BUQU 1130 – Business
Mathematics
Chapter 13
ANNUITIES DUE, DEFERRED ANNUITIES,
PERPETUITIES
SHORA EBRAHIMI
SHORA [email protected]

❶ Compute the future value, present value, periodic payment, term, and
interest rate for simple annuities due
❷ Compute the future value, present value, periodic payment, term, and
interest rate for general annuities due
❸ Compute the future value, present value, periodic payment, term, and
interest rate for ordinary deferred annuities
❹ Compute the future value, present value, periodic payment, term, and
interest rate for deferred annuities due
❺ Compute the present value, periodic payment, and interest rate for ordinary
perpetuities, perpetuities due, and deferred perpetuities
SHORA EBRAHIMI
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Objectives
BUQU 1130 - Chapter 13

Simple Annuities Due
If the periodic payment is made at the end of each
payment interval the annuity is an ordinary annuity
If the periodic payment is made at the beginning of each
payment interval, the annuity is an
annuity due
SHORA EBRAHIMI
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BUQU 1130 - Chapter 13

Simple vs. Ordinary
SHORA EBRAHIMI
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BUQU 1130 - Chapter 13

Simple Annuities Due - Future
Value
To calculate the future value of a simple annuity due
The formula for the FV of an ordinary annuity is then
multiplied by (1+i) to include the additional interest earned
for the payment made at the beginning of the first period
SHORA EBRAHIMI
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BUQU 1130 - Chapter 13

Simple Annuities Due - Future
Value
Formula:
SHORA EBRAHIMI
6
??
?
(
???
)
=
???
[
(
1
+
?
)
?
−
1
?
]
(
1
+
?
)
FV of Ordinary Simple
Annuity
Adjusted
One Periods
worth of Interest
BUQU 1130 - Chapter 13

Simple Annuities Due
“
Due
” means that the payments are made at the
beginning
of the payment interval
“
Simple
” means that the payment interval and interest
conversion interval are the
same.
SHORA EBRAHIMI
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BUQU 1130 - Chapter 13

Let’s see an example
Find the FV of $100 payments at 6% compounded
annually made at beginning of the year for four years
SHORA EBRAHIMI
8
$100
$100
$100
$100
Now-beginning of year 1
1
2
3
4
100(1+0.06)
4
100(1+0.06)
3
100(1+0.06)
2
100(1+0.06)
1
$106
$112.36
$119.10
$126.25
??
=
???
(
?
.
??
)
?
+
???
(
?
.
??
)
?
+
???
(
?
.
??
)
?
+
???
(
?
.
??
)
?
=
$
???
.
??
BUQU 1130 - Chapter 13

Let’s see an example
Using the formula
SHORA EBRAHIMI
9
??
?
(
???
)
=
???
[
(
1
+
?
)
?
−
1
?
]
(
1
+
?
)
??
4
(
???
)
=
100
[
(
1
+
0.06
)
4
−
1
0.06
]
(
1
+
0.06
)
??
4
(
???
)
=
100
[
(
1.06
)
4
−
1
0.06
]
(
1.06
)
??
4
(
???
)
=
100
(
4.637093
)
=
$
???
.
??
BUQU 1130 - Chapter 13

Let’s see an example
How much interest is earned?
Four payments of $100
$400
FV of the Ordinary Simple Annuity is $463.71
Interest earned: 463.71 – 400 = $63.71
SHORA EBRAHIMI
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BUQU 1130 - Chapter 13

Simple Annuities Due - Present
Value
Start with the present value formula for an ordinary
annuity
The formula is then adjusted to accommodate the
difference in the timing of the payment by multiplying by
the factor (1+
i
)
SHORA EBRAHIMI
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BUQU 1130 - Chapter 13

Simple Annuities Due - Present
Value
Formula:
SHORA EBRAHIMI
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??

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