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SIMPLE INTEREST FINANCE AND GROWTH Interest is based only on the original • Simple Interest sum of money • Hire Purchase Agreements Rand value of interest remains constant • Compound Interest Formula: A = P(1 + n.i)
• Inflation A = Total Amount • Population Growth P = Principle int erest _ rate • i Exchange Rates i = Interest Rate (where 100 ) n = Number of Years
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Example 1: R1500 is invested for 2 years at 18% per Example 2: R50 000 is invested for 18 years at 16% p.a. annum (p.a.) simple interest. simple interest. Determine: a) the value of the investment after 18 years Value of investment after 1 year: A = R1500 + 18% of R1500 b) the simple interest earned after 18 years
= R1500 + R270 P = R50 000 ; I = 0.16 p.a ; n = 18 years = R1770
a) A = P(1 + i.n) Value of investment after 2 years: A = R50 000 (1 + 0,16.18) A = R1770 + 18% of R1500 A = R194 000 = R1770 + R270 = R2040 b) Interest received Therefore value of interest remains constant. = A - P = R194 000 – R50 000 Using the Simple Interest formula: = R144 000. A = P(1 + i.n) A = R1500(1 + 0,18 x 2) A = R2040 3 4
Example 3: How much was invested 4 years ago if the Example 4: R1 600 accumlates to R3 000 after 4 years. value of the investment is currently worth R7 000? The Find the interest rate if the investment earned simple interest rate was 7% p.a. simple interest. interest.
A = R3000 ; P = R1600 ; n = 4 years A = R7000 ; n = 4 years ; i = 0.07 A = P(1 + i.n) A = P(1+i.n) R3000 = R1600(1 + i.4) R7000 = P(1 + 0,07 x 4) R3000 = R1600 + 4800i R7000 = P(l + 0,28) R3000 – R1600 = 4800i R7000 = P(1,28) R1500 = 4800i R1500 4800 = i R7000 i = 0,3125 P 1,28 r = i x 100 = 0,3125 x 100 P = R5468,75 = 31,25% Simple Interest Calculator 5 6
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EXERCISE 3. In 3 years’ time John wants to have saved R25 000 in order to visit his cousin who lives in England. He manages to receive an interest rate of 12% per annum 1. Sarah invests R5000 for 6 years at a simple interest simple interest. How much must he invest now in rate of 8% p.a. Determine: order to achieve his goal? (a) the future value of the investment. (b) the simple interest received after 6 years 4. Sandesh wants to invest a sum of money now so as to afford a new play station costing R5000, three years 2. Kabelo invests R15000 for 15 years at a simple from now. He receives a simple interest rate of 6% interest rate of 8% per annum. Calculate: p.a. Calculate how much Sandesh should invest? (a) the future value of the investment. (b) the simple interest received after 16 years 5. Calculate how long it would take an investment of R3900 to grow to a value of R5500 if the simple interest rate received is 12% per annum.
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6. Calculate how long it would take for an investment of R5800 to double if the simple interest rate is 14% p.a. HIRE PURCHASE AGREEMENTS A Hire Purchase Agreement (HP) is a short- 7. In order for R5000 to grow to R10000 over a period of term loan. Furniture, cars and household appliances, such as refrigerators, are 5 years, what simpleinterest rate would you need to generally bought on HP. secure?
Buyers sign an agreement with the seller to 8. Ronald has R5000 to invest and wishes to increase pay a specified amount per month. this amount to R6000 over a period of 2, 5 years. What simple interest rate will he need to receive in The interest paid on a hire purchase loan is order to achieve this? simple interest and it is calculated on the full value of the loan over the repayment period. 9. An amount of R2000 is invested at 14% p.a. simple interest. A further R1400 is invested at 12% p.a. Normally a deposit is paid initially and the simple interest. By when will both investments have balance is paid over a short time period the same accumulated values? 9 10
a) P = R7 800; n = 2 years (24 months); I = 14.5% p.a. Example 1: A = P(1 + i.n) Pumzile buys a DVD player for R7 800 .She takes A = R7 800 [1 + (0, 145)(2)] out a hire-purchase loan involving equal monthly A = R10 062 payments over 2 years. The interest rate charged is 14, 5% per annum simple interest. She also b) Interest paid = A – P takes out an insurance premium of Rl0, 35 per = R10 062 – R7 800 month, to cover the cost of damage or theft. = R2 262
Calculate: c) The monthly loan repayments: a) the actual amount paid for the DVD player. R10 062 24 months b) the interest paid. = R419,25 c) how much must be paid each month. Add in the monthly insurance premium, the total amount to be paid each month is: R419,25 + R10,35 = R429,60 p. m 11 12
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. Deposit : Example 2: = 0, 2 x R13 495 = R2 699 Jo - Ann buys a laptop priced at R13 495. She Balance on HP: takes out a 12-month hire purchase agreement. = R13 495 - R2 699 She pays a deposit of 20% and the interest = R10 796 charged on the balance is 15% per annum simple A=P(1+i.n) interest. A = R10 796(1 + 0, 15 x 1) A = Rl2 415,40 What will she actually pay for the laptop and what Jo - Ann will actually pay : will her monthly payments be? R2 699 + R12 415 = R15 114 for the laptop. Her monthly repayments will be: R15 114 12 = R1 259,50 p. m
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b) The 24 amounts of R45 that must be paid will add Example 3: up to R1 032, which is A, the final value. A car radio costs R960. Megan buys her car The deposit of Rl00 is paid at the start of the HP, so radio on HP and agrees to pay a deposit of the balance (P) to be paid is therefore R860. Rl00 and 24 monthly repayments of R45 p.m. P = R860 ; A = R 1 080; Calculate: n = 24 months = 2 years ; i = ? (a) the total simple interest paid. (b) the rate of simple interest. A = P (l + i.n) R1 080 = R860(1 + i x 2) a) Total amount paid: R1 080 R860 = 1 + 2i = Rl00 + 24 x R45 1,255813953 = 1 + 2i = R1 180 1,255813953 – 1 = 2i Total interest paid: 0.255813953 = 2i = R1 180 – R960 0,2558139532 2 = i = R220 r = 12,8 %
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3. A play station 11 costs R4593. Daniel buys a play station EXERCISE on HP and agrees to pay a deposit of R450 and 36 monthly payments of R150. Calculate the total simple 1. Mpho buys a furniture suite for R9895. He takes out a interest paid and the rate of simple interest. hire-purchase loan involving equal monthly payments over five years. The interest rate charged is 15, 25% per annum simple interest. He also takes out an 4. A hire purchase contract for a sound system requires insurance premium of RI2 per month to cover the cost of damage or theft. Calculate: OJ to pay a deposit of R550 and then make 6 monthly (a) the actual amount paid for the furnituresuite. payments of R 185. If the price of the sound system is (b) the interest paid. R1 401. Calculate the total simple interest paid and the (c) how much must be paid each month. rate of simple interest.
2. Greg buys a laptop priced at R12 535. He takes out a 5. Mr Sibanda bought a computer costing R22 253 on hire 24-month hire purchase agreement. He pays a deposit purchase. He traded in an old computer for R2000 and of 10% and the interest charged on the balance is paid a deposit of R1500. The balance was paid by means 16,125% per annum simple interest. What will he of monthly installments of RI 120,50 over two years. actually pay for the laptop and what will his monthly Calculate the total simple interest paid and the rate of payments be? 17 simple interest. 18
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Example 1: R5500 is invested for 2 years at 5% per annum (p.a.) compound interest. COMPOUND INTEREST Value of investment after 1 year: A R5500 0,05 R5500 Interest is based only on the original A R5500 R275 sum of money plus the interest earned A R5775
Rand value of interest increases Value of investment after 2 years: n A R5775 0,05 R5775 Formula: A = P(1 + i) A R5775 R288,75 A R6063,75
A = Total Amount Therefore value of interest remains constant. P = Principle int erest _ rate Using the Compound Interest formula: i = Interest Rate (where i ) 100 A P(1 i)n n = Number of Years A R5500(1 0,05)2
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Example 2: Petro invests R8000 for 9 years at 15%p.a. compounded annually. Find the final Example 3: Jamy takes out a loan in order to start his small car – wash business. Three years value of her investment after 9 years and the after granting the loan, he agrees to repay the interest she receives. loan by means of a once off payment of R405 000. The bank charges him an interest rate of A P(1 i)n 23% p.a. compounded annually. A R8000(1 0,15)9 How much money did he borrow from the bank?
A R28143,01 A P1 in Interest received 3 = A - P R405000 P(1 0.23) = R28 143,01 - R8000 R405000 P(1,23)3 = R20 143,01 P R217640,49 Calculate Compound Interest 21 22
Example 4: R4250 is invested for 6 years and EXERCISE grows to R14 740. Find the interest rate if interest is compounded annually. 1. Find the end value of R1300 invested for 6 years at: n A P1 i (a) 14, 78% p.a. compound interest. R14740 R4250(1 i)6 (b) 18, 70% p.a. compound interest. R14740 (1 i)6 R4250 1 2. Find the present value of an amount which R14740 6 1 1 i)6 6 accumulated to R2228,40 in 5 years at: R4250 1 (a) 11, 86% p.a. compound interest. R14740 6 1 i (b) 12,73% p.a. compound interest. R4250 1 R14740 6 1 i 3. Jana invests R100 500 in an account paying 17, 25% R4250 Graph of Compound i 0,230320023 vs Simple Interest p.a. compounded annually. Calculate the future value of his investment after 18 years. r 23,03% 23 24
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4. Patrick borrows money from a bank in order to finance his new business. The bank charges him an INFLATION interest rate of 14,75% p.a. compounded annually. Calculate the present value of the loan (the amount he originally borrowed), if he pays off the loan 8 Inflation is the steady compounded years from now with a payment of R500 000. increase in prices over time throughout 5. R6000 is invested for 5 years and grows to R9000. the economy. Find the interest rate if interest is compounded
annually. Example 1: A motorcar currently costs R800 000. If the rate of inflation is 12% p.a. compounded 6. Find the annual compound interest rate that makes R2500 double in 5 years. annually, how much will this car cost in 5 years’ time? 7. R50 000 is invested at 10% p.a. simple interest for 3 A P(1 i)n years. Thereafter, the total amount is reinvested in a 5 different financial institution at 25% p.a. compound A 800000(1 0.12) interest for 2 more years. What is the future value of A 1409873,35 the investment after the five-year period? 25 26
n (a) A P(1 i) 15 Example 2: You decide to invest R50 000 now A 50000(1 0,09) and you secure an interest rate of 9% p.a. A 182124,12
Compounded annually. The inflation rate is (b) A P(1 i)n currently running at 12% p.a. 15 A 50000(1 0,12) A 273678,29 (a) What will the future value of your money be in 15 years’ from now? (c) The buying power of your money will have declined (b) Due to inflation, what money will have the by R273 678,29 - Rl82 124,12 = R91 554,17 same buying power as R50 000 in 15 years’ (d) A P(1 i)n time? 15 (c) By how much will the buying power of your A 50000(1 0,18) money have declined after 15 years? A 598687,39 (d) Had you rather secured an interest rate of 18% p.a., by how much would the buying power The buying power of your money will have of your money have increased? increased by = R598 687,39 — R273 678,29 = R325 009,10 27 28
POPULATION GROWTH EXERCISE
The growth of a population is exponential, therefore 1. Determine the value of your money if you invest it is compounded year-on-year. R2000 for 4 years. Inflation is at 7% p.a.
Example: At the start of a bacterial culture there are 5 2. How much will a loaf of bread cost in 2 years time, if organisms. How many organisms will be present in the it currently costs R10,80 and inflation is at 6% p.a. culture after 20 minutes, if the population grows at a rate of 10% per minute? 3. If a TV costs R6 400 now and inflation is at 6.7% p.a., then how much will you pay for a TV in 5 years’ n time? A P(1 i) A 5(1 0,10)20 A 33,64 A 33 29 30
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EXCHANGE RATES EXERCISE Remember! The higher the exchange rate, the weaker the value of your money e.g. R8,50 : $1 1. A DVD player costs $256 in California. What would it cost in South Africa if the Rand/dollar exchange rate Example: Mabel is on a trip to England to visit her family. The current rand/pound exchange rate is is Rl0,45 to the US dollar? R12,50 to the British pound. She has R20 000 to spend in England. How many pounds does she have to spend in England? 2. You want to buy a book online costing £35 in R12,50 London. How much will it cost you in rand? 1£ = R12, 50 1£ divided by R12, 50 = 12,50 The rand/pound exchange rate is RI3 to the pound.
£ 0, 08 = R1 3. You want to buy a box of Japanese cigarettes £ 0, 08 20 000 = R1 20 000 costing 50 yen. If the rand/yen exchange rate is one rand to 16, 78 yen, then calculate the cost of the £ 1600 = R20 000 sweets in rand. Mabel has £1600 to spend in England. 31 32
4. Andrew is visiting a friend in New York for a week. He has R2500 to spend and will exchange the money for US dollars. How many dollars will he have to spend if the rand/dollar exchange rate is R10, 38 to the US dollar?
5. Sandesh wants to purchase an electric guitar from a music dealership in London. The guitar costs £3000 and Sandesh has saved R40 000. The rand/pound exchange rate is R13, 24 per one pound. Will he have enough money, in rand, to buy this guitar?
6. One yen costs R0, 22. How much yen can you spend in Japan if you have R3000 available to you?
7. A certain watch costs €350 in Germany or £245 in England. Which price is better for the South African buyer if the exchange rates are R9, 24 to the euro and R12, 06 to the pound? 33
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