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!

Suzie wins $ 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$ x 0.08 $800 $ x 0.08$864 $ x 0.08$ $

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

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

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

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( ) n. They’re both the same thing.

$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 = $

$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 = $ These have to be put in terms of half-years

$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 = In this question, time periods are ¼ years (3-monthly) n = 20 x 4 = 80 A = 1600( /100) 80 = 1600 x ( ) 80 = $ Interest earned = = $

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

I invest $8000 with interest compounding daily for 5 years, and end up with $ 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 = = 8000(1 + r/365) 1825 A =

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

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)

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

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

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

$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

$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!!)

$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: $

$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 - $

$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 - $

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!

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

Wilma and Betty decide to go on a cruise together in two years time. They each estimate their total costs (per person) will be $ 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

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 $ , how much did he invest at the start?

(1) $ (2) $28500 Go to next question