme row, fid he ieres o he previous balace usig he simple ieres formula 4. Calculae he deb payme (pricipal porio) = he repayme amou - he ieres porio 5. Calculae he ew balace = previous balace - he pricipal porio 6. Repea Seps 3, 4 ad 5 for all he oher paymes from payme 2 owards 7. For he las payme, he pricipal porio = he previous balace 8. Whe all he paymes have bee made he fial balace is 0.00 Explaaio of he amorisaio schedule Payme # oal payme Ieres porio Pricipal porio Balace 0 Loa amou Fixed repayme amou calculaed usig Amorisaio - loas ad morgages formula Simple ieres o he previous balace Repayme amou - ieres porio Previous balace - his payme s pricipal porio 2 Fixed repayme amou calculaed usig Amorisaio - loas ad morgages formula Simple ieres o he previous balace Repayme amou - ieres porio Previous balace - his payme s pricipal porio Las Fixed Repayme = pricipal porio + ieres porio Simple ieres o he previous balace Previous balace 0.0 Page 6 of 52

17 Fiacial Mahs Sum of he prese values = Loa amou Usig JC kowledge of compoud ieres o show he ousadig balace a he ed of each payme period, ad he ieres ad pricipal porio of each payme for example 3 page 8: Payig back a loa of 0,000 over 5 years a a APR of 6% i fixed aual repaymes of 2, Page 7 of 52

18 Fiacial Mahs Sum of he prese values = Loa amou Page 8 of 52

19 Fiacial Mahs Sample Paper Q6 LCHL 20 Quesio 6 (50 marks) Pádraig is 25 years old ad is plaig for his pesio. He ieds o reire i fory years ime, whe he is 65. Firs, he calculaes how much he was o have i his pesio fud whe he reires. The, he calculaes how much he eeds o ives i order o achieve his. He assumes ha, i he log ru, moey ca be ivesed a a iflaio-adjused aual rae of 3%. Your aswers hroughou his quesio should herefore be based o a 3% aual growh rae. (a) Wrie dow he prese value of a fuure payme of 20,000 i oe years ime. (b) Wrie dow, i erms of, he prese value of a fuure payme of 20,000 i years ime. (c) Pádraig was o have a fud ha could, from he dae of his reireme, give him a payme of 20,000 a he sar of each year for 25 years. Show how o use he sum of a geomeric series o calculae he value o he dae of reireme of he fud required. Pádraig plas o ives a fixed amou of moey every moh i order o geerae he fud calculaed i par (c). His reireme is 40 2 = 480 mohs away. (i) Fid, correc o four sigifica figures, he rae of ieres per moh ha would, if paid ad compouded mohly, be equivale o a effecive aual rae of 3%. (ii) Wrie dow, i erms of ad P, he value o he reireme dae of a payme of P made mohs before he reireme dae. (iii) If Pádraig makes 480 equal mohly paymes of P from ow uil his reireme, wha value of P will give he fud he requires? (iv) If Pádraig wais for e years before sarig his pesio ivesmes, how much will he he have o pay each moh i order o geerae he same pesio fud? Page 9 of 52

20 Fiacial Mahs Sample Paper Q6 LCHL 20 Suggesed soluio (a) Wrie dow he prese value of a fuure payme of 20,000 i oe years ime. A: P = 20000/.03= 9,47.48 (Noe: 9,47.48 would icrease o i oe year a a 3% aual growh rae.) (b) Wrie dow i erms of, he prese value of a fuure payme of i years ime. A: P=20000/(.03) (c) Padraig was o have a fud ha could, from he dae of his reireme, give him a payme of 20,000 a he sar of each year for 25 years. Show how o use he sum of a geomeric series o calculae he value o he dae of reireme of he fud required. A: The amou of moey i he fud o he dae of reireme = sum of he prese values of all he paymes up o he dae of reireme. Page 20 of 52

21 Fiacial Mahs Sample Paper Q6 LCHL 20 Suggesed soluio Prese value of he firs payme Prese value of he secod payme Prese value of he hird payme 2 (.03) Prese value of he firs payme (.03) Fud geeraed o he dae of reireme (.03) (.03) Fud of a geomeric series S a( r ) S where a 20000, r, 25 r.03 S 20000( ) , Sudes oe ha his is less ha 20000*(25). Page 2 of 52

22 Fiacial Mahs Sample Paper Q6 LCHL 20 Suggesed soluio 6(d)(i) Pádraig plas o ives a fixed amou of moey every moh i order o geerae he fud calculaed i par(c). His reireme is 40 2 = 480 mohs away mohs away. Fid correc o four sigifica figures he rae of ieres per moh ha would, if paid ad compouded mohly, be equivale o a effecive aual rae of 3%. Suggesed soluio: (+i) 2 =.03 (+i) =.03 /2 = i= rae of ieres per moh = = % 6(d)(ii) Wrie dow i erms of ad P, he value o he reireme dae of a payme of P made mohs before reireme dae. Suggesed soluio: The fuure (fial) value of a payme of P paid mohs before reireme dae is P( ) 6(d)(iii) If Pádraig makes 480 equal mohly paymes of P from ow uil his reireme, wha value of P will give he fud he requires? Suggesed soluio: The fuure value of he auiy , i , mohly payme P P( i) P( i)... P( i) P( i) P i P i ( ) ( )... P( i) P( i) (i reverse order) Righ had side is S of a geomeric series ar ( ) S where a P ( i ), r ( i ), 480 r 480 P( )( ) P(99.38) P Page 22 of 52

23 Fiacial Mahs Sample Paper Q6 LCHL 20 Suggesed soluio 6(e) If Pádraig wais for e years before sarig his pesio ivesmes, how much will he he have o pay each moh i order o geerae he same pesio fud? Suggesed soluio: Fuure value of he auiy = 358, years = 20 mohs reireme is 360 mohs away P P P P P Page 23 of 52

24 Fiacial Mahs Pre - Leavig 20 (a) A graduae is seig up his ow compuer compay. He borrows he 5000 for se-up coss for 6 mohs a a fla rae of % per moh (compouded mohly). He was o arrage o pay his off i equal mohly isalmes. (i) Calculae he mohly repayme amou. (ii) Make a schedule showig he mohly payme, he mohly ieres o he ousadig balace, he porio of he payme coribuig oward reducig he deb, ad he ousadig balace. Suggesed soluio: (i) i( i) A P ( i) A (ii) Payme # Payme Ieres Deb Payme Balace (b) Afer 5 years he compay eeds o raise moey o expad. I proposes o issue a 0-year 2,000 bod ha will pay 00 a he begiig of every year. If he curre marke ieres rae is 5% per aum, wha is he fair marke value of his bod? Explai your aswer ad jusify ay assumpios you make. (Bod: A bod is a cerificae issued by a goverme or a public compay promisig o repay borrowed moey a a fixed rae of ieres a a specified ime.) Suggesed Soluio: This bod promises wo higs. The bod holder ges 2000 whe he bod maures i 0 years ime. I he meaime hey ge 00 per year a he begiig of each year. The quesio beig asked is wha would be a reasoable price for he compay sellig he bod o charge he perso buyig he bod. Fiacial Mahs Page 24

25 Fiacial Mahs Pre - Leavig 20 The prese value of 2000 due i 0 years ime is calculaed as follows: Prese Value (P) of 2000 due i 0 years = (.05) The prese value of e paymes of 00 per year for e years: I order o be able o pay he bod holder 00 a he begiig of he firs year, he compay will ow eed o charge hem 00. I order o be able o pay hem 00 i year s ime (a he begiig of he secod year) he compay will eed o charge hem he prese value of 00 due i year calculaed as follows: P= ( 0.05).05 I order o be able o pay hem 00 i 2 years ime (a he begiig of he hird year) he compay will eed o charge hem he prese value of 00 due i 2 years ime calculaed as follows: P ( 0.05).025 The paer coiues i his way. The oal amou ha he ivesor will eed o pay he compay i order o be able o ge 00 a he begiig of each year for he ex 0 years is he sum of he prese values of all he paymes of S The righ had side of his equaio is a Geomeric series. a = 00, r =.05 S 0 ad = The compay eeds o charge he bodholder ow i order o be able o pay him/her 00 per year for 0 years. Hece i order o be able o pay he ivesor 00 per year for he ex 0 years ad pay 2000 a he ed of he 0 years, he fair marke value of he bod is = Noe: This aswer is based o he fac ha he 00 is paid a he begiig of every year. (Hece you ge he firs 00 immediaely afer he ivesme is made.) Fiacial Mahs Page 25

26 Fiacial Mahs Sample quesios Se A. Regular savigs (Fuure value of a auiy) Sample quesio: A he ed of each moh a deposi of 500 is made io a accou ha pays a AER of 8% compouded mohly. Wha will he fial amou be afer 5 years? 2. Regular savigs (Fuure value of a auiy) Sample quesio: Soya deposis 300 a he ed of each quarer i her savigs accou. If he moey ears 5.75% (EAR), how much will his ivesme be worh a he ed of 4 years? 3. Regular savigs (We someimes hear he phrase sikig fud i.e. he fuure value has bee decided ad you sik regular amous io he fud o reach his arge. Fid he value of he regular payme.) Sample quesio: Rober eeds 5000 i hree years. How much should he deposi a he ed of each moh i a accou ha pays 8% (EAR) i order o achieve his goal? 4. Prese value of a auiy wih a fiie umber of erms Sample quesio: You have wo a prize i a loery. The prize eiles you o,000 per moh, paid a he ed of each moh, for he ex 20 years. However, you prefer o have he eire amou ow i a lump sum. If he EAR is 8%, how much will you accep as a lump sum? 5. Prese value of a auiy wih a fiie umber of erms Sample quesio: Jack wo a prize i a loery. He has bee give a choice of wo opios: Opio A: Receive a auiy of 500, a he begiig of each moh for 25 years. Opio B: Take a lump sum isead. Jack decides o ake Opio B. a) Wha lump sum should he accep assumig a AER of 4%? b) He ivess he lump sum he receives, for 20 years, i a accou ha pays 4% (AER). How much will Jack s ivesme have amoued o afer he 20 years? See also - Q6 20 LCHL Mahemaics Paper 6. Prese value of a auiy wih a ifiie umber of erms Sample quesio: Suppose ha you expec o receive a payme of 00 oce per year for a idefiiely log ime. Wha is he prese value of his auiy? (A auiy is a sum of moey o be paid i regular iervals.) 7. Savigs Bods Sample quesio: New Horizos Compuer Compay eeds o raise moey o expad. I issues a 0-year bod of,000 ha pays 30 a he ed of every six moh period for he 0 years. If he curre marke aual ieres rae 7.2% (AER), wih ieres added a six-moh iervals, wha is he fair marke value of he bod? 8. Prepayig a loa Sample quesio: Mr. Mooey bough his house i 975. He obaied a loa from he bak for 30 years a a ieres rae of 9.8% APR. His mohly payme a he ed of each moh was 260. I 995, Mr. Mooey decided o pay off he loa. Fid he balace of he loa he sill owed a ha ime. Fiacial Mahs Page 26

27 Fiacial Mahs Sample Quesios Se A, quesio - Suggesed soluio Sample quesio LCHL Regular savigs (Fuure value of a auiy) A he ed of each moh a deposi of 500 is made io a accou ha pays a AER of 8% compouded mohly. Wha will he fial amou ( i.e. fuure value) be afer 5 years? Sice his is a quesio o savigs/ivesmes i is a good idea o look a fuure values. Suggesed soluio: 59 2 Fuure value of he firs 500 = 500((.08) ) Fuure value of he secod 500 = 500((.08) ) Fuure value of he hird 500 = 500((.08) ) Fuure value of he secod las 500 = 500((.08) 2 ) Fuure value of he las 500 = 500 as i is ivesed a he ed of he las payme period The sum of all he fuure values of all he regular savigs of 500 per moh is (.08) (.08) 500(.08) 500(.08) The sum of hese erms is S of a geomeric series : ar S a r r S.08 2 ( ) wih 500, (.08) 2 ad , Fiacial Mahs Page 27

28 Fiacial Mahs Sample Quesios Se A, quesio 2 - Suggesed soluio Sample quesio 2 LCHL Regular savigs (Fuure value of a auiy) Soya deposis 300 a he ed of each quarer i her savigs accou. If he moey ears 5.75% (EAR), how much will he ivesme be worh a he ed of 4 years? Sice his is a quesio o savigs/ivesmes i is a good idea o look a fuure values. Suggesed Soluio Fuure value of he firs 300 = 300(.0575) Fuure value of he secod 300 = 300(.0575) Fuure value of he hird 300 = 300(.0575) Fuure value of he secod las 300 = 300(.0575) Fuure value of he las 300 = 300 The sum of all he fuure values of all he regular savigs of 300 per moh is (.0575) (.0575) 300(.0575) 300(.0575) This is a geomeric series wih a 300, r (.0575) ad 6 S ar ( ) ,34.56 r Fiacial Mahs Page 28

29 Fiacial Mahs Sample Quesios Se A, quesio 3 - Suggesed soluio Sample quesio 3 LCHL Regular savigs ( Sikig fud he fuure value has bee decided ad you sik regular amous io he fud o reach his arge. Fid he value of he regular payme.) Rober eeds 5000 i hree years. How much should he deposi a he ed of each moh i a accou ha pays 8% (EAR) i order o achieve his goal? Sice his is a quesio o savigs/ivesmes i is a good idea o look a fuure values. The fuure/fial value ( FV) of his ivesme eeds o be Assume ha he makes a regular savig of x per moh = sum of he fuure values of all he regular paymes of x io he fud. Fuure value of he firs x = x(.08) 35 2 Fuure value of he secod x = x(.08) Fuure value of he hird x = x(.08) Fuure value of he secod las x = x(.08) Fuure value of he las x = x The sum of all he fuure values of all he regular savigs of x per moh is x x(.08)... x(.08) x(.08) x(.08)...equaio (i) This is a geomeric series wih a x, r (.08) ad a( r ) x((.08) ) ) x r x Noe: 2 (.08) i Fiacial Mahs Page 29

30 Fiacial Mahs Sample Quesios Se A, quesio 3 - Suggesed soluio Aleraive view (Sample quesio 3): We could also view his as he prese value of 5000 = prese value of all he mohly paymes. I oher words fid he value of each of he equal mohly repaymes eeded o pay back a loa equal o he prese value of (We ca see his from equaio (i) o he previous page if we divide boh sides by (+i) 36 = (.08) 36/2.) x x(.08)... x(.08) x(.08) x(.08)...equaio(i) 5000 x x... x (.08) (.08) (.08) (.08) PV of 5000 = Prese value of las payme Prese value of firs payme We ca sum he righ had side usig a geomeric series ad hece fid x. Aleraively we ca use he "amorisaio- moragages ad loas" formula o fid he amou of he regular payme x,( A i he formula) where P i he formula is he prese value of he 5000, ad i is he mohly ieres rae. Usig he amorisaio formula (o fid A) Amorisaio formula Page 3 Tables ad Formulae: i( i) A P ( i) Firs we fid he prese value of 5000 due i 3 years ime assumig a EAR of 8% 5000 P Subsiuig his for P io he amorisaio formula we ca fid A, he 3 (.08) amou which should be deposied each moh. 2 (.08) i i( i) ( )(.08) A P ( i) ((.08) ) Fiacial Mahs Page 30

31 Fiacial Mahs Sample Quesios Se A, quesio 4 - Suggesed soluio Sample quesio 4 LCHL Prese value of a auiy wih a fiie umber of erms You have wo a prize i a loery. The prize eiles you o,000 per moh a he ed of each moh for he ex 20 years. However, you prefer o have he eire amou ow i a lump sum. If he EAR is 8%, how much will you accep as a lump sum? We ca look a his quesio i differe ways. As he quesio looks for he lump-sum you will accep righ here righ ow, le us iiially use he cocep of prese value. Firs, fid he rae of ieres per moh ha would, if paid ad compouded mohly, be equivale o a effecive aual rae (EAR) of 8%. (+ i) 2 =.08 (+ i) =.08 /2 = i = rae of ieres per moh = = % You will accep a lump - sum x which saisfies he followig crierio: The lump-sum x = he sum of he prese values of he regular paymes x... ((.08) ) ((.08) ) ((.08) ) ((.08) ) ((.08) ) Muliplyig boh sides by ((.08) ) (avoidig fracios i he S formula x ((.08) ) 000((.08) ) 000((.08) ) ((.08) ) 000 * ): Reversig he righ had side of his equaio (also o avoid fracios i he S formula) : x((.08) ) ((.08) ) ((.08) ) 000((.08) ) The righ had side of his equaio represes a geomeric series wih a 000, r.08 ad ((.08) ) ) (.08) ) = ((.08) ) x x 22, Assumig ha paymes sar i oe moh's ime, he rae is fixed, ad here is zero risk, he prese value of he auiy of 000 per moh for 20 years is 22, based o a EAR of 8%. This is he miimum lump - sum you will accep. Fiacial Mahs Page 3

32 Fiacial Mahs Sample Quesios Se A, quesio 4 - Suggesed soluio Usig he amorisaio formula o solve Sample quesio 4. Oe could also use he amorisaio formula o page 3 of he Formulae ad Tables bookle. Give ha A= 000 fid P. 2 (.08) ) i( i) ( i) A P P A , ( i) i( i) 2 ( ) (.08) (Whe he loery pays you 000 each moh for 20 years hey are really geig a loa of 22, from you a 8% (APR) for 20 years.) Aleraive view - usig he cocep of fuure value While he quesio is lookig for he prese value of a auiy of 000 per moh for 20 years we could also look a i i erms of fuure value. Le s say you accep a lump sum of x. The he x deposied a 8% (EAR) for 20 years should yield he same fial value as he fial or fuure value of he 240 mohly paymes of 000 paid o you a he ed of each moh for 20 years ad he ivesed by you each moh a 8% (EAR). (This is for he purposes of workig ou he lump sum oly, as i is ulikely ha you would sped oe of he moey durig he 20 years ad ives i all.) 240 Fuure value (FV) of he lump sum = Fuure value (FV) of he auiy The fuure value (FV) of he auiy of 000 per moh for 20 years paid a he ed of each moh: = (The FV of he firs payme is as i ears ieres for 239 mohs. The FV of he las payme paid a he ed of he 240 h moh is 000 as i ears o ieres.) Check usig S 240 of a geomeric series ha he fuure value of his auiy is 568, Fuure value (FV) of he lump sum x is x (.08)...(ii) Equaig (i) ad (ii) leads o he equaio below which is exacly he same as equaio * o he previous page x(.08) 000((.08) ) 000((.08) ) ((.08) ) 000 * Hece, i agreeme wih he oher mehods: x 22, Fiacial Mahs Page 32

33 Fiacial Mahs Sample Quesios Se A, quesio 5 - Suggesed soluio Sample quesio 5 LCHL Lookig for he prese value of a auiy wih a fiie umber of erms Jack wo a prize i a loery. He has bee give a choice of wo opios: Opio A: Receive a auiy of 500, a he begiig of each moh for 25 years. Opio B: Take a lump sum isead. Jack decides o ake Opio B. a) Wha lump sum should he accep assumig a AER of 4%? b) He ivess he lump sum he receives for 20 years i a accou ha pays 4% (AER). How much will Jack s ivesme have amoued o afer he 20 years? As i Sample quesio 4 ca look a his quesio i differe ways. As he quesio looks for he lumpsum you will accep righ here righ ow, le us iiially use he cocep of prese value. Noe ha he paymes are made i his case a he begiig raher ha a he ed of each moh. Firs, fid he rae of ieres per moh ha would, if paid ad compouded mohly, be equivale o a effecive aual rae (EAR) of 4%. (+ i) 2 =.04 (+ i) =.04 /2 = i = rae of ieres per moh = = % You will accep a lump - sum x which saisfies he followig crierio: The lump-sum x = he sum of he prese values of he regular paymes x ((.04) ) ((.04) ) ((.04) ) ((.04) ) ((.04) ) A his poi, i Sample quesio 4, we avoided fracios. However his ime we will omi his sep ad fid S of a geomeric series for he righ had side of he above equaio as i is. a 500, r, r.04 S300 a , r 2.04 (Use of he memory faciliy o he calculaor o sore he reciprocal of (.04) /2 is recommeded here.) Fiacial Mahs Page 33

34 Fiacial Mahs Sample Quesios Se A, quesio 5 - Suggesed soluio Aleraive view - usig he cocep of fuure value a) The fuure value (FV) of he lump sum Jack acceps should be he same as he fuure value of he auiy. The FV of a auiy of 500 per moh for 25 years paid a he begiig of each moh is: Wriig his i reverse order: FV of he auiy = 500(.04 ) 500(.04 )...500(.04 ) /2 /2 2 /2 300 This is a geomeric series wih a r /2 /2 500(.04), (.04) ad = 300 FV of he auiy = ).04 2 FV of he lump sum x x(.04) ) x(.04) FV of he lump sum x FV of he auiy ((.04) ) ) 2 2 x(.04) 500(.04) (.04) x 287, Assumig ha paymes sar a he begiig of each moh, he rae is fixed, ad here is zero risk he prese value of he auiy of 500 per moh for 25 years is 287, based o a EAR of 4%. This is he amou of moey ha, if he loery board ivesed i a 4% (AER), would produce aual paymes of Aleraively use he formula o page 3 of he ables. ( A 500, i (.04),fid P) You eed o muliply he aswer from he formula by ( i) as he formula assumes ha he auiy is paid a he ed ad o he begiig of he payme period. Hece muliply P by ( i). (See page 2) b) Jack s ivesme of will have amoued o (.04) 20 = 629, Fiacial Mahs Page 34

35 Fiacial Mahs Sample Quesios Se A, quesio 6 - Suggesed soluio Sample quesio 6 Prese value of a auiy wih a ifiie umber of erms Suppose ha you expec o receive a payme of 00 oce per year, a he ed of each year, for a idefiiely log ime. Wha is he prese value of his auiy? (A auiy is a sum of moey o be paid i regular iervals.) Receivig 00 a year from ow is worh less o you ha a immediae 00, because you cao ives he moey uil you receive i. I paricular, he prese value of a 00 oe year i he fuure is 00 i where i is he aual equivale rae of ieres expressed as decimal or a fracio. 00 Similarly, a payme of 00 wo years i he fuure has a prese value of 2 ( i). Therefore, he prese value of receivig 00 per year for a idefiiely log ime ca be expressed as a ifiie series i ( i) ( i) ( i) 00 This is a ifiie geomeric series wih a i ad r i a S i i (See Tables o page 22 for he formula.) r i i i i i i For example, if he yearly ieres rae is 0% (i= 0.0), he he auiy has a prese value of 000. Fiacial Mahs Page 35

36 Fiacial Mahs Sample Quesios Se A, quesio 7 - Suggesed soluio Sample quesio 7 LCHL Savigs Bods The New Horizos compuer compay eeds o raise moey o expad. I issues a 0 year bod of,000 ha pays 30 a he ed of every six moh period for he 0 years. If he curre marke aual ieres rae is 7.2% (AER), wih ieres added a six-moh iervals, wha is he fair marke value of he bod? Quesios o bods provide a ice applicaio of he cocep of prese value. We may kow he value of he bod i x years ime bu are ieresed i wha is value is righ ow. The bod cerificae i his quesio promises wo higs a amou of,000 o be paid i 0 years ad a half-yearly payme of 30 for e years. Therefore, o fid he fair marke value of he bod, we eed o fid he prese value of he lump sum of,000 we are o receive i 0 years ime, as well as he prese value of he 30 half-yearly paymes for he 0 years. 000 Prese value of 000 due i 0 year's ime P 0 (.072) Prese value of regular paymes of 30 = P P P ** Reversig he righ had side o avoid havig r as a fracio: P Righ had side is a geomeric series wih a 30, r.072, 20 S 2 20 ((.072) ) ) 2 (.072) ((.072) ) ) 2 (.072) 30 2 P 2 30 P (.072) The prese value of he lump-sum of,000 due i 0 years = The prese value of he 30 semi-aual paymes for 0 years = Therefore, he fair marke value of he bod = = Fiacial Mahs Page 36

37 Fiacial Mahs Sample Quesios Se A, quesio 7 - Suggesed soluio Use of he amorisaio formula o fid he value of P 2 : i( i) ( i) A P P 2 A ( i) i( i) Ierpreig ** o he previous page To fid he prese value of he semi aual paymes of 30 is he same as askig wha lumpsum payme ow would have he same fuure value if ivesed a 7.2% AER as a auiy of 30 paid semi-aually for 0 years give a AER of 7.2%. The fuure value (FV) of he lump sum of P 2 is give by P (i) The fuure value (FV) of he auiy of 30 paid semi-aually for 0 years paid a he ed of each 6 moh period: Fiacial Mahs Page 37

38 Fiacial Mahs Sample Quesios Se A Quesio 8 - Suggesed soluio Sample quesio 8 LCHL Pre-payig a loa i.e. payig off a loa early, before he loa s erm is over i.e. payig off he upaid balace Mr. Mooey bough his house i 975. He obaied a loa from he bak for 30 years a a ieres rae of 9.8% APR. His mohly payme a he ed of each moh was 260. I 995, Mr. Mooey decided o pay off he loa. Fid he balace of he loa, i.e. he amou he owed, a ha ime. Noe: We do o kow or eed o kow he origial loa amou. Sice his is a quesio o loa repaymes i is a good idea o look a he prese value(s). Mr. Mooey has made paymes for 20 years (240 mohs) so he sill has 20 paymes o make so he bak should charge him he prese value of hese paymes. Mehod : The amou owig o a loa a ay ime is he prese value of he remaiig repaymes. We ca hik i erms of geig he prese value of 20 paymes of 260 due o be paid a he ed of each moh for 20 mohs x... (i) (.098) (.098) (.098) (.098) 260 The righ had side is a geomeric series wih a, 20 ad r 2 2 (.098) r x S20 a r 2 (.098) Evaluae r ad sore i i he calculaor's memory. Recall i each ime i is eeded i calculaio. x 97, Fiacial Mahs Page 38

39 Fiacial Mahs Sample Quesios Se A Quesio 8 - Suggesed soluio Aleraive way of calculaig x from equaio (i) above: x... (i) (.098) (.098) (.098) (.098) 2 Muliply boh sides by (.098) o ge: x(.098) 260 (.098) 260 (.098) 260 (.098) ** x(.098) (.098) 260 (.098) (.09 8) 20 2 (.098) 20 r 2 x (.098) a 260 r 2 (.098) x 97, ** To raioalise his lie, we could ask he quesio wha lump sum x ivesed for 20 mohs will give he same fial value as a regular ivesme of 260 per moh for 20 mohs? The fuure value (FV) of he lump sum of x is give by x (.098) (ii) The fuure value(fv) of he auiy of 260 paid a he ed of each moh for 0 years: (.098) 260 (.098) 260 (.098) (iii) Equaio (ii) above is wha he bak would ear if i ivesed lump sum x, which is he prese value of he auiy sill owed o he bak. The righ had side shows how much he bak will ge if Mr. Mooey coiues o make he paymes from ow o for he ex 0 years a he ed of each moh. Equaig hese wo fuure values (ii) ad (iii) is wha we have i ** above. Fiacial Mahs Page 39

40 Fiacial Mahs Sample Quesios Se A Quesio 8 - Suggesed soluio Usig he amorisaio formula for Sample Q8 LCHL Se A: 2 (.098) ) i( i) ( i) A P P A , ( i) i( i) 2 2 ((.098) ) (.098) 20 Fidig ou he amou of moey borrowed by Mr. Mooey If you wished o, alhough i is o ecessary for he above quesio, how would you calculae Mr. Mooey s loa? Mehod Loa = sum of he prese value of all he repaymes Loa =... ( i) ( i) ( i) ( i) ( i) ( i) Loa ( i) = 260( i) 260( i) ( i) 260 Reversig he order of he righ had side of he above equaio - Loa ( i) = ( i) ( i) +260( i) Righ had side is S of a geomeric series wih a 260 ad r ( i) ad (( i) ) 260( ) Loa ( i) = i Loa = = Usig he amorisaio morgages ad loas formula (.098) i( i) ( i) A P P A 260 5, ( i) i( i) (.098) (.098) Fiacial Mahs Page 40

41 pricipal porio i euro Ieres porio Fiacial Mahs Sample Quesios Se A Quesio 8 - suggesed soluios Mehod 2: You ca fid he upaid balace by lookig a a amorisaio schedule, which requires kowig how much was borrowed. Mehod 3: If a amorisaio schedule is o available you ca approximae he upaid balace (quie accuraely) as follows: Upaid balace = Curre value of he loa amou curre value of he auiy (I oher words, if he bak had give Mr Mooey he loa bu had ivesed his amou isead, for 240 mohs, a he same ieres rae as hey are chargig Mr. Mooey for he loa, wha would be he differece bewee he curre value of he ivesed loa amou ad he value of he acual ivesme of 260 every moh from Mr. Mooey which hey have ivesed a he same ieres rae for he 240 mohs. This is wha Mr. Mooey owes hem o clear his deb.) Whe a loa is paid off, he upaid balace is zero ad hece he curre value of he loa amou equals he curre value of he auiy. Upaid balace = curre value of he loa amou - curre value of he auiy ( i) = P( i) pym where P is he loa amou, pym is he loa payme, i is he periodic i ieres rae, is he umber of periods from he begiig of he loa o he prese ( ) Upaid balace =5,348.36( ) , , Ieres porio of he payme Payme umber, Pricipal porio of he payme, Fiacial Mahs Page 4

42 Fiacial Mahs Sample Quesios Se B LCHL. (a) Show ha he gross rae before ax (ad afer ax) of 0% i he 3 year savigs bod referred o i his adveriseme is equivale o a AER (aual equivale rae) of 3.23%. (b) Show ha he gross rae of 2% o a 5.5 year savigs bod is equivale o a AER of 3.53%. 2. Alice ad Ke ope a accou wih a ieres rae of 3.4% AER. They deposi 000 a he ed of each year for 0 years. Wha is he fuure (fial) value of heir auiy? How much will hey ear o heir ivesme? 3. The maageme compay of a aparme block esimaes ha hey will eed 30,000 i 4.5 years ime o repai he ouside of he buildig ad commo areas. If regular paymes are made o a (sikig) fud earig 2.75% AER calculae (i) The rae of ieres per moh, ha would if paid ad compouded mohly, be equivale o a effecive aual rae of 2.75%. (ii) How much mus be deposied i he fud a he ed of each moh o mee his arge? (iii) How much ieres will be eared i he 4.5 years? 4. Gradpares Joe ad Melissa wa o sar a regular savigs accou for heir ew gradchild so ha o his 8 h birhday he will have 20,000 o help fud his educaio. How much will hey deposi a he ed of each moh o achieve his arge if hey avail of a regular savigs scheme a 2.5% AER? 5. Eddie plas o deposi 400 a he ed of each moh for 3 years i a accou earig 3.25% AER. Wha sigle sum of moey would Eddie eed o ives ow o achieve he same fuure /fial value? Wha does your aswer mea? 6. Jea wo,000,000 i a loery o be received i four aual paymes of 250,000. She will receive he firs payme i exacly oe year from ow. Wha is he prese value of he four paymes if he ieres rae is 4.2% AER? Did i cos he loery,000,000 o pay Jea her prize moey? Explai. 7. Maria coribued 00 a he ed of each week for 20 years o a pesio (superauaio) fud earig 4.6% AER. Take year o be 52 weeks. (a) Fid he rae of ieres per week, which would if paid ad compouded weekly, be equivale o a effecive aual rae of 4.6%. Fiacial Mahs Page 42

43 Fiacial Mahs Sample Quesios Se B LCHL (b) How much was her lump sum payme (o he eares euro) whe she reired? (c) Fid he rae of ieres per moh, which would if paid ad compouded mohly, be equivale o a effecive aual rae of 3.8%. (d) Maria used her lump sum o buy a auiy a 3.8% AER givig her a allowace a he ed of each moh for he ex 20 years. How much is her mohly allowace o he eares euro? Loa repaymes: A series of loa repaymes is a auiy where he amou borrowed is he prese value of he series of repaymes. The amou owig o a loa a ay ime is he prese value of he remaiig repaymes. 8. Elle ad Mike ge a loa of 200,000 o be repaid a he ed of each moh i a series of equal paymes over 25 years. The ieres rae for he loa is 8.00% APR. Calculae: (i) (ii) (iii) (iv) (v) he rae of ieres per moh, ha would if paid ad compouded mohly, be equivale o a effecive aual rae of 8.00% he amou of each mohly repayme he oal amou o be repaid he oal ieres o be paid If hey had paid forighly, wha would he repayme amou be ad wha would he oal ieres paid be i his case? (Possible exesio quesio: How much would Elle ad Mike save by payig every forigh isead of every moh?) 9. A graduae paid 400 a he ed of each moh for five years o pay back a loa he borrowed while i College a a APR of 8.30%. How much did he borrow? Also: Q6 20 LCHL Mahemaics Paper Fiacial Mahs Page 43

44 Fiacial Mahs Sample Quesios Se B LCHL - Suggesed soluios Suggesed soluios o he above quesios. (a) For every you ives you ge back. i 3 years ime. F P( i). ( i) 3 i (.) 3 i i EAR = 3.23% (as adverised) (b) For every you ives you ge back.2 i 5 years ime. F P( i) ( i) ( i) i (.2) i i EAR = 3.53% (as adverised) 2. Alice ad Ke ope a accou wih a ieres rae of 3.4% AER. They deposi 000 a he ed of each year for 0 years. Wha is he fuure (fial) value of heir auiy? How much will hey ear o heir ivesme? (You kow he regular payme you do kow he fuure value) The sum of all he fuure values of all he regular savigs of 000 per moh is FV (.034) (.034) 000(.034) 8 9 This is a geomeric series wih a300, r (.0575) ad ar ( ).034 S, r.034 They ear o heir ivesme. 4 Fiacial Mahs Page 44

45 Fiacial Mahs Sample Quesios Se B LCHL - Suggesed soluios 3. (i) For every you ives you ge back.0275 i 2 mohs. F P( i).0275 ( i) 2 i (.0275) 2 i i (ii) 30,000 ( ) ( ) ( )... ( ) ( ) P i P i P i P i P i P 30,000 P P( i) P( i)... P( i) P( i) ,000 S of a geomeric series wih a P, r ( i) , 54 30, 000 S P (iii) 54 P Ieres 30, F P( i).025 ( i) 2 i (.025) 2 i i ,000 ( ) ( ) ( )... ( ) ( ) P i P i P i P i P i P 20,000 P P( i) P( i)... P( i) P( i) ,000 S of a geomeric series wih a P, r ( i) , 26 20, 000 S P P Fiacial Mahs Page 45

46 Fiacial Mahs Sample Quesios Se B LCHL - Suggesed soluios 5. Fuure value of he lump - sum deposied a 3.25% AER = fuure value of a auiy of 400 per moh a 3.25% AER F P( i).0325 ( i) 2 i (.0325) 2 i i x(.0325) 400( i) 400( i) 400( i) ( i) 400( i) 400 i o he righ had side of his equaio is he mohly rae of a EAR of 3.25% 3 2 x(.0325) ( i) 400( i) x(.0325) S36 of a geomeric series wih a P, r ( i) , 36 3 x(.0325) S x(.0325) S x 3, ( i) 400( i) 400( i) Eddie would have o have 3,72.45 ow o ives a 3.25% AER for hree years o achieve he same fial value as ivesig 400 per moh for hree years a 3.25% AER. 6. Prese value of he four paymes of 250, , , , , 000 i i i i , 000 S of a geomeric series wih a, r, 4, i , , i i Payig i four isalmes of 250,000 saved he loery 96, Fiacial Mahs Page 46

47 Fiacial Mahs Sample Quesios Se B LCHL - Suggesed soluios Aleraively oe could use he amorisaio formula from Page 3 of he formula ad ables bookle usig P as he subjec of he formula. P 7. (a) A i i i F P( i).046 ( i) 52 4 i (.046) 52 i i (b) Fial (fuure) value 00( i) 00( i) 00( i)... 00( i) 00 i o he righ had side of his equaio is he weekly rae of a EAR of 4.6% F i i ( ) 00( ) ( i) 00( i) 00( i) F S of a geomeric series wih a 00, r ( i) , 040 F S S (c) F P( i) , ( i) 2 i (.038) 2 i i Fiacial Mahs Page 47

48 Fiacial Mahs Sample Quesios Se B LCHL - Suggesed soluios (d) A A A A 68, i i i i A Righ had side is S of a geomeric series wih a, i , r ad , A i i A o he eares euro Aleraively use he amorisaio ad loas formula o page 3 of he formula ad ables bookle i( i) ( ) A P 68, ( i) ( ) o he eares euro 8. F P( i).08 ( i) 2 i (.08) 2 i i Usig he amorisaio ad loas formula o page 3 of he formula ad ables bookle 300 i( i) ( ) A P ( i) ( ) (Aleraively use S of a geomeric series.) (iii) The oal amou o be repaid = , (iv) Ieres paid = 252, Fiacial Mahs Page 48

49 Fiacial Mahs Sample Quesios Se B LCHL - Suggesed soluios 9. Loa = The sum of he prese values of all he repaymes P... i i i i i The righ had side is S of a geomeric series wih a, r, i (.083), 60 i i r 400 i S a = 9, r i i Sore i he calculaor memory ad recall as eeded o faciliae calculaio. i Aleraively use he amorisaio ad loa formula o page 3 of he ables ad formula bookle o calculae P, give A 400, i (.083), 60 P A i i i 9, Fiacial Mahs Page 49

50 Fiacial Mahs Coiuous compoudig ad e Coiuous compoudig ad he umber e For a give omial rae of ieres (Page 32, formulae ad ables bookle) he oly facor ha iflueces he fial value is he umber of compoudig periods per year. To see he effec of differe umbers of compoudig periods per year we ca look a he followig siuaio. Suppose you ives for oe year, a a omial ieres rae of 00% per year, compouded imes durig he year. Wha will be he fial value afer year, if he ieres is added: i. a he ed of he year ii. every 6 mohs, iii. every 3 mohs iv. every moh v. every week, day, hour, miue, secod The resuls are show i he able below: How ofe ieres is compouded Fial value Yearly F ( ) =2 Every 6 mohs 2 F ( ) = Every 3 mohs 4 F ( ) = Every moh 2 F ( ) = Every week 52 F ( ) = Every day 365 F ( ) = Every hour 8760 F ( ) = Every miue F ( ) = Every secod F ( ) = Coclusio: The fial value ges bigger ad bigger bu he rae a which i is growig slows dow ad seems o be geig closer ad closer o some fixed value close o I fac, he will o grow o more ha 2.72 durig he year, regardless of how ofe he ieres is compouded. As, he umber of compoudig periods ges larger ad larger, he urouded amou is geig closer ad closer o he umber called e. Fiacial Mahs Page 50

51 Fiacial Mahs Coiuous compoudig ad e We say ha he Lim e represes he urouded amou o which would grow i a year if he omial rae were 00% ad he umber of compoudig periods per year icreased wihou limi; ha is whe he ieres is compouded coiuously. Wha if we coiuously compouded P for years a a omial aual ieres rae i (i = aual ieres rae expressed as a decimal or a fracio)? i I oher words wha is he Lim P? Wih a lile help from algebra - Le m ad mi i m i mi m Lim P Lim P m m For a fixed i, as m, m as. i mi m m Thus Lim P Lim P P Lim Pe m m m m m A() Pe i he coiuously compouded ieres formula i i i i Fiacial Mahs Page 5

52 Fiacial Mahs Amorisaio Schedule Appedix Formulae for he spreadshee for he amorisaio schedule o page 5 Fiacial Mahs Page 52