CHAPTER 11 Financial mathematics

Transcription

1 CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula to calculate the iterest rate (i) Use the simple iterest formula to calculate the ivestmet period () Compare offers from differet baks Calculate iterest usig the compoud iterest formula (A = P(1 + i) ) Use the compoud iterest formula to calculate iterest Calculate differet time periods ad their correspodig iterest rates Calculate compoud iterest over differet time periods Compare iterest rates Use the compoud iterest formula to calculate the pricipal (P) Use the compoud iterest formula to calculate the iterest rate (i) Use the compoud decrease formula to calculate fial amouts Use the compoud decrease formula to calculate populatio decreases Calculate repaymets o hire purchase agreemets Calculate percetage profit ad percetage loss This chapter covers material from Topic 5: Fiacial Mathematics SUBJECT OUTCOME 5.2: Use simple ad compoud iterest to explai ad defie a variety of situatios Learig Outcome 1: Differetiate betwee simple ad compoud iterest ad extrapolate the advatages ad disadvatages of each i specific situatios Learig Outcome 2: Calculate simple ad compoud iterest over differet periods at specific rates Learig Outcome 3: Do calculatios usig computatioal tools efficietly ad correctly ad verify solutios i terms of the cotext Learig Outcome 4: Use solutios to calculatios effectively to defie the chages that occur over a period 190

2 11.1 THE SIMPLE INTEREST FORMULA We calculate the amout of simple iterest eared by usig the formula where: SI = simple iterest, P = the pricipal (the moey ivested), i = the iterest rate p.a., writte as a decimal, ad = the umber of time periods If we wat to calculate the fial amout received at the ed of the time period, we use the formula A = P + SI where: A = the fial amout received at the ed of the time period, P = the pricipal or the amout ivested ad SI = simple iterest. Note that p.a. meas per aum or per year. Note also that we always roud amouts of moey off to 2 decimal places. S 1) Write each of the followig as a decimal: 9 a) 9% 9% = 100 = 0,09 2) b) 14,5% 14,5% = 14, 5 = 0, P = R5 430 = 5 year a) Calculate SI, the simple iterest eared whe R5 430 is ivested for 5 years at a iterest rate of 9,5% p.a. b) Calculate the fial amout received at the ed of 5 years. Exercise ) Write the followig percetages as decimals: i = 9,5% = i = 9, S = 0,095 p.a. = R ,095 5 = R2 579,25 A = P + SI = R R2 579,25 = R8 009,25 a) 11,25% b) 10,4% c) 8,9% 2) A pricipal of R7 395 is ivested for 6 years at a iterest rate of 10,25% p.a. a) Calculate SI, the simple iterest eared P = = i =. b) Calculate A, the amout received at the ed of 6 years. A = P + SI 191

3 11.2 USING THE SIMPLE INTEREST FORMULA TO CALCULATE THE PRINCIPAL ( P ) A sum of moey is ivested at a simple iterest rate of 10,7% p.a. for 5 years. 1) How much eeds to be ivested i order to make R550 iterest? P =? 2) What amout is received at the ed of 5 years? A =? i = 10,7 % p.a. = 10,7 p.a.= 0,107 p.a. 100 = 5 years S I = R550 P =? 550 = P 0, P 0,107 5 = 0, ,107 5 R1 028,04 = P So R1 028,04 eeds to be ivested. A = P + SI = 1 028, = 1 578,04 R1 578,04 is received at the ed of 5 years. Exercise ) A sum of moey is ivested at a iterest rate of 11,6 % p.a. simple iterest for 6 years. a) How much eeds to be ivested i order to make R1055 iterest? i = = S I = So. eeds to be ivested. b) What amout will be received at the ed of 6 years? A = P + SI 2) A sum of moey is ivested at a iterest rate of 10,75% p.a. for 4 years. a) How much must be ivested i order to make R950 S I o the ivestmet? i = = S I = So. eeds to be ivested. b) How much is saved i total at the ed of 4 years? A = P + SI 192

4 11.3 USING THE SIMPLE INTEREST FORMULA TO CALCULATE THE INTEREST RATE ( i ) R is ivested for 6 years. At the ed of this time, a amout of R is paid out. P = R A = R = 6 years 1) Calculate the simple iterest, S I, received S I =? A = P + SI = S I S I = S I = R ) At what % iterest rate p.a. was the moey ivested? Aswer correct to 2 decimal places. i =? = i i 6 = i = 0,02436 i = 0, % i = 7,79 % p.a. Exercise ) Covert the followig decimals to a percetage (correct to 2 decimal places); a) 0,08 b) 0,12 c) 0,325 d) 0,615 e) 0,022 5 f) 0, ) R is ivested for 4 years. At the ed of this time, a amout of R is paid out. a) Calculate the simple iterest eared. A =... P =.. =. S I =.. A = P + SI... b) Calculate the iterest rate ad write your aswer as a percetage correct to 2 decimal places

5 11.4 USING THE SIMPLE INTEREST FORMULA TO CALCULATE THE INVESTMENT PERIOD ( ) R6 058 is ivested at a simple iterest rate of 9,25% p.a. How may years will it take i order to receive a amout of R10 540,92 at the ed of the ivestmet period? 9, 25 i = 9,25% p.a. = 100 p.a. = 0,0925 p.a. A = R10 540,92 P = R6 058 A = P + SI ,92 = S I , = S I S I = R4 482, ,92 = , , ,0925 = , ,0925 = 8 The moey must be ivested for 8 years. Exercise ) R is ivested at a simple iterest rate of 9,35% p.a. How may years will it take i order to receive a amout of R at the ed of the ivestmet period? A =... P =.. i = ) R8 990 is ivested at a simple iterest rate of 10,25% p.a. How may years will it take i order to receive a amout of R12 675,90 at the ed of the ivestmet period? A =... P =.. =. S I =

6 11.5 COMPARING OFFERS FROM DIFFERENT BANKS Mpho has R5 980 to ivest for 6 years. Bak A offers him a fixed simple iterest rate of 9,5% p.a. PLUS a bous of 5% of the pricipal. Bak B offers him a fixed simple iterest rate of 10,5% p.a. Which bak offers the better ivestmet? BANK A BANK B P = R5 980 = 6 years P = R5 980 = 6 years i = 9,5 % p.a. = 9,5 p.a. = 0,095 p.a. i = 10,5 % p.a. = 10,5 p.a. = 0,105 p.a = ,095 6 = R3 408,60 Bous = 5% of R5 980 = 0, = R299 At the ed of 6 years Mpho would receive R3 408,60 + R299 = R3 707,60 from Bak A. = ,105 6 = R3 767,40 At the ed of 6 years Mpho would receive R3 767,40 from Bak B, which is more tha the R3 707,60 he would receive from Bak A So Bak B offers the better ivestmet. Exercise 11.5 Patricia has R6 400 to ivest for 4 years. Bak A offers her a fixed simple iterest rate of 9,75 % p.a. plus a bous of 2% of the pricipal. Bak B offers her a fixed simple iterest rate of 10% p.a. I which bak should she ivest her moey? BANK A P =... =.. i =.. BANK B P =... =.. i =.. 195

7 11.6 MIXED EXERCISE Remember that ad A = P + SI 1) Jea ivests R for 5 years at 9,5% p.a. S I. a) How much iterest will she receive at the ed of this period? b) How much will she have saved at the ed of this period? 2) How much must Sipho ivest at 9,5% S I p.a. for 6 years i order to ear iterest of R2 500? 3).. a) Calculate the umber of whole years eeded for a ivestmet of R5 990 to grow to R7 876,85 if the iterest rate is 9% S I p.a. b) At what iterest rate (correct to 2 decimal places) must it be ivested for R5 990 to grow to R8 500 after 5 years? 4) For how log (whole years) must Joh ivest R at 9,5% p.a. S I to save R i total? A =. P =... i = =... S I = A =. P =... i = =... S I = A =. P =... i = =... S I = A =. P =... i = =... S I = 196

8 11.7 COMPOUND INTEREST Whe a perso is paid iterest at regular itervals o a sum of moey ivested, without the iterest beig added to the sum ivested, the ivestmet is called simple iterest. The formula for calculatig simple iterest is. We use the formula A = P + S I to calculate the fial amout (A). Whe the iterest is added to the sum ivested, ad iterest is calculated o this ew amout, the ivestmet is called compoud iterest. The formula used for calculatig compoud iterest is A = P( 1+ i ), where A is the fial amout obtaied after the ivestmet period P is the pricipal or the amout ivested i is the iterest rate (writte as a decimal) is the umber of time periods of the ivestmet James ivests R4 000 at 8% p.a. for 5 years. Should he choose: 1) Simple iterest or 2) Compoud iterest? SIMPLE INTEREST P = i = 8% p.a. = 0,08 p.a. = 5 years = ,08 5 = R1 600 A = P + SI = = R5 600 P = i = 8% p.a. = 0,08 p.a. = 5 years A = P( 1+ i ) COMPOUND INTEREST = (1 + 0,08) 5 = (1,08) 5 = R5 877,31 James should choose compoud iterest as he would ed up with a larger fial amout. Exercise 11.7 Paula ivests R5 500 at 10% p.a. for 4 years. Should she choose simple iterest or compoud iterest? SIMPLE INTEREST COMPUND INTEREST CONCLUSION:... P = i =. = 197

9 11.8 USING THE COMPOUND INTEREST FORMULA TO CALCULATE INTEREST ( i ) We use the formula A = P (1 + i ) to calculate the fial amout received at the ed of the ivestmet period. We use the formula C I = A P to calculate the iterest obtaied at the ed of the ivestmet period. 1) Use the CI formula to calculate the amout received at the ed of 8 years if R7 000 is ivested at 5% p.a. 2) How much iterest is received o this ivestmet? P = R7 000 = 8 i = 5 % = 0,05 p.a. A = P(1 + i) = 7 000(1 + 0,05) 8 = 7 000(1,05) 8 = R10 342,19 C I = A P = R10 342,19 R7 000 = R3 342,19 Exercise ) Use the CI formula to calculate the amout paid out at the ed of a ivestmet of R6 955 at a iterest rate of 10,4% p.a. compouded aually for 7 years. P = = i =. 2) How much of this amout is iterest? 198

10 11.9 TIME PERIODS AND INTEREST RATES Iterest o a ivestmet is calculated at the ed of a give time period. This time period is ot ecessarily oe year. The iterest ca be compouded (calculated) aually, half-yearly, quarterly, mothly ad sometimes daily. To fid the iterest rate whe the iterest is compouded more tha oce per year, divide the iterest rate p.a. by the umber of time periods i oe year To fid the umber of time periods, multiply the umber of years by the required umber of time periods i oe year Suppose the ivestmet period is 6 years at a iterest rate of 9% p.a. Suppose the iterest is compouded more tha oce per year, we divide the iterest rate up ito the differet time periods we adjust the time periods accordigly. TIME PERIOD INTEREST RATE PER TIME PERIOD WRITTEN AS A DECIMAL (i) NUMBER OF TIME PERIODS () aually 9% p.a. = 0,09 per year = 6 1 = 6 years half-yearly 9% p.a. = 0,090 2 quarterly 9% p.a. = 0, mothly 9% p.a. = 0, = 0,045 per half year = 6 2 = 12 half-years = 0,0225 per quarter = 6 4 = 24 quarters = 0,0083 per moth = 6 12 = 72 moths Exercise ) Complete the table for a ivestmet period of 7 years ad a iterest rate of 4% p.a. Time period Iterest rate per time period as a decimal (i) Number of time periods () aually half-yearly quarterly mothly 2) Complete the table for a ivestmet period of 8 years ad a iterest rate of 12,5% p.a. Time period Iterest rate per time period as a decimal (i) Number of time periods () aually half-yearly quarterly mothly 199

11 11.10 CALCULATING COMPOUND INTEREST OVER DIFFERENT TIME PERIODS R8 550 is ivested at a fiacial istitutio P = R8 550 for 6 years at a iterest rate of 7,2 % p.a. i = 7,2% p.a. = 0,072 p.a. compouded quarterly. = 0, 072 per quarter = 0,018 per quarter 4 = 6 4 = 24 quarters 1) What amout is due after 6 years? A = P(1 + i) = 8 550(1 + 0,018) 24 = 8 550(1,018) 24 = R13 119,36 2) How much of this is iterest? CI = A P = R13 119,36 R8 550 = R4 569,36 Exercise ) A ivestmet of R8 550 is made at a fiacial istitutio. The iterest rate is 7,2 % p.a. compouded aually for 6 years. a) What amout is due after 6 years? P = i = = b) How much is iterest?. c) Compare your aswers with those i the example above. Do you ear more iterest by compoudig aually or compoudig quarterly?. 2) Use the umbers i questio 1 to compoud the ivestmet half-yearly. a) What amout is due after 6 years? b) How much is the iterest?.. 3) Is compoudig a ivestmet more frequetly a advatage? 200

12 11.11 COMPARING INTEREST RATES Joe wats to ivest R for 6 years. Bak A offers him 9,6% p.a. iterest compouded mothly. Bak B offers 10% p.a. iterest compouded quarterly. 1) Calculate the amout received from each bak. 2) How much iterest does each optio yield? 3) Which offer should Joe take? BANK A BANK B P = R P = R i = 9,6% p.a. = 0,096 p.a. i = 10% p.a. = 0,1 p.a. = 0, 096 per moth = 0,008 per moth = 0, 1 per quarter = 0,025 per quarter 12 4 = 6 12 = 72 moths = 6 4 = 24 quarters 1) A = P(1 + i) 1) A = P(1 + i) = (1 + 0,008) 72 = (1 + 0,025) 24 = (1,008) 72 = (1,025) 24 = R23 871,55 = R24 327,36 2) CI = A P 2) CI = A P = R23 871,55 R = R24 327,36 R = R10 421,55 = R10 877,36 3) Joe should take the offer by Bak B, sice he receives more iterest there. Exercise Vusi wats to ivest R9 560 for 5 years. Bak A offers him 10,2% p.a. iterest compouded mothly. Bak B offers 10,8 % p.a. iterest compouded quarterly. BANK A 1) Calculate the P =.. amout received from each bak. i =.... 2) How much iterest will each optio yield? 3) Which offer should Vusi take?.... = BANK B P =.. i = =

13 11.12 USING THE COMPOUND INTEREST FORMULA TO CALCULATE THE PRINCIPAL ( P ) How much moey must be ivested i order to receive a amout of R i 5 years 1) at a iterest rate of 10,4 % p.a. compouded aually? 1) at a iterest rate of 10,4 % p.a. compouded quarterly? 1) A = R ) A = R i = 10,4 % = 10, 4 = 0,104 p.a. i = 0,104 p.a. = 0, 103 = 0,026 per quarter = 5 = 5 4 = 20 quarters A = P(1 + i) A = P(1 + i) = P(1 + 0,104) = P(1 + 0,026) = P(1,104) = P(1,026) P(1,104) P(1, 026) = 5 5 = (1,104) 1, , ,41 = P R8 902,41 must be ivested ,87 = P R8 737,87 must be ivested. Key Sequece x 5 = Key Sequece x 20 = Exercise A ivestmet amouted to R24 987,50 at the ed of 5 years. 1) How much was the iitial ivestmet at a) 9,5% p.a. compouded aually? b) 9,5% p.a. compouded half-yearly? 2) For which oe would you have to ivest less? 202

14 11.13 CALCULATING THE INTEREST RATE ( i ) A ivestmet of R grows to R i 5 years. 1) What was the percetage iterest rate p.a. compouded aually (correct to 2 decimal places)? 2) What was the percetage iterest rate p.a. compouded half-yearly (correct to 2 decimal places)? 1) P = R A = R = 5 years i =? % p.a. A = P(1 + i) = (1 + i) = (1 + i) , = (1 + i) 5 5 1,47... = 5 (1 + i) 5 1, = 1 + i 1, = 1 + i 1 0, = i i = 0, % = 8,02 % 2) P = R A = R = 5 2 = 10 half-years i =? % p.a. A = P(1 + i) = (1 + i) = (1 + i) , = (1 + i) , = 10 (1 + i) 10 1, = 1 + i 1, = 1 + i 1 i = 0, per half year i = 0, % = 3,9319 % per half year = 7,86 % p.a. Key Sequece = Key Sequece = Exercise A ivestmet of R5 000 icreases to R7 500 i 6 years Calculate the iterest rate as a percetage per aum, correct to 1 decimal place 1) whe iterest is compouded aually 2) whe iterest is compouded half-yearly. 203

15 11.14 A MIXED EXERCISE 1) Calvi's overseas trip i five years time will probably cost R How much must he ivest ow at 12,25% p.a. compouded aually to fud the trip? 2) Doris has R to ivest ad requires R at the ed of 6 years. What iterest rate compouded half-yearly will she eed? Aswer correct to 1 decimal place. 3) Sophie ivested R ad received R13 546,70 i iterest at the ed of 5 years. a) How much has she saved i total after 5 years? b) At what iterest rate did she ivest her moey if it was compouded mothly for 5 years? 204

16 11.15 THE COMPOUND DECREASE FORMULA A ivestmet which grows or icreases i value, over time, is said to appreciate i value. A ivestmet which decreases i value, over time, is said to depreciate i value. Office equipmet ad motor cars are ivestmets that depreciate with time ad become worth less tha what was paid for them. The purchasig power of your moey decreases owig to iflatio. The price of most goods ad services show a steady, compouded icrease, from year to year. For depreciatio we use the formula A = P ( 1 i) where A = the amout received at the ed of the ivestmet time P = the pricipal (the moey ivested) i = the iterest rate p.a. writte as a decimal = the umber of time periods of the ivestmet A ew car cost R i The car depreciates by 11% of its value each year 1) How much will the car be worth i 2008? Give the aswer correct to the earest rad. P = R i = 11 % p.a. = 0,11 = = 4 years A = P( 1 i) = (1 0,11) 4 = (0,89) 4 = The car is worth R i ) By how much will the car have depreciated? The car has depreciated by R R = R i four years. Exercise A motorcar cost R i July Its value depreciates by 10% p.a. compouded aually. 1) How much will the motorcar be worth i July 2008? 2) By how much will the car have depreciated?

17 11.16 POPULATION DECREASE The compoud decrease formula may also be used for calculatig populatio decrease. A tow has a populatio of i Jauary A aual average rate of compoud decrease of 6% p.a. is expected i the populatio after this date. 1) What will be the expected populatio i Jauary 2012? P = i = 6% p.a. = 0,06 p.a. = = 4 years A = P( 1 i ) A = (1 0,06) 4 = (0,94) 4 = The expected populatio i Jauary 2012 is ) Calculate the populatio decrease. Populatio decrease = = Exercise ) A tow has a populatio of i Jauary A aual average rate of compoud decrease of 8% p.a. is expected i the populatio after this date. a) What will be the expected populatio i Jauary 2010? b) Fid the populatio decrease. 2) The populatio of a city i July 2008 is people. The city has experieced a aual average rate of compoud decrease of 7% p.a.. a) What was the populatio i July 2004? b) Calculate the populatio decrease. 206

18 11.17 MIXED EXERCISE 1) Equipmet depreciates at 11% p.a. I Jue 2008, the equipmet was worth R a) What will the equipmet be worth i Jue 2011? b) By how much will it have depreciated? ) If the iflatio rate is steady at 6% p.a. for 5 years compouded aually, what is the purchasig power of R100 after 5 years? 3) Machiery worth R depreciates at a rate of 12% p.a. How much is it worth after 6 years if the rate of depreciatio is compouded aually? 207

19 11.18 HIRE PURCHASE (H.P.) AGREEMENTS Goods bought o HP are take from the store before the full purchase price is paid. A deposit may be paid at the time of purchase ad iterest is charged o the balace. The equal repaymets made may be weekly or mothly. These agreemets ofte charge S I ( ) o the full loa for the full period, ot just o the amout still to be paid. Musa buys goods for R o H.P. He puts dow a 15% deposit ad repays the balace at 16% p.a. S I i equal mothly istalmets over 5 years. 1) Calculate the deposit 2) Calculate the balace owig after iterest has bee icluded 3) How much will he pay per moth? 1) Deposit = 15% of R = 0,15 R = R ) Balace owig after payig deposit = R R5 190 = R P = R = 5 years i = 16% p.a. = 0,16 p.a. = ,16 5 = R Total amout owig, A = P + SI = R R = R ) = 5 years = 5 12 = 60 moths Mothly repaymets = 60 = R882,30 per moth Exercise Neo purchases a motorcar for R She pays a deposit of 15% ad pays the rest off i equal mothly istalmets at a iterest rate of 12,5% p.a. SI over 5 years. 1) How much is her deposit? 2) How much does of the origial price does she still owe? 3) How much does she owe i total oce iterest has bee added? 4) What is her mothly repaymet? 208

20 11.19 PROFIT AND LOSS The profit (P) or the loss (L) is the differece betwee the sellig price (SP) ad the cost price (CP). The profit will be a positive amout ad the loss will be egative. P = SP CP L = CP SP The profit or loss is usually calculated as a percetage of the cost price. P L Percetage Profit = 100 % Percetage Loss = % CP CP 100 1) A baker sells biscuits for R12,55 per bag. He calculates that each bag of biscuits costs him R6,50 to produce. a) What is his profit o each bag? b) What is his percetage profit (correct to 2 decimal places) o each bag? 2) The baker sells his old stock at half-price, at R6,28 per bag. a) What is his loss per bag? b) What is his percetage loss (correct to 2 decimal places) o each bag? a) P = SP CP = R12,55 R6,50 = R6,05 b) Percetage Profit 6,05 = ,50 = 93,08% a) L = CP SP = R6,50 R6,28 = R0,22 His loss per bag is 22c b) Percetage Loss = 0,22 100% 6,50 = 3,38% Exercise ) A bookshop sells a book marked R149,50. The cost of the book to the shop is R75,50. a) Calculate the profit made o each book sold. b) Fid the percetage profit (correct to 1 decimal place) made by the shop o each book sold.... 2) Owig to poor sales, the bookshop decides to sell the book i 1) for R109,20. a) What discout are they givig o the origial sellig price? b) Calculate the profit made o each book sold at the ew sellig price c) Calculate the ew percetage profit, correct to the earest whole umber. 209