Download year 12 – annuities help……

YEAR 12 – ANNUITIES HELP……..
Annuities are fairly complex. The first and most important step is to be able to identify what formula (method)
to use. In each question, look for keywords that might help you decide what method to use.
TYPE 1
HOW MUCH WILL I END UP WITH if I save a set amount periodically that earns interest.
M[(1  r)n  1]
Use the FUTURE VALUE FORMULA A 
r
EXAMPLE To ensure that it has sufficient funds for a new delivery van, a bakery intends to invest $3000 at
the end of each year, at 8% p.a. compounding annually for 6 years.
(a) How much will they have after 6 years (ie. At the end of the time) ?
A  3000
[(1  0.08)6  1]
0.08
A = $22 007.79
they will end up having $22007.79
(b) How much interest will they have earnt ?
Sum of all payments made = $3000  6  $18000
Final Value (F.V) = $22007.79
 Interest = Final Value - Payments
$22007.79  $18000  $4007.79
TYPE 2
I NEED TO END UP WITH AN AMOUNT (eg. $7000) AT THE END OF x YEARS, HOW MUCH DO I
NEED TO SAVE PERIODICALLY.
M[(1  r)n  1]
Use A 
to find M.
r
‘A’ is the amount you end up with.
EXAMPLE: Anna wants to go on holidays and needs $7000 for the trip. She has 24months to save for the trip.
How much will she need to save every month at 6%p.a. compounding monthly to end up with the $7000 she
needs ?
7000  M 
[(1  0.005)24  1]
0.005
NOTE: Convert rate and periods to monthly.
 7000  M  25.43
 M  7000  25.43
 M  $275.27
She should save $275.27 per month
TYPE 3
HOW MUCH do I need TO INVEST TODAY (IN A SINGLE LUMP SUM) TO END UP WITH AN
AMOUNT ?
A
‘A’ is the amount you want to end up with
(1  r)n
EXAMPLE: How much will Joe need to invest today in a lump sum at 8% p.a. compounding yearly to give
him $70000 after 20 years?
Use Present Value formula N 
70000
(1  0.08)20
 N  $12575.77
N
He needs to invest $12575.77 TODAY
TYPE 4
HOW MUCH DO I NEED TO INVEST TODAY (IN A LUMP SUM) TO END UP WITH THE
EQUIVALENT OF INVESTING WITH AN ANNUITY?
Use Present Value N 
M[(1  r)n  1]
r(1  r)n
‘M’ is the periodical payment made in the annuity
EXAMPLE: Ellen has an investment account that pays 6% p.a. compounding annually into which she
deposits $1500 per year. She plans to make regular deposits for the next 12 years. What single investment
could she make today that would produce an equivalent amount in 12 years time, assuming the same rate.
N  1500 
[(1  0.06)12  1
0.06(1  0.06)12
N = $12575.77
TYPE 5
HOW MUCH DO I NEED TO INVEST TODAY (IN A LUMP SUM) IN ORDER TO BE ABLE TO
WITHDRAW REGULAR PAYMENTS?
M[(1  r)n  1]
Use N 
r(1  r)n
to find N.
‘M’ is the required periodical withdrawal
EXAMPLE: How much must Anna invest today at 8% p.s. so that she can withdraw $3000 every 6 months
for the next 10 years?
N  3000 
[(1  0.04)20  1]
0.04(1  0.04)20
N = $40 770.98
TYPE 6
HOW MUCH WILL THE LOAN REPAYMENTS BE ON A LOAN ?
M[(1  r)n  1]
Use N 
r(1  r)n
to find M.
[(1  r)n  1]
to find M, divide N by
r(1  r)n
EXAMPLE: Linda borrows $2000 at 8% p.a. repayable quarterly over 2 years. Calculate her quarterly
repayments.
[(1  0.02)8 ]
2000  M 
0.02(1  0.02)8
2000  M  7.325
 M  2000  7.325
 M  $273.02