1. This is how nearly all investments operate. It is a better option than simple interest because: With simple interest you earn exactly the same amount year after year but With compound interest you earn interest on your interest! Over several years, this is a really good option!

2. Suzie wins $10 000 in Lotto and decides to invest it in an account paying 8% p.a. compound interest. How much has she after 3 years ? Number of years Calculate the interest Interest earnedTotal 0$10000 1 2 3 10000 x 0.08 $800 $10800 10800 x 0.08$864 $11664 11664 x 0.08$933.12 $12597.12

3. So we can summarise the previous results as follows: Number of yearsAmount 010000 110800 211664 312597.12 But what if I needed to know more years? After 1 year, she will have 10000 + 0.08 x 10000 which may be rewritten as 1 x 10000 + 0.08 x 10000 in other words, 10000 x 1.08

4. And after two years.... Number of yearsAmount 010000 110800 211664 312597.12 After 1 year, she has After 2 years, she has 10000 x 1.08 (10000 x 1.08) + 0.08 of (10000 x 1.08) = (10000 x 1.08) (1 + 0.08) = (10000 x 1.08 x 1.08 = 10000 x 1.08 2 = 1(10000 x 1.08) + 0.08(10000 x 1.08) After 2 years, she has Do I see a PATTERN?? WE GOT THIS FROM THE PREVIOUS SLIDE

5. And after three years.... Number of yearsAmount 010000 110800 211664 312597.12 After 2 years, she has After 3 years, she has 10000 x 1.08 2 (10000 x 1.08 2 ) + 0.08 of (10000 x 1.08 2 ) = (10000 x 1.08 2 ) (1 + 0.08) = (10000 x 1.08 2 x 1.08 = 10000 x 1.08 3 = 1(10000 x 1.08 2 ) + 0.08(10000 x 1.08 2 ) After 3 years, she has WE GOT THIS FROM THE PREVIOUS SLIDE So after 7 years she would have??? 10000 x 1.08 7

6. If $P is invested for n time periods (years, months, weeks, quarters, half-years etc), at an interest rate of r% per time period, then this original $P amounts to $A where A = P (1 + r/100) n You can also write this as A = P(1 + r) n but remember that if you use this formula, r is a decimal rather than a percentage. So if the interest rate is 7%, then the top formula will be A = P( 1 + 7/100) n and the second formula would be A = P( 1 + 0.07) n. They’re both the same thing.

7. $2000 is invested for 15 years at an interest rate of 12% per year, calculate how much this will grow to. A = P (1 + r/100) n P = 2000 r = 12 In this question, time periods are YEARS n = 15 A = 2000(1 + 12/100) 15 = 2000 x (1.12) 15 = $10947.13

8. $5000 is invested for 10 years at an interest rate of 8% per year, calculated at half-yearly (6-monthly) intervals. Calculate how much this will grow to. A = P (1 + r/100) n P = 5000 r = 8  2 = 4 In thisquestion, time periods are half-years (6-monthly) n = 10 x 2 = 20 A = 5000(1 + 4/100) 20 = 5000 x (1.04) 20 = $10955.62 These have to be put in terms of half-years

9. $1600 is invested for 20 years at an interest rate of 11.5% per year, calculated at quarterly (3-monthly) intervals. Calculate how much this will grow to and the interest earned A = P (1 + r/100) n P = 1600 r = 11.5  4 = 2.875 In this question, time periods are ¼ years (3-monthly) n = 20 x 4 = 80 A = 1600(1 + 2.875/100) 80 = 1600 x (1.02875) 80 = $15449.27 Interest earned = 15449.27 - 1600 = $13849.27

10. I invest some money at 5.75% per annum compounding fortnightly for 8 years, and end up with $19790.87. How much did I initially invest? A = P (1 + r/100) n P = ? r = 5.75/26=.22115 In thisquestion, time periods are fortnights (26 in a year) n = 8 x 26 = 208 19790.87 = P(1 + 0.22115/100 ) 208 19790.87 = P x 1.0022115 208 P = $12500 A = 19790.87 Ans: I originally invested $12500

11. I invest $8000 with interest compounding daily for 5 years, and end up with $11352.16. What is the annual interest rate? A = P (1 + r/100) n P = 8000 r = ? In thisquestion, time periods are days (365 in a year) n = 5 x 365 = 1825 11352.16 = 8000(1 + r/365) 1825 A = 11352.16

12. Now raise both sides to power of 1/1825 The reason for this is that the powers on the right side now cancel leaving it easier to get r as the subject Ans: Interest rate is 7% pa

13. An important skill – solving equations where the unknown is in the power. Example – Solve 2 n = 512 Solution Ans n = 9 (Check 2 9 = 512)

14. Example – Solve 1.25 n = 6.35 Solution Ans n = 8.28 (Check 1.25 8.28 = 6.35)

15. I invest $7000 with interest compounding monthly at 8% p.a. and end up with $19000. For how many months do I need to invest? A = P (1 + r/100) n P = 7000 r = 8/12 = 0.66667 In thisquestion, time periods are months (12 in a year) n = ? 19000 = 7000(1.0066667) n A = 19000

16. n = 150.27 The money would need to be invested for 151 months i.e. 12 years & 7 months Note we round UP!!

18. $2000 is invested for 15 years at an interest rate of 12% per year, calculate how much this will grow to. APPS Choose (1) - FINANCE Choose (1) – TVM SOLVER

19. $2000 is invested for 15 years at an interest rate of 12% per year, calculate how much this will grow to. Number of time periods in total Interest rate – always YEARLY Principal (Present Value Payment. Set to 0 Number of time periods in a YEAR. If we’re talking years, then this is 1. If we’re talking 6-monthly, then this is 2. If monthly, then this is 12. etc.. Ignore. Leave on “END” Ignore. changes automatically FUTURE VALUE (THE ANSWER!!)

20. $2000 is invested for 15 years at an interest rate of 12% per year, calculate how much this will grow to. Now put cursor on FV Hit ALPHA ENTER (SOLVE) ANS: $10947.13

21. $5000 is invested for 10 years at an interest rate of 8% per year, calculated at half-yearly (6-monthly) intervals. Calculate how much this will grow to. 10 X 2 = 20 LOTS OF 6 MONTHS ALWAYS YEARLY INTEREST. DON’T CHANGE THERE ARE TWO 6-MONTH PERIODS IN EACH YEAR Answer - $10955.62

22. $1600 is invested for 20 years at an interest rate of 11.5% per year, calculated at quarterly (3-monthly) intervals. Calculate how much this will grow to. 10 X 2 = 20 LOTS OF 6 MONTHS ALWAYS YEARLY INTEREST. DON’T CHANGE THERE ARE FOUR 3-MONTH PERIODS IN EACH YEAR Answer - $15449.27

23. After shopping around, the best deal for investment I can get is with the Hungry Bank, which pays 9.6%p.a. compounding daily. How long do I need to invest my money for before it doubles ? First enter interest 9.6 Let PV be any random amount ($100) Then FV will be double PV ($200) 365 days in a year! We’re trying to find N Note either PV or FV needs to be negative!

24. Cursor on N and ALPHA SOLVE It will take 2636 days for my money to double! We’re trying to find N

25. Wilma and Betty decide to go on a cruise together in two years time. They each estimate their total costs (per person) will be $12 000. Wilma has worked out that she needs to invest $8000 in cash in an account which pays interest quarterly. Betty, being a bit smarter, realises daily interest is a better deal. How much does she need to invest in an account paying the same interest as Wilma’s but compounding daily – so she also has $12 000? Answer

26. Lazarus decides to invest some money in an account paying 11.3% p.a. simple interest. At the end of 3 ½ years he takes this out and deposits it into another account paying 9.6% compounding monthly. After a further 2 years he then places all his funds in another account at 9.95% daily. If the total life of his investment was 9 years, and he ends up with $68210.30, how much did he invest at the start?

27. (1) $7917.77 (2) $28500 Go to next question