FINANCIAL MATHEMATICS : GRADE 12

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Rolf Lester
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1 FINANCIAL MATHEMATICS : GRADE 12 Topcs: 1 Smple Iterest/decay 2 Compoud Iterest/decay 3 Covertg betwee omal ad effectve 4 Autes 4.1 Future Value 4.2 Preset Value 5 Skg Fuds 6 Loa Repaymets: 6.1 Repaymets wth Future Value Formula. 6.2 Preset value formula 7 Balace o a loa 8 Calculato of Tme Perod 9 Mcroleders 10 Pyramd Schemes

2 Itroducto: No busess ca exst wthout the formato gve by fgures. Borrowg, usg ad makg moey s the heart of the commercal world thus the prcple of terest ad terest rate calculatos are extremely mportat. Ths leads to a examato of the prcples volved assessg the value of moey over tme ad how ths Iformato ca be utlzed the evaluato of alterate facal decsos. Remember that the facal decso area s a mefeld the real world, full of tax mplcatos, deprecato allowaces, vestmet ad captal allowaces etc. The basc prcples facal decso makg are establshed through the cocept of terest ad preset value: Defto of terest: Iterest s the prce pad for the use of borrowed moey Iterest s pad by the user of the moey to the suppler of t. It s calculated as a fracto of the amout borrowed or saved over a certa perod of tme. Ths fracto s also kow as terest rate ad s expressed as a percetage per year (per aum). Preset Value of moey s the value of the tal vestmet:.e. P V (Preset Value) = P (Prcple) PV or P t = term r = terest FV or S = P(1 + rt) PV = Preset Value or Prcple FV = Future Value or Sum Assured

3 Smple terest Smple terest: s computed o the prcple for the etre term of the loa ad s thus due at the ed of term. Growth Decay S I = Prt A P( 1 ) A P( 1 ) where: I s the terest pad or eared P s the prcple or Preset value r s the terest rate per aum t s the tme or term of loa s the umber of years Compoud Iterest Compoud terest arses whe, a trasacto over a Exteded perod of tme, terest due at the ed of a paymet perod s ot pad, but added to the prcpal. Thus terest also ears terest.e. t s compouded. The amout due at the ed of trasacto perod s referred to as the compouded amout or accrued prcpal. Iterest perods ca vary : daly, mothly, quarterly, half-yearly or yearly. Formula Compoud Growth: A P( 1 ) Compoud Decay: A P( 1 ) OR r Fv P( 1 ) 100s s F = Amout or Future Value P = Prcpal or Ital value r = rate of terest per aum = umber of years vested OR r Fv P( 1 ) 100s s s = umber of tme perods terest s calculated ( aum, quarterly, half yearly, mothly or daly)

4 Further Formulae 1. Fdg Prcple: P A( 1 ) or 2. Fdg the terest rate: r P Fv(1 ) 100s s 1 Fv s s 1 P NB: To get rate (r), multply by 100. Nomal Iterest rates: 1.1. I cases where terest s calculated more tha oce a year, the aual rate quoted s the omal aual rate or omal rate. Effectve Iterest rates: 1.2. If the actual terest eared per year s calculated ad expressed as a percetage of the relevat prcpal, the the so-called effectve rate s obtaed. Ths s the equvalet aual rate of terest that s, the rate of terest eared oe year f compoudg s doe o a yearly bass. Covertg Nomal Rate to Effectve Rate: Formula r eff =100[(1+ r om 100s ) 1 ] where: r eff : effectve rate (percetage) r : omal rate (percetage) s : o. of perods oe year e.g. Calculate the effectve rate correspodg to omal rate of terest of 22% p.a. compouded baually. r om =22 s = 2 (baually) r eff = 100[( (2) )2-1] = %

5 Autes Defto: A auty s a sequece of equal paymets at equal tervals of tme. The paymet terval of a auty s the tme betwee successve Paymets whle term s the tme from the begg of the frst paymet terval to the ed of the last paymet terval. Types of Autes: Ordary auty- auty where paymets are made at the ed of each paymet terval. Auty due- auty where paymets are made at the begg of the paymet terval. Future Value Autes Defto: Regular paymets to a savg accout. Iterest grows the vestmet. The formula for the sum of a geometrc seres s used facal maths to calculate values of autes. S a( r 1) r 1 a s the frst term or paymet made r s the commo rato ( 1 ) s the umber of paymets or o of terms the seres. Future value formula As well as the sum of geometrc seres formula, a more useful formula s: where: F = x[(1 + ) 1] x : paymet amout : terest rate : umber of paymets The above formula F ca oly be used f there s a fal paymet at the ed whch does ot ear terest (ordary auty).

6 Example: 1) Suppose R1000 s vested every moth, startg oe moth from ow for 10 moths. Iterest rate of 18% p.a. compouded mothly, calculate the accumulated amout T 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 Paymet starts 1 moth from ow Last paymet does ot ear terest F = x[(1+) 1] = = 0.18 = 1000[( )10 1] = ) Km deposts R2000 mmedately to a savgs accout, cotug to make mothly paymets at the ed of each moth for 10 years. Iterest rate s 24% p.a. compouded mothly T 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10. T 118 T 119 T 120 Paymet mmedately Last paymet does ot ear terest Note: There wll be a total of 121 paymets. Why? wll be 12 moths X 10 years =120 but oe more paymet ( the mmedately paymet) must be added. Paymet starts mmedately, so ths s the extra paymet at T 0. There wll be a mmedate paymet ad addtoal paymets at the ed of each moth. Therefore there wll be 121 paymet total. F = x[(1+) 1] = = 0.24

7 [(1 + = 12 )121 1] = R Summary: Cases whe F = x[(1+) 1] formula may be used Case 1: Whe paymets made at the ed of the moth. Case 2: Whe paymets made oe moth from ow (.e. Oe tme perod from ow). Case 3: Whe paymet mmedately ad cotued at the ed of each tme perod. (NB: Oe extra paymet) Case 4: Start ad ed o brthday. NB: Oe extra paymet. All these cases have oe thg commo: the fal paymet does ot ear terest. Wheever ths s the case we ca use the above formula. Whe the fal paymet does ear terest, we use a dfferet formula. See Gaps below. Gaps Rule: Whe paymet begs mmedately ad whe there s a gap betwee the last paymet ad the ed of the year.e. whe paymets made at the begg of the tme perod (Auty due), the the followg formula s used: F = x(1+)[(1+) 1] Example Te equal paymets of R8000 are made to a savgs accout aually at the begg of each year ( effectve mmedately). Calculate the total accumulated amout at the ed of 10 years f a terest rate of 9% compouded aually s appled T 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 Paymet starts mmedately Gap at the ed, last paymet ears terest T 1 represets the begg of the 2 d year ad the ed of the 1 st year. Smlarly T 9 represets the begg of the 10 th year as well as the ed of the 9 th year. 10 paymets are made at the begg of each year (startg mmedately), the last paymet wll therefore be made at the begg of the 10 th year, but terest wll be accumulated utl the ed of the 10 th year.

8 Therefore there s a gap at the ed betwee the last paymet ad utl the terest stops accumulatg. Because of ths gap we use the formula: F = x(1+)[(1+) 1] The soluto to the prevous example wll therefore be: F = x(1+)[(1+) 1] = (8000(1+0.09)[(1+0.09)10 1]) 0.09 = R ,35 = = 0.09 Preset Value Autes Defto: Regular paymets to a loa accout, terest accumulated s the eemy. Preset Value Formula: where: P = x[1 (1 + ) ] P: preset value : terest rate : umber of paymets x: paymets of auty Note: The P formula ca oly be used f there s a gap betwee the loa ad the frst paymet. The gap must be oe perod. Skg Fuds Compaes ofte purchase equpmet ad use t for a specfed tme perod. The old equpmet s the sold at scrap value ad ew, upgraded equpmet s bought. I order to face the purchasg of the ew equpmet, the compay wll, advace, have set up a auty called a skg fud. Importat terms: Book Value of a asset s ts value after deprecato has bee take to accout. Scrap Value s the book value of a asset at the ed of ts useful lfe. A skg fud s a fud set up to replace a asset at the ed of ts useful lfe.

9 Example A school buys a photocopyg mache that cost R It deprecates at 22% p.a. reducg balace. Its useful lfe s 5 years, ad a ew mache wll flate at 19% p.a. effectve. Old mache wll be sold at scrap value 5 years ad the proceeds wll be used together wth a skg fud to buy a ew mache. The school wll pay mothly ad ear at a 14.4 % p.a. compouded mothly rate. The frst paymet wll be made mmedately ad the last at the ed of the 5 year perod. 1) Scrap value (Use compoud decay formula ad deprecate mache for 5 years). A=P(1-) = (1-0.22) 5 = R ) Cost of ew mache 5 years(use compoud terest to flate mache for 5 years) A=P(1+) = (1+0.19) 5 = R ) Amout requred skg fud 5 years skg fud= cost of ew equpmet scrap value = = ) Fd equal mothly paymets(nb: There wll be oe extra mmedate paymet) No. of paymets: 61 Solve for x F = x[(1 + ) 1] = x[( )61 1] x = R =14.4/100=0.144 Loa Repaymets Loa repaymets usg the Future-Value formula: Whe usg the future value: Load(plus terest)= Repaymet(plus terest) James buys a car for R ,00 ad takes a loa from the bak. Calculate hs mothly repaymets f the loa s for 5 yrs. The bak charges 9% terest p.a. compouded mothly. Repaymets start a moth after the loa s draw.

10 T0 T1 T2 T59 T60 R x x x x I ths soluto we calculate the loa wth the terest accrued at the ed of the 5 yrs. The repaymets are also calculated at T 60 together wth the terest eared. The repaymets plus the terest eared are the equal to the loa plus the terest at the ed of the 5 yrs. Loa ( plus terest) = repaymets (plus terest) 0, ,09 x , = R 2 491,00.e 0,09 0, x 60 0, I geeral : m t erest P m m m tme perods x Where: m P loa amout 1 1 m x mothly repaymets Notes: 1 I geeral loas start oe moth after loa has bee draw. 2 I some stuatos, such as home loas, t ca be arraged that the repaymets start 3 moths after loa s draw etc. Loa repaymets usg the Preset-Value formula: P A(1 ) or A P ( 1 ) or P v A ( 1 ) v

11 It s more commo to use the above formula to calculate repaymets ad balaces tha the Futurevalue formula. Example: A loa s take out to buy a TV set wth surroud soud. The loa s repad wth two equal paymets of R5000,00. The frst paymet s made oe year after he bought the set, ad the secod oe year later. Iterest s calculated at 6% p.a. effectve. Calculate the tal value of the loa. Method 1: Use the Preset-value formula. The repaymets represet the future value ad clude the terest eeded to repay the loa. The preset value of the frst repaymet s: P A(1 ) 5000(1 0,06) 1 R4716,98 The P v of the 2 d loa repaymet s: P A(1 ) 5000(1 0,06) 2 R4449,98 Total value of the loa = R4716,98 + R4449,98 = R9166,96 I oe calculato: 5000(1 0,06) (1 0,06) 2 = R9166,96 Method 2: Usg the future value formula (1 0,06) x 2 (1 0,06) x R9166,96

12 Example 2: A loa of R90 000,00 s take out ad repad by equal mothly stallmets over a 3 year perod. Iterest s set at 7,5% p.a. compouded mothly. Calculate the mothly repaymet. x x P ,075 0, , = R2799,56 Balace o a loa balace outstadg =loa (plus terest) repaymets(plus terest) Example: A car s purchased for R ,00. 10% depost s pad ad the balace s faced through a bak loa. The loa s for 6 years at a terest rate of 10,5% compouded mothly. Task: a) Determe the mothly repaymets. b) Determe the balace owed o the loa at the ed of two years mmedately after the 24 th paymet. Soluto: a) Loa s for R ,00 (.e 90% of R ,00.) T0 T1 T2 T71 T72 R x x x x 0, ,105 0,105 x x1 x paymets

13 0,105 0, x 72 0, = R3042,19 72 b) 0, , , , R199673,37 R80853,309 R118820, NB: The calculato ca be splt to two sectos: 1. 0, = R , , , , =R80 853,309 Fal Aswer: 1 2 = R ,07 If you have a ew calculator.e the Caso fx 82ES or later verso the the calculato ca be doe as oe calculato BUT t would be prudet to wrte dow the equato used to show the examer. I log term loas (.e.home Loas) the repaymets the early stages cover mostly terest wth a small amout towards captal reducto. Ths reverses the latter stages of the loa.

14 Calculato of tme perod (see log rules for more o logs) Example 1: To calculate the tme perod logarthms are used. A vestmet of R24500 at a terest rate of 9%. The vestmet grows to R 48817,78 after years. Calculate : A P( 1 ) 48817, (1 0,09) 48817,78 (1 0,09) ,09 1,9925 log1,09 log1,9925 log1,09 7, yrs log1,9925 Example 2: R ,00 s deposted to a savgs accout. Iterest s pad at 8,5% p.a. Compouded aually. How log wll t take for the prcple to double? A P( 1 ) P = thus A = = 0,085 log1,085 log 2 log1,085 8,4965 log 2 0, yrs 182days 8yrs 6moths

15 Mcroleders They offer short term loas at very hgh terest rates. The terest s calculated upfrot usg smple terest based o the full amout of the loa over the repayable perod. There s o advatage early settlemet. The followg s from a advert for stat face: Do you eed to borrow moey urgetly? You could have stat cash your had wth 24 hours. You have bee specally selected to receve ths loa from FSP. Whether you wat to add value to your homewth some reovatos, or spol yourself to a dream holday, stat cash up to R s avalable to you rght ow. Amout 24 moths 36 moths 48 moths 60 moths R8000 R500 R388 R333 R300 R16000 R960 R737 R626 R560 R20000 R1183 R905 R767 R683 R25000 R1479 R1132 R958 R854 Use ths hady stallmet table to choose the loa that wll sut your budget ad crcumstaces. Fd the loa amout you eed ad choose the repaymet perod that offers you a mothly repaymet you feel comfortable wth. Choce of loa term: Fxed terest rate: Cash to use as you choose: Up to 5 years to repay your loa. For the full term of your loa. For aythg that s mportat to you. BEWARE OF THESE TYPES OF SCHEMES AS THEY CAN COST YOU AN ARM AND A LEG. EG: R25000 over 5 yrs = R854 X 60 = R51240 Iterest rate: A P( 1 )

16 (1 5) , , % Pyramd Schemes A pyramd scheme s a scam that reles o ew vestors to provde moey to those above them the pyramd. These schemes have bee aroud for a log tme ad exst oe form or aother all over the world. They rely o gullblty ad greed. The followg s a theoretcal example of how a pyramd scheme works. Each vestor pays R5000 ad recruts oly 2 others to vest to the scheme. Oce the vestor gets 2 more recruts the fud wll be a further R He s the refuded hs R5000 that he pad to the scheme ad the other R5000 s shared by the other vestors above hm the pyramd. As the pyramd grows below hm so does hs share of the spols.

GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 3 (LEARNER NOTES)

GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 3 (LEARNER NOTES) MATHEMATICS GRADE SESSION 3 (LEARNER NOTES) TOPIC 1: FINANCIAL MATHEMATICS (A) Learer Note: Ths sesso o Facal Mathematcs wll deal wth future ad preset value autes. A future value auty s a savgs pla for