Introductory Mathematics
& Statistics
Chapter 7
Annuities
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-1
Learning Objectives
• Understand and apply annuities
• Distinguish between future and present values of annuities
• Solve problems involving the future value of an annuity
• Calculate the present value of an annuity
• Calculate the periodic payment of a present value annuity
(amortisation)
• Calculate the periodic payment of a future value annuity
(sinking fund)
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-2
7.1 Introduction
•
An annuity is a series of equal payments, often made under
contract, paid at equal intervals (e.g. quarterly or monthly)
Definitions
1. The rent period (or payment period) is the interval of time
between the payments of an annuity
2. The term of an annuity is the time from the beginning of
the first payment period to the end of the last payment
period
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-3
7.1 Introduction (cont…)
• The two main types of annuities are simple and general
• A simple annuity is where interest is compounded at
the same times as the annuity payments
• A general annuity is where interest is compounded at
times that are either greater or smaller than when the
annuity payments are made
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7.1 Introduction (cont…)
 Simple annuities can be classified into four types
1. An ordinary annuity is one where the payments are made at
the end of each period
2. An annuity due is one where the payments are made at the
beginning of each period
3. A deferred annuity is one where the payments do not
commence until a period of time has elapsed
4. A perpetuity is an annuity in which the payments continue
indefinitely
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-5
7.1 Introduction (cont…)
• Ordinary annuities have the following aspects
1. The future value of an annuity: this is the value of the annuity
at the end of its term
2. The present value of an annuity: this is the value of the
annuity at its beginning
3. Amortisation: this is the making of periodic payments to repay
a debt (including the principal and interest)
4. Sinking fund: this is a fund into which the periodic payments
necessary to realise a given sum of money in the future are
made
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-6
7.2 Future value of an annuity
•
•
The future value (or accumulated value) of an annuity is the
amount due at the end of the term
That is, it is the sum of all the periodic payments made and interest
accrued up to and including the final payment period
n

1  i  1
S R
i
Where:
S = future value of the annuity
R = amount of the annuity payment made per period
i = interest rate per payment period
n = total number of payments
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-7
7.2 Future value of an annuity (cont…)
The formula can also be written as
S  Rs n
i
Where

1  i

n
sn i
1
i
The value of sn i (or sn at i %) may be thought of as the
future value at an interest rate of i% of $1 paid at the end
of each period for n periods.
Copyright  2010 McGraw-Hill Australia Pty Ltd
PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
7-8
7.2 Future value of an annuity (cont…)
Example
A customer deposits $250 every 3 months into a building society
account that pays interest at a rate of 8% per annum convertible
quarterly. How much money will be in the account at the end of
10 years?
Solution
0.08
R  $250, i 
 0.02, n  4  10  40
4
s 40 0.02
40

1  0.02  1

0.02
 60.40198318
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-9
7.2 Future value of an annuity (cont…)
Solution (cont…)
S  R  sn i
 $250  60.40198318
 $15100.50
Hence, the customer will have $15 100.50 in the account at
the end of 10 years.
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-10
7.3 Present value of an annuity
• The present value (or discounted value) of an annuity is its
value at the beginning of the initial rent period.
• That is, it is the sum of the compound present values of all the
payments one period before the initial payment.
1 1 i 
A R
n
i
Where:
A = present value
R = annuity payment per period
i = interest rate per period
n = number of payments
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-11
7.3 Present value of an annuity (cont…)
•
Another way of writing the formula
A  R  an i
•
Where
1  1  i 
an i 
i
n
•
The value of a n i (or an at i %) may be thought of as the present
value at an interest rate of i % per period of $1 paid at the end
of each period for n periods
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-12
7.3 Present value of an annuity (cont…)
Example
Melinda took out a loan from a credit union in order to purchase
a home computer. She was to repay the loan in monthly
instalments of $120 for 5 years. Calculate the present value of
these repayments if the interest rate was 9%, convertible
monthly
Solution
0.09
R  $120, i 
 0.00785, n  12  5  60
12
From Table 3:
a60 0.0075  48.17337352
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-13
7.3 Present value of an annuity (cont…)
Solution (cont…)
A  R  an i
 $120  48.17337352
 $5780.80
That is, the repayments are worth $5780.80 at the beginning
of the loan
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-14
7.4 Amortisation
•
When an individual or business pays a debt (including interest)
by making periodic payments at regular intervals, the debt is
said to be amortised.
•
An amortisation problem involves finding the periodic payment
that will discharge a debt.
•
In particular, as each periodic payment is made, this amount
covers part of the principal and interest on the balance of the
principal.
•
In turn, there is a reduction in the remaining principal such that
at the end of the term of the annuity the debt is extinguished.
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-15
7.4 Amortisation (cont…)
Example
A family undertakes a mortgage of $40 000 from a bank in order to
buy its new home. The bank charges interest at a rate of 12% per
annum, compounded quarterly over 20 years. What quarterly
payments will the family have to make on this loan?
Solution
A  $40000, i 
0.12
 0.03, n  4  20  80
4
From Table 3:
a80 0.03  30.20076345
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7.4 Amortisation (cont…)
Solution (cont…)
A  R  an i
40000  R  a80 0.03
$40000
30.20076345
 $1324.47
R
Hence, the family must repay the loan at the rate of $1324.47 per quarter.
Note that the family will, in fact, be paying back to the bank a total of
$1324.47 × 80 = $105 957.60, considerably more than the original $40 000
loan!
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-17
7.5 Sinking funds
•
•
A sinking fund is a fund into which periodic payments are made
so as to accumulate a nominated amount of money at the end of
a specified period
It is assumed that each deposit earns compound interest until the
end of the period
R
S
1  in  1
or
S
R
sn i
i
Where:
S = future value
R = annuity payment per period
i = interest rate per period
n = number of payments
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7.5 Sinking funds (cont…)
Example
Connie is planning to spend her Christmas holidays in 2 years
time in the United States. She estimates that she will need an
amount of $6000 to pay for her airfares, accommodation and
other expenses. She is going to save her money in an account
that attracts 10% interest per annum, compounded quarterly.
How much will Connie have to deposit into her account at the
end of each quarter, to have the desired amount at the end of 2
years?
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7.5 Sinking funds (cont…)
Solution
0.10
S  $6000, i 
 0.025, n  2  4  8
4
s8 0.025
So
8

1  0.025   1

0.025
 8.73611590
$6000
R
8.76311590
 $686.80
If Connie deposits $686.80 every 3 months over the 2 years,
she will have the required $6000 at the end of that time
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-20
Summary
• We looked at understanding and applying annuities
• We found the difference between future and present value of
annuities
• We solved problems involving the future value of an annuity
• We calculated the present value of an annuity
• We calculated the periodic payment of a present value
annuity (amortisation)
• We calculated the periodic payment of a future value annuity
(sinking fund)
Copyright  2010 McGraw-Hill Australia Pty Ltd
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7-21