ECON1202 Lecture 3
Geometric Progressions and Annuities
UNSW Economics
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Agenda
1 Background: the geometric progression and series;
2 Annuities – present, future value;
3 Annuities due;
4 Perpetuities.
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Consider the following number sequences:
2 -2 2 -2 2 -2 (-1)
1.00 0.60 0.36 0.216 0.1296 (0.6)
0.5 1 2 4 8 16 32 64 (2)
... in each, we have,
• An INITIAL VALUE, a; and
• A RATIO OF TERMS, q,
q =
xi+1
xi
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Definition: Geometric Progression
A (finite) GEOMETRIC PROGRESSION (or sequence) is a list of numbers
where the first number a is chosen, and subsequent numbers are given
by multiplying the preceeding term by a constant factor q,
a, aq, aq2, aq3, . . . , aqn−1, aqn, aqn+1, ... (1)
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Definition: Geometric Series
A GEOMETRIC SERIES is simply the sum of the first n terms of a
geometric progression, i.e.,
s = a + aq + aq2 + aq3 + · · ·+ aqn−1. (2)
We can show that this sum simplifies to become
s =
a(1− qn)
1− q
. (3)
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Example
Find the sum of the series,
s = 2+ 4+ 8+ 16+ 32+ 64+ 128+ 256+ 512
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Example
Find the sum of the series,
s = 2+ 4+ 8+ 16+ 32+ 64+ 128+ 256+ 512
Applying formula (3), with a = 2, q = 2 and n = 9,
s =
2(1− 29)
1− 2
=
2(1− 512)
−1
= 1022
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Example
Find the sum of the series,
s = 2+ 4+ 8+ 16+ 32+ 64+ 128+ 256+ 512
Applying formula (3), with a = 2, q = 2 and n = 9,
s =
2(1− 29)
1− 2
=
2(1− 512)
−1
= 1022
Check (by hand): 2+ 4+ 8+ 16+ 32 = 62, 64+ 128+ 256+ 512 = 960:
960+ 62 = 1022.
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Problem: choosing mobile-phone contracts
You have recently signed a contract for a new mobile phone. The
repayments required per month for the phone alone are $17 per month
(calls are extra). The contract says you’ve got to pay this for two years,
with the first payment due in a month.
How should this payment plan be valued? In particular, how would
you compare the plan to an alternative deal where you just buy the
phone outright?
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Definition: Annuity
An ANNUITY is a sequence of agreed payments made at fixed
intervals, called the PAYMENT PERIOD, over a given length of time,
called the TERM of the annuity.
Example sequence – repay a loan at $250 with a payment period of 2
months, for a term of 12 months:
2 4 6 8 10 12
t (months)
Deposits: $250 $250 $250 $250 $250 $250? ?
Question then: how is this different to one lump-sum payoff at the
outset? At the end?
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Example (Present Value of Annuity)
Show that the present value A of an annuity of agreed payments R, paid at the
end of each of n periods, with r interest rate (per period), is given by (hint: use the
geometric progression result in (3)),
A = R ·
1− (1+ r)−n
r
.
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Example (Present Value of Annuity)
Show that the present value A of an annuity of agreed payments R, paid at the
end of each of n periods, with r interest rate (per period), is given by (hint: use the
geometric progression result in (3)),
A = R ·
1− (1+ r)−n
r
.
Writing out the present value of the annuity components we have,
A = R(1+ r)−1 + R(1+ r)−2 + · · ·+ R(1+ r)−n ,
which is just the same as a geometric series with initial value R(1+ r)−1, and constant
factor (1+ r)−1, using our formula (3),
A =
R(1+ r)−1
[
1− (1+ r)−n
]
1− (1+ r)−1
=
R
[
1− (1+ r)−n
]
(1+ r)
[
1− (1+ r)−1
] = R
[
1− (1+ r)−n
]
(1+ r)− 1
= R ·
1− (1+ r)−n
r
.
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The previous example leads to the following definition,
Definition: Present Value of an Annuity
The present value of an ordinary annuity having regular payments R
at the end of each payment period for n payments with an interest rate
of r per period is given by,
A = R ·
1− (1+ r)−n
r
. (4)
• Payments are assumed to be made at the end of each payment
period;
• Likewise, interest is assumed to be calculated at the end of each
payment period.
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Example (Back to the Mobile-phone)
You have signed a contract to make $17 payments on your mobile phone
every month for two years, with the first payment in one month. What is the
present value of these payments? (Assume an interest rate of 8 % per year.)
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Example (Back to the Mobile-phone)
You have signed a contract to make $17 payments on your mobile phone
every month for two years, with the first payment in one month. What is the
present value of these payments? (Assume an interest rate of 8 % per year.)
First, we identify the information: the repayment period is one month so the interest
rate per period is 0.08/12 = 0.0067; also, the number of periods is 2× 12 = 24. Next
we use formula (4),
A = R ·
1− (1+ r)−n
r
= 17 ·
1− (1+ 0.0067)−24
0.0067
= $376 .(try your own phone...)
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Example (Car loan)
Suppose you are considering purchasing a car. The model you want will cost you
$13,450. You have access to a loan through your bank, who would charge a
nominal interest rate of 9.50%. What would the monthly repayments be if the
term of the loan was 5 years?
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Example (Car loan)
Suppose you are considering purchasing a car. The model you want will cost you
$13,450. You have access to a loan through your bank, who would charge a
nominal interest rate of 9.50%. What would the monthly repayments be if the
term of the loan was 5 years?
What we have, is an annuity, but this time we know the present value ($13,450) and just
have to find the fixed repayments that give this amount. So, rearranging,
A = R ·
1− (1+ r)−n
r
R = A ·
r
1− (1+ r)−n
substituting in our information, (periodic rate 0.095/12 = 0.0079)
R = 13, 450 ·
0.0079
1− (1+ 0.0079)−5×12
= $282 per month .
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Interpretation
• Each payment is for the same nominal amount;
• However, due to the time-value-of-money, payments at different
times have different values.
• (Think about this as: one dollar today is worth more than one
dollar next year, because the dollar today can be invested to earn
interest in the meantime.)
• It is for this reason that we need to work out the real value of the
stream of payments.
2 4 6 8 10 12
t (months)
Deposits: $250 $250 $250 $250 $250 $250
↑
focal date
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Notice, we can of course, work out the future value of the loan, just as
easily as the present value,
2 4 6 8 10 12
t (months)
Deposits: $250 $250 $250 $250 $250 $250
↑ focal date
• Notice: the final payment is made on the focal date itself, so
requires no adjustment;
• Thus, previous payments need adjustment up till the n− 1
payment,
S = R + R(1+ r) + R(1+ r)2 + · · ·+ R(1+ r)n−1
• Another geometric series.
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Definition: Future value of an annuity
The FUTURE VALUE OF AN ANNUITY (ordinary) with payments of R
per payment period for n periods at interest rate r per period is,
S = R ·
(1+ r)n − 1
r
(5)
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Alternatively, rather than an ordinary annuity (payments are at the
end of each payment period), suppose the first payment is at the start
of the period (like a magazine subscription, say).
0 1 2 3 4 5 6 7
t (months)
Deposits: $50 $50 $50 $50 $50 $50 $50
term
This scheme is called an ANNUITY DUE and changes our definitions
slightly...
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Definition: Present value of annuity due
The PV of an annuity due is the same as the value of an ordinary
annuity one period before the present, brought to the present. So the
PV of an annuity due is just the PV of an annuity, with one period’s
interest added (accumulated),
A = (1+ r)
[
R ·
1− (1+ r)−n
r
]
(6)
0 1 2 3 4 5 6 7
t (months)
$50 $50 $50 $50 $50 $50 $50
Ordinary Annuity term
PV−1
PV0
(1+ r)×
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Definition: Future value of annuity due
As with the PV of an annuity due, the FV is the same as the future
value of an ordinary annuity which starts one period before the
present, brought to the present. So the FV of an annuity due is just the
FV of the annuity, with one period’s interest added (accumulated),
S = (1+ r)
[
R ·
(1+ r)n − 1
r
]
(7)
0 1 2 3 4 5 6 7
t (months)
$50 $50 $50 $50 $50 $50 $50
Ordinary Annuity term
FV−1
FV0
(1+ r)×
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Example
Suppose you put $100 on the first of each month into a savings account which
pays 5.5% accumulated monthly. What would be size of the account after 45
years?
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Example
Suppose you put $100 on the first of each month into a savings account which
pays 5.5% accumulated monthly. What would be size of the account after 45
years?
Using formula (7) for the future value of an annuity due, we have (with monthly rate
0.055/12 = 0.0046..,
S = (1+ r)
[
R ·
(1+ r)n − 1
r
]
= (1+ 0.0046..)
[
100 ·
(1+ 0.0046..)45×12 − 1
0.0046..
]
= $237, 038.79
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Perpetuities
One more variation: an ordinary annuity with payments (at the end of
every period) that continue forever, n = ∞.
0 1 2 3 . . . k . . .
t (months)
Payments: $250 $250 $250 $250
This scheme is called a PERPETUITY.
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Perpetuities
What is the present value of a perpetuity?
The easiest way to work this out is to start with the PV of an ordinary
annuity,
S = R ·
1− (1+ r)−n
r
,
and let the number of periods n approach infinity; so
S →n→∞ R ·
1− 0
r
=
R
r
.
Definition: Present value of perpetuity
The present value of a perpetuity is S = R/r.
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Example
Anton has set up an annual math competition, the “Kolotilin Kup”, starting
in one year and running forever. Anton has asked you to make a one-time
contribution into the prize fund today, which will be invested at an interest
rate of 10% compounded annually. What contribution must you make given
that the annual prize will be $1000?
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Example
Anton has set up an annual math competition, the “Kolotilin Kup”, starting
in one year and running forever. Anton has asked you to make a one-time
contribution into the prize fund today, which will be invested at an interest
rate of 10% compounded annually. What contribution must you make given
that the annual prize will be $1000?
Notice that the sequence of annual prizes as a perpetuity.
The present value of the perpetuity is exactly your one-time contribution today.
So:
S = R/r = 1000/0.1 = $10000.
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