INTERESTS

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FUTURE VALUE ANNUITIES
(INSTALMENT SAVINGS)
X-Kit Textbook
Chapter 7
Information29 April 2013
• Homework Task 8 on X-Kit Chapter 7 due by
Thursday 2 May. Complete solutions (step-
by-step) in homework task.
• Class Test 8 on Thursday 2 May on X-Kit
Chapter 7.
EXAMPLEON PRESENTVALUE ANUITIES
Ahmed has a home loan for R255 000 which he has to repay
in equal monthly instalments over 25 years at 15.5% per
annum compounded monthly.
1. What are his monthly instalments?
2. What part of his first instalment is interest and how
much is repayment of capital?
3. His last (300th) instalment reduces the amount he owes
to zero. How much did he owe at the beginning of the
last month, just after paying his 299th instalment?
4. What part of his last instalment is interest and how
much is repayment of capital?
A LENDER’SSCHEDULE FOR REPAYMENTSOF R240
PER MONTH
Month Owing at
the
beginning
of the
month
Interest
added at
the end of
the month
Total
owing at
the end of
the
month
Instalment
subtracted
New
amount
owing at
the end of
the month
1 750.00 169.17 919.17 240.00 679.17
2 679.17 153.20 832.37 240.00 592.37
3 592.37 133.62 725.99 240.00 485.99
4 485.99 109.62 595.61 240.00 355.61
5 355.61 80.21 435.82 240.00 195.82
6 195.82 44.17 239.99 240.00 - 0.01
CONTENT
Future
Value
Annuity
Deposits
in arrear
Deposits
in
advance
FUTUREVALUE ANNUITIES
• To save money by making regular deposits into
an account.
• Manage finances carefully – plan for future
expenses.
• Remember assets depreciate.
• To replace expensive assets (machinery or
equipment) in future, we need to set up a
FUTURE VALUE ANNUITY or SINKING FUND.

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EXAMPLE
Trevor stopped smoking. To reward
himself he deposited the R400 per month
that he used to spend on cigarettes into an
account earning interest at a rate of 9.5%
per annum compounded monthly. After
one year will he have the R5000 that he
would like to spend on sports equipment?
CONTENT
•A FUTURE VALUE ANNUITY is the whole
process of saving by making a sequence of
payments at regular intervals in time.
•The FUTURE VALUE of an annuity is the
value just after the last payment of such a
sequence of payments. This is the sum of
all future values of 𝒏 payments.
•For example a retirement annuity
MATHEMATICALFORMULA
𝑭𝑽 = 𝑷
𝟏 + 𝒊 𝒏
− 𝟏
𝒊
𝐹𝑉 = future value of all the payments (the amount saved)
𝑃 = amount of each payment
𝑛 = number of payments
𝑖 = rate of interest per interest period (written as a
decimal fraction)
Payments in arrear, payments at the end of each
interest period
MATHEMATICALFORMULA
𝑭𝑽 = 𝑷
𝟏 + 𝒊 𝒏
− 𝟏
𝒊
𝟏 + 𝒊
Payments in advance, payments at the
beginning of each interest period
STEPS
1. Draw a time line.
2. Write down a list of the variables.
3. Write down the correct formula
(payments at the end or in the beginning).
4. Substitute the variables and calculate the
answer.
EXAMPLE: INVESTINGMONEY
When income tax was reduced, David
decided to invest his monthly tax saving of
R290 into an account earning 8.75% per
annum compounded monthly. How much
will he have in his account after 3 years?
Assume that payments are in arrear.

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EXAMPLE: SINKINGFUND
A company bought a machine for R1 000 000 on 1
January 2004 and they expect that it will be worn
out by the end of 2011. They plan to replace it with
a new machine on 1 January 2012. They expect
that the value of their present machine will
depreciate at 20% p. a. and they want to trade it in
for its scrap value when they buy a new one. To
provide for the replacement, the company started a
sinking fund, with a fixed annual payment at the
end of each year from 2004 until 2011 inclusive.
QUESTIONS
1. What is the scrap value of the old machine at the
end of 2011?
2. Calculate the expected cost of the new machine
at the beginning of 2012 if the annual inflation
rate is 8% over these years.
3. What is the future value of the sinking fund that
will be needed on 1 January 2012 if the old
machine is traded in for the new one?
4. Find the annual payment into the sinking fund if
the account pays 10.4% interest per annum.
EXAMPLE: MISSINGA DEPOSIT
Mr Forgetful was supposed to make 100
monthly deposits of R500 at the end of 100
consecutive months, but he forgot to make the
75th deposit. How much was there in his
account just after he made the last deposit if the
interest rate is 9.8% p.a. compounded monthly?
EXAMPLE: INVESTINGIN REPAIRS
Ben bought a house for R110 000. It required a lot of
repair and maintenance. Ben decided to do the work
himself during weekends over 18 months. He
calculated that it would cost him R3 000 per month,
starting one month after he bought the house, to do
the necessary repairs. At the end of 18 months he is
offered R220 000 for the house. If he accepts the offer
for his house, will he make more money on the deal
than if he had invested the purchase price and the
monthly cost of repairs in an account earning 12.6%
per annum compounded monthly?
EXAMPLE: SAVINGSPLAN
At the same time each year Lulu invests
R1 000 into a savings plan that pays
interest at the rate of 9% per annum.
What is the value of her savings plan on
the 10th anniversary of the first
investment if she makes her last deposit
then?
EXAMPLE: DIFFERENTDEPOSITS
James makes a deposit of R1 000 and then
makes monthly deposits of R300 in arrear for 3
years. The account earns 7% interest per annum
compounded monthly. How much will James
have in his account:
1. Immediately after his last deposit?
2. Two years after his last deposit?

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EXAMPLE: MISSEDDEPOSIT
Suppose James, in the previous example,
missed his 24th monthly payment. How
much will he have in his account
immediately after his last deposit?
EXAMPLE: SINKINGFUND
A machine bought now costs R11 500 and is expected
to depreciate at 10% p.a. The replacement machine is
expected to increase in price by 12.5% p.a. The
machine is to be replaced in 7 years’ time and the
owners establish a sinking fund for the difference
between the replacement price of the machine and its
depreciated value after 7 years. Calculate the size of
the monthly deposits that the owners must make at the
end of each month starting one month from now and
ending when the new machine is bought. The account
earns interest at 8.25% p.a. compounded monthly.
FINANCIALMATHEMATICS - CONTENT
1. Simple & Compound Interest
2. Depreciation (straight line &
reducing balance)
3. Inflation
4. Nominal & Effective Interest
Rates
5. Time Lines
6. Sinking Funds
7. Annuities (Investments &
loans)
8. Hire Purchase
9. Money Lenders
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EXAMPLE: HIRE PRUCHASE
A motorcycle costs R36 000. Simon pays a 25%
deposit and will pay the balance over a period of
two years, in equal monthly payments.
Find the size of the monthly payment if:
1. Simon borrows the balance at 20.7% p.a.
compounded monthly.
2. Simon borrows the balance on hire purchase,
where the simple interest is 13% per annum.
EXAMPLE: MONEYLENDER
A money lender grants a loan of R21 000 to
Sandy at 24% p.a. compounded monthly.
Sandy agrees to pay the loan over a period of
one year. She pays R6 000 after 3 months.
R5 000 after 6 months and R8 500 four
months later. Calculate the amount of
Sandy’s final payment at the end of the year.
EXAMPLE: TIMELINES
Lindy invested R5 000 at 10% per annum
compounded semi-annually. After 2 years
the interest rate changed to 12% per
annum compounded monthly. Find the
value of the investment after 5 years.