Financial Mathematics Written by : T Remias

Financial Mathematics Written by : T Remias Page 1

CONTENTS PAGE CONTENTS PAGE Financial Maths (def)..... 3 Types of growth / interest.... 3 Appreciation..... 7 Depreciation..... 7 Nominal interest rate..... 11 Effective interest rate.... 11 Future value of an annuity.... 14 Sinking fund..... 15 Present value of an annuity.. 18 Outstanding balance. 21 Bibliography. 24 Page 2

What is Financial Mathematics? Finance may be defined as the study of how people allocate scarce resources over time. Financial mathematics is a collection of mathematical techniques that find applications in finance, e.g. Asset pricing: derivative securities Hedging and risk management Portfolio optimization Structured products In the business world, information is given by figures. Borrowing, using and making money it the heart of the commercial world thus the principle of interest and interest rate calculations are extremely important. Short History of Financial Mathematics: In 1990: Bachelier used Brownian motion as underlying process to derive option prices. In 1973: Black and Scholes published their PDE-based option pricing formula. On 1980: Harrison and Kreps introduced the martingale approch into mathematical finance. Financial Mathematics has been established as a separate academic discipline only since the late eighties, with a number of dedicated journals Types of growths 1 Simple growth/simple interest (linear growth) The simple growth formula is given by: Also note that : ( ) ( ) 2 Compound growth/compound interest (exponential growth) The compound growth formula is given by: Also note that: ( ) Page 3

Where: ( ) Note that : A is the accumulated (future) amount (value after investment/loan period) P is the present (initial) amount (value at the beginning of the investment period) i is the interest rate as a decimal n is the number of years of investment/loan period/depreciation period/interest bearing period If interest is compounded then divide the interest rate (i) with the intervals in which the interest is compounded and the number of years is multiplied by the interest bearing periods. Suppose the interest is compounded monthly then divide (i) by 12 and multiply n by 12 EXAMPLES QUESTION 1 Compounded monthly i = and n = 12n Compounded quarterly i = and n = 4n Compounded half yearly i = and n = 2n Compounded daily i = and n = 365n Damiel wants to invest R3450 for 5 years. Absa offers a savings account which pays simple interest at a rate of 12,5% per annum and Capitec offers a savings account paying compound interest at a rate of 10.4% per annum. Which bank account would give Damiel the greatest accumulated balance at the end of the 5 year period SOLUTION Absa offer P = 3450. i = 0,125.n = 5 Capitec offer A = P(1 + in) A = 3450 ( 1 + (0,125)(5)) = R5606,25 P = 3450 Page 4

. i = 0,104.n = 5 A = P(1 + i) n A = 3450(1 + 0,104) 5 = R5658,02 The greatest accumulated balance is at Capitec QUESTION 2 Micky decides to put R30 000 in an investment account. What is the compound interest rate must the investment account achieve for Micky to double his money in 6 years? SOLUTION A = 60 000 P = 30 000. i =?.n = 6 A = P(1 + i) n 60 000 = 30 000(1 + i) 6.i = 0,12246.... Activity QUESTION 1 Mr Remias would like to buy a house and need a loan of R500 000. He applied to four different financial institutions, and they offer him the following : ABSA A simple interest rate of 18% p.a. over a period of 5 years. The loan must be repaid in 60 equal instalments. Page 5

FNB An interest rate of 24% per annum, compounded monthly, repayable in equal instalments over 4 years CAPITEC An interest rate of 20% p.a. compounded monthly, repayable in equal monthly payments over 5 years STANDARD BANK An interest rate of 18% p.a. compounded quarterly, repayable in equal monthly payments over 6 years Determine : 1.1 The monthly instalments for each option (6) 1.2 The total amount repaid for each option (3) 1.3 The amount of interest paid for each option (3) 1.4 Which bank offers the best option? Why? (2) QUESTION 2 2.1 A man invests R20 000. After 5 years the value of the investment is R37 543,75. Calculate the interest rate if the interest was compounded half yearly. (4) 2.2 R9 723,94 is invested at Absa by Mr Remias with interest compounded semi-yearly. Calculate the interest rate if after 4years the amount is R14 764,12 (4) 2.3 How long does it take for an amount to double if the interest rate is 4,57 % compounded daily (4) 2.4 An amount is invested and accumulates to R52 794,21 after 7years at 23% compounded yearly. Calculate the amount invested (4) 2.5 A vehicle is bought for R3,2 million and after 4years it is sold for R1,01 million. Calculate the rate of depreciation if depreciation was done on reducing balance method (4) [20] QUESTION 3 3.1 R5 000 is invested at 9.6% p.a., interest is compounded quarterly. After how many years will the investment be worth R35 000? (4) 3.2 At what annual percentage interest rate, compounded quarterly, should a lump sum be invested in order for it to double in 6 years (4) 3.3 Sipho invests R5 000 at an interest rate of 8.4% per annum, compounded annually. After how years will he be able to withdraw R8 793 (4) [12] [14] Page 6

QUESTION 4 4.1 Sandra invests R60 000 for 18 months at 11,2% p.a. compounded annually. Calculate the interest she earned after 18 months. (3) 4.2 The bank offers katlego a loan at a nominal rate of 15% p.a. compounded monthly. Calculate the effective annual interest rate that he has to pay. (3) 4.3 Denny invests R3 000 in a savings account. The interest rate for the first 5 years is 9% p.a. compounded monthly, thereafter the interest rate is changed to 10% p.a. compounded semi-annually for the next 7 years. Determine the amount of money that Denny had in his savings account at the end of the investment period. (5) 4.4 At what annual percentage interest rate, compounded quarterly, should a lump sum be invested in order for it to double in 6 years? (5) 4.5 R20 000 is invested at 14% p.a. compounded monthly. After how many years will the investment be worth R52 987,69? (4) 4.6 Determine the time taken in years, for a sum of money to double if the interest rate is QUESTION 5 12.64% p.a., compounded half-yearly. (4) R5 000 000 is to be invested for the next 5 years. A grade 11 mathematics learner requires advice on the following investment options: - An investment fund retail bond offering 12% per annum compounded half yearly. - A bank offering 10% per annum compounded quarterly. 5.1 Compare by means of calculations the two options (4) 5.2 Which option will be the best for the learner? (2) APPRECIATION It is the increase in value of an asset Use the formula but sometimes one can use that is if required in the question Normally addressed when dealing with sinking funds DEPRECIATION [24] [6] It is the loss in value of an asset while being used Book value / Residual value / Scrap value value of an asset after depreciation Page 7

There are two types, namely 1 Reducing balance method / Compound decay use the formula Also note that: (( ) ) ( ) 2 Straight line method / linear decay Use the formula Also note that: ( ( ) ) ( ( ) ) EXAMPLES QUESTION 1 A car is valued at R240 000. If it depreciates at 15%p.a. using straight line depreciation, calculate the value of the car after 5 years SOLUTION P = 240 000. i = 0,15.n = 5 A = P(1 in) Page 8

A = 240 000 (1 (0,15)(5)) QUESTION 2 = 240 000(0.25) = R60 000 The value of the car after 5 years is R60 000 A school buys a bus for R950 000, which depreciates at 13,5% per annum. Determine the value of the bus after 3 years if the depreciation is calculated on a reducing balance method SOLUTION Activity P = 950 000. i = 0,135.n = 3 A = P(1 i) n QUESTION 1 A = 950 000 (1 (0,135)) 3 = 950 000(0.865) 3 = R614 853,89 The value of the bus after 3 years is R614 853.89 1.1 A vehicle is bought for R3,2 million and after 4years it is sold for R1,01 million. Calculate the rate of depreciation if depreciation was done on reducing balance method (4) 1.2 After how many years will a vehicle originally valued at R980 000 decrease to R147 000 if the rate of depreciation is 10 % per annum on a reducing balance. (5) 2.3 A firm buys computers. After 5 years the value of the computers will be R12 000. Calculate the value of the computers if 1.3.1 the computers decrease at a rate of 12% on a straight-line basis (3) 1.3.2 the computers decrease at a rate of 15% on reducing balance basis (3) [15] 1.4 After 4 years of reducing balance depreciation, an asset has a quarter of its original value. The original value was R86 000. Calculate the depreciation interest rate, as a percentage (4) [8] 1.5 A photocopier valued at R12 000 depreciates at a rate of 9% p.a. on the reducing balance method. After how many years will its value be R7 500? (4) Page 9

1.6 Determine how long in years, it will take for the value of a motor vehicle to decrease to 25% of its original value if the rate of depreciation based on the reducing balance is 12% per annum. (4) 1.7 Genesis purchased a new laptop for R16 176. The laptop depreciated on the reducing balance QUESTION 2 method to a value of R9 388,05. Determine his laptop s rate of depreciation after 3years if the interest is compounded monthly. (4) 2.1 The number of cormorants at the vaal river mouth is decreasing at a compound rate of 8% p.a. If there are now 10 000 cormorants, how many will there be after 18 years time (3) 2.2 The population of the metropolitan city of Johannesburg decreases at a reducing balance rate of 9,5% per annum as people migrate to small towns. Calculate the decrease in population over a period of 5 years if the initial population was 4 001 937 (3) 2.3 A 20kg watermelon consists of 98% water. If it is left outside in the sun it loses 3% of its water QUESTION 3 each day. How much does it weigh after a month of 31 days? (3) 3.1 Six years ago, Peter s drum kit cost him R12 500. It has now been valued at R2 300. What rate of simple depreciation does this represent? (3) 3.2 A business buys a truck for R560 000. Over a period of 10 years the value of the truck depreciates to R0 using the straight line method. What is the value of the truck after 8 years? (3) 3.3 Tanya bought a new car 7 years ago for double what it is worth today. At what yearly compound QUESTION 4 rate did her car depreciate? (3) The total number of AIDS deaths in South Africa reached 314 196 in 2005. Over the next 4 years there was a decline in the number of deaths, so that in 2010 there were 270 107 reported Aids deaths in South Africa. Using the straight line method of depreciation, determine the rate per annum at which the Aids deaths had decreased over these 4 years, correct to one decimal place [4] [30] [9] [9] Page 10

QUESTION 5 Allen leaves a bowl of soup to cool. The temperature of the soup decreases on a reducing balance method at a rate of 4% per minute, and it is now 65 0 C. Calculate the temperature of the soup when she served it 6 minutes ago. QUESTION 6 Rhino poaching is a serious problem, and has resulted in rhinos becoming an endangered species, particularly the black rhino. Statistics suggest that there were 60 000 black rhino in 1970 but that only 4 200 remained in 2011. 6.1 Determine the annual rate of depreciation that these statistics represent if the rate is : 6.1.1 Simple decay (2) 6.1.2 Compound decay (3) 6.2 Using your answer to 6.1.2; determine the number of black rhino there would be in 2050 if this rate of compound decay continues (4) 6.3 Melissa has just bought her first car. She paid R145 000 for it. The car s value depreciates on the straight line method at a rate of 17% per annum. Calculate the value of Melissa s car 5 years after she bought it (3) QUESTION 7 7.1 Calculate how long it would take for an investment to become one third of its original value at a rate of simple decay of 12% p.a. (4) 7.2 Kevin has a motorbike that costs him R8 000. He wishes to replace it with the equivalent model in 5 years time. If depreciation on his motorbike is at 15% per annum on the reducing balance method, and inflation is at 5,3%p.a., calculate: 7.2.1 The expected cost of the equivalent model in 5 years time (3) 7.2.2 The expected book value of his motorbike in 5 years time (3) 7.2.3 The amount of money Kevin is expecting to have to pay in if he sells his old motorbike to offset the total cost of the new one. (2) Nominal interest rate Is the rate of interest that banks and financial institution quote Effective interest rate [4] [12] [12] Is the actual interest rate received per year It is more than nominal interest rate Page 11

NOTE: In cases where interest is calculated more than once a year, the annual rate quoted is the nominal annual rate or nominal rate. If the actual interest earned per year is calculated and expressed as a percentage of the relevant principal, then the so-called effective rate is obtained. This is the equivalent annual rate of interest that is, the rate of interest earned in one year if compounding is done on a yearly basis. The Nominal Interest Rate is the interest rate that is quoted when interest is compounded more or less frequently than the time period specified in the interest rate. Example: Interest rate of 8% per annum, compounded semi-annually. The Effective Interest Rate is a converted nominal interest rate, so that the compounding frequency is the same as the interest rate quoted. The formula ( ) ( m is the number of interest bearing periods eg compounded monthly m = 12) The left hand side is effective interest rate & the right hand side represent nominal interest rate NOTE ( ) ( ( ) ) EXAMPLES QUESTION 1 Interest on a credit card is quoted as 23% p.a. compounded monthly. What is the effective annual interest rate? Give your answer to two decimal places SOLUTION.m = 12.i nom = 0.23 ( ) ( ) QUESTION 2 Determine the nominal interest rate compounded quarterly if the effective interest rate is 9% per annum. Give your answer to two decimal places Page 12

SOLUTION.m = 4.i eff = 0,09 ( ( ) ) ( ) Activity QUESTION 1 1.1 When the nominal interest rate is 12% per annum, compounded monthly, what is the effective interest rate? (4) 1.2 Mr Remias can invest his money at ABSA at 13% per annum, compounded monthly, and at FNB at 15% per annum compounded quarterly. Calculate the effective interest rate in each case. Which bank offers the best investment? (4) 1.3 If the effective interest rate 13% on an investment, what is the nominal interest rate if the interest QUESTION 2 is compounded monthly? (4) 2.1 Calculate the effective yearly rate if an investment offers a nominal rate of 13,5% p.a. compounded: 2.1.1 Quarterly (3) 2.1.2 Monthly (3) 2.2 Calculate the nominal interest rate if the effective interest rate is: 2.2.1 15%, while interest is compounded quarterly (3) 2.2.2 13.5% while interest is compounded monthly (3) 2.3 Emma s investment grows from R7 000 to R9 304,60 in a period of three years. Interest is compounded quarterly. Calculate: 2.3.1 the nominal interest rate on her investment (3) [12] Page 13

2.3.2 the effective interest rate on her investment (3) QUESTION 3 3.1 The effective interest rate is 14,47%. Calculate the nominal annual rate of interest if interest is compounded quarterly. (3) 3.2 R9 723,94 is invested at Absa by Mr Remias with interest compounded semi-yearly. Calculate the interest rate if after 4years the amount is R14 764,12 (4) [7] QUESTION 4 Consider the following 16,8% p.a. compounded annually 16,4% p.a. compounded monthly 16,5% p.a. compounded quarterly 16.6% p.a. compounded half yearly 4.1 Determine the effective annual interest rate of each of the nominal rates listed above (4) 4.2 Which is the best interest rate for an investment? (4) 4.3 Which is the best interest rate for a loan? (4) 4.4 Mpho deposits R15 000 into savings account to save for a holiday in 3 years time. The account offers her 10%p.a. compounded monthly. 4.4.1 How much will she have accumulated at the end of 3 years? (4) 4.4.2 What was the effective interest rate that Mpho received each year? (3) 4.4.3 How much interest had Mpho received at the end of 3 years (2) Future Values of an Annuity Suppose you would like to save money for a specific reason [buying a television or buying a car] you invest a single initial amount but deposit fixed monthly amounts into a savings account or an investment fund. Compound interest is paid on the money deposited into the fund. Therefore the future value of an annuity is the sum of all the deposits made plus all the interest earned. Formula [24] [21] Page 14

Also note: ( ) If payments are made immediately at the beginning of investment period (month) and ends at the beginning of the last month then the formula is If payments are made immediately at the beginning of investment period (month) and ends at the end of the last month then the formula is. Where: = Future Value = Fixed payments per period = Interest rate as a decimal = number of payments If the last payment is not made at the end of the period therefore use the formula for the sum of a geometric series. With annuities, the last payment is usually made at the end of the period and the first payment at the end of the first time interval. However if a payment is made at the end of the first interval e.g. at the beginning of the first month remember to add that extra payment to the value of. SINKING FUNDS Is money that a company sets aside in order to make provision for a large capital expense or to replace equipment after a certain period of time. A sinking fund is simply a savings plan or a future value of an annuity. Use the future value of an annuity formula when calculating the monthly deposit and /or the sinking fund period and/or interest rate under sinking fund In some cases sinking fund is equal to value of asset in future after appreciation minus the value of old asset after depreciation Page 15

EXERCISE QUESTION 1 1.1 A company plans to replace equipment in 5 years time at an expected cost of R100 000. Calculate the equal monthly payments that they would need to make into a sinking fund to be able to buy the new equipment. The interest rate is 18% per annum, compounded monthly. (5) 1.2 Keletso starts saving money for a car. She opens a savings account and immediately deposits R1 000. After that she deposits R1 000 each month, and continues to do so for the next three years. The interest rate is 13% per annum, compounded monthly. How much money will she have after 3 years, to buy a car? (4) QUESTION 2 Kathu game reserve buys a tractor for R1 450 000. The tractor depreciates at a rate of 12% per annum on a reducing balance. The cost of a tractor is expected to increase by 9% per annum. The game reserve sets up a sinking fund to be able to buy a new tractor after 8 years. The sinking fund as well as the trade-in value of the old tractor will be used to buy the new tractor. 2.1 Calculate the value of the old tractor after 8years. (3) 2.2 What will the new tractor cost after 8 years? (2) 2.3 Calculate the value of the sinking fund that the game reserve need to establish. (2) 2.4 If the interest rate is on the sinking fund is 12% p.a. compounded monthly, what will the game QUESTION 3 reserve s monthly contributions be in order to be able to buy the new tractor. (4) Mr Bauer is a farmer needing new farm equipment. He buys a brand-new tractor costing R250 000. It is expected to have a useful life of 8 years. He accordingly sets up a sinking fund to enable him to buy a new tractor without having to use credit facilities in 8 years time. He decides that he will not sell this tractor in 8 years time when he buys a new one. 3.1 What is the predicted inflation rate if a new tractor is expected to cost R407 565,51 in 8 years time? (4) 3.2 What equal monthly payments (96 of them, starting at the end of the first month) must be paid into the fund, at an interest rate of 12% p.a., in order for the fund to reach the necessary value of R407 565,51? (5) [9] [9] [9] Page 16

QUESTION 4 A small company plans to replace its operating equipment in 5 years time. The present value of this equipment is R57 000. To guard against inflation, they plan to implement a sinking fund which will take price increases related to a 9% p.a. inflation rate into account. They estimate that they will be able to sell off their old equipment for R10 000 4.1 What is the total amount (correct to the nearest R1 000) that the sinking fund must accrue to?(4) 4.2 If the company is offered an interest rate of 12% p. a. compounded monthly, calculate how much the equal monthly payments into the sinking fund will be. Assume that money is deposited at the beginning of each month, starting immediately, and at the end of the last month before the new equipment is purchased. (5) [9] QUESTION 5 Top class printers bought a printing machine for R500 000. They expect to replace this machine with a new one in 6 years time. The rate of depreciation is 20% per annum on the reducing balance whilst the cost of a new machine escalates by 15% per annum. 5.1 What is the scrap value of this machine in six years time? (3) 5.2 What will it cost top class printers to replace this machine if the old machine is traded in? (3) 5.3 Top class printers sets up a sinking fund to cover the replacement cost of this machine. Payments QUESTION 6 are made into a banking account paying interest at 18% per annum compounded monthly. Find the value of the monthly payments if the payment commences at the beginning of the month in which the machine was bought and ends at the beginning of the last month. (6) The cost of a bus is R1,2 million. It is expected that the value of this bus will depreciate to R491 520 on a reducing balance per annum in 4 years time. The price of a new bus is expected to increase by 15 % per annum 6.1 Calculate the percentage annual rate of depreciation of the bus (4) 6.2 If the bus needs to be replaced in 4 years time, calculate the value of the sinking fund that needs to be set up to pay for the new bus. Assume that the old bus will be traded in at its depreciation value of R491 520 (2) 6.3 Calculate the amount that must be deposited monthly into the sinking fund if interest is charged at 13% compounded monthly for 4years (5) [12] [16] Page 17

QUESTION 7 A school buys a bus for R480 000. The bus depreciates at the 12% per annum on a reducing balance basis. 7.1 Calculate the value of the bus after 5 years. (4) 7.2 The school can invest money at an interest rate of 12% per annum, compounded half-yearly. If the school wants to buy a new bus for R600 000 after 5 years and the old bus is traded in, how much money will have to be invested monthly to be able to buy the new bus after 5 years? (4) Present Value of an Annuity or Loan Repayments When money is borrowed from a financial institution for example to buy a house, a car, furniture. A fixed amount is paid each month until the loan (borrowed money) is repaid in full as a result of the effect of compound interest, the full amount you eventually pay back is a lot more than the initial amount you borrowed. The present value of an annuity is the value of the loan before interest has been added. Formula [8] Also note that : ( ) Where = Present value of the loan (Amount borrowed) = Fixed repayments = Interest rate as decimal = Number of payments When comparing investment and loan options use the formula for present value of an annuity or loan repayment. Page 18

EXERCISE QUESTION 1 1.1 Kagiso plans to buy a car for R125 000. He pays a deposit of 15% and takes out a bank loan for the balance. The bank charges 12,5% p.a. compounded monthly. Calculate: 1.1.1 The value of the loan borrowed from the bank (2) 1.1.2 The monthly repayment on the car if the loan is repaid over 6 years (4) 1.2 Lesego is paying R2000 per month towards an initial loan of R100 000. The interest rate is 18% per annum, compounded monthly. How long will it take for the loan to be repaid. (4) 1.3 A newly married couple buy a house for R950 000. They pay 10% as a deposit and negotiate a bond for the balance. The bond is to be repaid over 20 years. If the interest rate is on the bond is 15% per annum, compounded monthly, calculate the monthly repayments on the bond. (4) 1.4 Keletso starts saving money for a car. She opens a savings account and immediately deposits QUESTION 5 R1 000. After that she deposits R1 000 each month, and continues to do so for the next three years. The interest rate is 13% per annum, compounded monthly. How much money will she have after 3 years, to buy a car? (4) Jabu invests a sum of money for 5 years. She receives interest of 12% per annum compounded monthly for the first two years. The interest rate changes to 14% per annum compounded semi- annually for the remaining term. The money grows to R75 000 at the end of the 5-year period. 5.1 Calculate the effective interest rate per annum during the first year (3) 5.2 Calculate how much money Jabu invested initially (4) [7] QUESTION 6 A car that costs R130 000 is advertised in the following way: No deposit necessary and first payment due three months after date of purchase. The interest rate quoted is 18% p.a. compounded monthly. 6.1 Calculate the amount owing two months after the purchase date, which is one month before the first monthly payment is due. (3) 6.2 Herschel bought this car on 1 March 2009 and made his first payment on 1 June 2009. Thereafter he made another 53 equal payments on the first day of each month. [18] Page 19

6.2.1 Calculate his monthly repayments (3) 6.2.2 Calculate the total of all Herschel s repayments. (1) 6.3 Hashim also bought a car for R130 000. He also took out a loan for R130 000, at an interest rate of18% p.a. compounded monthly. He also made 54 equal payments. However, he started payments one month after the purchase of the car. Calculate the total of all Hashim s repayments. (5) 6.4 Calculate the difference between Herschel s and Hashim s total repayments (2) QUESTION 7 Andrew wants to borrow money to buy a motorbike that costs R55 000 and plans to repay the full amount over a period of 4 years in monthly instalment. He is presented with two options: option 1: the bank calculates what Andrew would owe if he borrows R55 000 for 4 years at a simple interest rate of 12.57% p.a. and then pays that amount back in equal monthly instalments over 4 years option 2: He borrows R55 000 from the bank. He pays the bank back in equal instalments over 4 years, the first payment being made at the end of the first month. Compounded interest at 20% p.a. is charged on the reducing balance. 7.1 If Andrew chooses Option 1, what will his monthly instalment be? (4) 7.2 Which option is the better option for Andrew? Justify your with appropriate calculations (4) 7.3 What interest rate should replace 12,57% in option 1 so that there is no difference between the two options? (3) [11] QUESTION 8 Kevin has a motorbike that costs him R8 000. He wishes to replace it with the equivalent model in 5 years time. If depreciation on his motorbike is at 15% per annum on the reducing balance method, and inflation is at 5,3%p.a., calculate: 8.1 The expected cost of the equivalent model in 5 years time (3) 8.2 The expected book value of his motorbike in 5 years time (3) 8.3 The amount of money Kevin is expecting to have to pay in if he sells his old motorbike to offset the total cost of the new one. (2) [8] [14] Page 20

Micro Loans Short term loans with high interest rates. Interest is calculated over full period of the loan at a simple interest rate. OUTSTANDING BALANCE(OSB) When calculating the outstanding balance one can use the formula Where z = months that are not yet paid up p = outstanding amount i = interest rate x = amount paid monthly ( amount on instalment) Or OSB = ( ) Where P = amount actually borrowed i = interest rate k = the number of months already paid x = amount paid monthly (amount on instalment) OSB = outstanding balance In cases where the last instalment is less than the normal instalment, the outstanding balance is calculated using OSB = ( ) NOTE : Use in an investment where there is one initial deposit, in appreciation or when you are looking for the future value for amount after incurring interest for some given period Use monthly payments when dealing with loan and bond matters and there are instalments or Use deposits when dealing with an investment and or sinking fund and there are monthly ACTIVITY QUESTION 1 Marks wants to buy a house for R700 000,00. He has a deposit of R50 000,00 and takes out a loan for the Page 21

balance at a rate of 18% p.a. compounded monthly. 1.1 How much money must Marks borrow from the bank? (2) 1.2 Calculate the monthly payment if he wishes to settle the loan in 15 years (4) 1.3 Marks won a lottery and wishes to settle the loan after the payment. What is the outstanding QUESTION 2 balance? (4) 2.1 Marry deposited R8 000,00 into a savings account at the beginning of 2007. She further deposited R5000,00 at the beginning of 2009. The bank offers an interest of 6% p.a compounded monthly. How much must Mary expect in her savings account at the end of 2015, if no further deposits were made? (3) 2.2 Mr Bengu buys a car for R125 000,00 and he pays 20% in cash. He takes out a bank loan for the balance. If the bank charges 10.5% p.a. interest compounded monthly and he plans to pay the loan off in 5 years time QUESTION 4 2.2.1 Calculate the amount taken from the bank as a loan (1) 2.2.2 Calculate the monthly repayment on the car loan (4) 2.2.3 If he makes monthly repayment of R3000,00, calculate the loan balance after three years. (5) [13] Nomma purchases a house for R780 000 on a mortgage for 20 years with interest at 17 % p.a. compounded monthly. The mortgage payments are made at the end of each month. 4.1 Show that Nomma s monthly instalment is R11 441,04 (4) 4.2 Calculate the outstanding balance after 10 years (3) 4.3 How much has been paid into the bond account and how much capital was paid off on this loan after the 10 years (3) [15] QUESTION 5 Cherry wants to buy a three bedroom house worth R930 000 for her family. She takes out a bank loan for 20 years at an interest rate of 9,2% p.a. compounded monthly. She repays the loan by means of equal monthly instalments starting one month after the granting of the loan. 5.1 Calculate Cherry s monthly instalments. (4) [10] Page 22

5.1 Due to some unexpected financial difficulty, Cherry was unable to pay any instalments at the end of the 50 th, 51 th and 52 nd months. Calculate the amount outstanding after the 49 th instalment was made. (4) 5.2 Calculate her increased monthly payment which comes into effect from the 53 rd month onward in order for her to still pay off the loan in 20years. (4) [16] QUESTION 6 A father decided to buy a house for his family for R800 000. He agreed to pay monthly instalments of R10 000 on a loan which incurred interest at a rate of 14% p.a. compounded monthly. The first payment was made at the end of the first month. 6.1 Show that the loan would be paid off in 234 months. (4) 6.2 Suppose the father encountered unexpected expenses and was unable to pay any instalments at the end of the 120 th, 121 th, 122 th and 123 th months. At the end of 124 th month he increased his payments so as to still pay off the loan in 234 months by 111 equal monthly payments. Calculate the value of his new instalment. (7) [11] QUESTION 7 Mrs Remias wishes to retire when she is 60 years old. She has 35 years to build up her savings and would like to have R 1 000 000 saved as a lump sum by the time she retires. She decides to invest in an ordinary annuity, with interest given at 6.5% per annum, compounded monthly. 7.1 Determine the amount of the monthly contribution which she will require to make to achieve her goal if she contributes monthly for 35 years. (5) 7.2 The value of the money will depreciate over the next 35 years and affect the real value of her savings. Experts estimate that money will devalue at 4% per annum. Determine the real value of R1 000 000 in 35 years time, if this depreciation rate is applied (3) QUESTION 8 Mr and Mrs Remias want to buy a house which is valued at R750 000. They have R326 000 left from the sale of their present house, after paying lawyers fees, so will take out a bond to cover the rest of the cost 8.1 Looking at their joint salaries, they work out that they can afford to pay R8 000 each month in bond payments. They take out a bond which charges interest at 9.2% compounded monthly. How long will it take them to pay off the amount of the money they have borrowed if the interest rate remains the same over the whole repayment period? Give answer in years correct to 2 decimal places (5) [8] Page 23

8.2 The Remias family decide to repay the loan in full after 4 years of making payments of R8 000 per month. Determine the balance outstanding on the loan at this time. (4) Bibliography Advanced Level Mathematics by C.J. Tranter and C.G. Lambe CAPS Document 2011 DOE Examination Question Papers Answer Series Grade 10, 11 and 12 Platinum Mathematics Maskew Miller Longman Mathematics Exam Practice Book Study and Master Mathematics Wikipedia www.everythingmaths.com www.ixl.com www.kutasoftware.com [9] Page 24