Future Value of an Annuity - Annual Payments
Dr. Craig Ruff
Department of Finance
J. Mack Robinson College of Business
Georgia State University
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© 2014 Craig Ruff
Future Value of an Annuity - Annual Payments
Definition: An annuity is a series of equal payments made at regular intervals for a
finite period of time.
A simple example of an annuity is the payment stream associated with a standard car
loan. If you have a five-year car loan, then the equal monthly payments for 60 months
are an annuity.
Another typical example is a standard fixed-rate mortgage; a mortgage obligates the
borrower to make a series of equal monthly payments (since most mortgage payments
also include real estate taxes and homeowners’ insurance, your actual payments will
change over time; also, most mortgages in the U.S. allow you to payoff the outstanding
balance of the mortgage at any time.)
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Future Value of an Annuity - Annual Payments
Please note:
• In finance, unless stated otherwise, annuities are assumed to be paid at the end of the
period, with the first payment being made at the end of the first period. This assumption
is consistent with setting our calculator to ‘END.’
•In contrast, an “annuity due” refers to an annuity where the payments are made at the
start of the period, with the first payment being made at the start of the first period.
One way to handle annuities due is to set your calculator to ‘BGN.’
•A perpetuity is annuity that last forever.
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Future Value of an Annuity - Annual Payments
If you understand the future value of a single sum, then the future value of an annuity is not much
of an intellectual leap. Let’s start with a simple example.
Example: Suppose you want to determine the future value of a $100 annuity/year that lasts for 3
years. The compounding rate is 10%, compounded annually.
On a time-line, it would look something like this, with the first payment being made at the end of
the first year:
0
1
2
3
100
100
100
1st payment
made at end
of 1st year
2nd payment
made at end
of 2nd year
3rd payment
made at end
of 3rd year
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Future Value of an Annuity - Annual Payments
The Mythical Bank Account (aka “The Imaginary Bank Account”) - When I think about the future
value of an annuity, I find it useful to imagine a mythical bank account. Every time an annuity
payment is made to you, pretend you deposit that money into your mythical bank account. The
rate your money is earning in the mythical account is equal to whatever is the assumed
“compound rate” associated the future value calculation.
Thus, when you think about the typical future-value-of-an-annuity question, what you are really
asking is: how much money is in my mythical bank account at the end of the life of the annuity
(after the final payment is made)?
0
1
2
3
100
100
100
In calculating the future value of this
annuity, our goal is to find the balance of
the “mythical bank account” here (the
instant after the final payment is made.).
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Future Value of an Annuity - Annual Payments
To get started on calculating the FV of this annuity, imagine that when the first $100 arrives at
the end of year 1, you place that $100 in the mythical bank account for the next two years,
earning 10%, compounded annually. By the end of the third year (t=3), this $100 would have
grown to $121.
0
1
2
3
100
100
100
100 (1.1) 2
121
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Future Value of an Annuity - Annual Payments
Next, imagine that when the second $100 arrives at the end of the second year, you place that
$100 in the mythical bank account for the remaining one year, earning 10%, compounded
annually. By the end of the third year (t=3), this $100 would have grown to $110.
0
1
2
3
100
100
100
121
100 (1.1)2
100 (1.1)1

110

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Future Value of an Annuity - Annual Payments
Finally, when the third and final $100 payment is made, that amount is also placed in the
mythical bank account. Thus, the balance in the mythical bank account at the end of the third
year is $331 ($100 +$121 +$110). This is the future value of the annuity at t=3.
0
1
2
3
100
100
100
121
100 (1.1)2
100 (1.1)1

110

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Future Value of an Annuity - Annual Payments
Notice that the three pieces used to compute this future value can be written (and then
rewritten) as:
FV3 =100 (1.1) 2  100 (1.1)1  100 (1.1) 0  331
FV3 =100 (1.1) 0  100 (1.1)1  100 (1.1) 2  331
This can be rewritten in summation notation as:
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FV3 = 100 (1.1) t  331
t =0
You may not have to worry about the summation notation at an introductory level;
however, you will likely run into this notation later in life if you continue on in the finance
area.
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Calculating the Future Value of an Annuity - Annual Payments
There is also a computational formula for calculating the future value of an annuity with
annual payments (assuming that the first payment comes at the end of the 1st year).
(1 +r) N 1
FVN = A 

r



Where A is the annual annuity payment, r is the annual compounding rate and N is the
number of payments.
Using the computational formula, the future value of this annuity is:
(1.1) 3 1
FV3 = 100
 331
 .1 

You are more likely to use your TVM buttons to find the future value of an annuity;
however, the computational formula can be useful in spreadsheet applications.
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Calculating the Future Value of an Annuity - Annual Payments
Continuing on with this example: Suppose you want to determine the future value of a $100
annuity/year that lasts for 3 years. The compounding rate is 10%, compounded annually.
On the calculator, you would enter this as:
Buttons
Numbers
to Enter
PV
0
FV
????
I
10
N
3
PMT
100
331
Initially, it may seem odd to enter zero here. But try to keep in mind
what you are telling your calculator. By entering that N=3 and
PMT=100, you are telling the calculator that there are three
payments of $100. And, in this case, that is all of the payments; thus,
the PV=0. Simply, there is no payment at t=0.
By entering N=3 and PMT=100, you are telling your calculator that
there are three payments of $100. By having your calculator set on
“END,” you are telling the calculator that the first payment is due at
the end of the 1st period.
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Examples
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Example: Future Value of an Annuity - Annual Payments
As an example, suppose you deposit $10,000 per year for the next fifty years into an account
paying 5%, compounded annually. Assume the first deposit is made in one year. What is the
FV of this annuity at t=50?
Buttons
Numbers
to Enter
PV
0
FV
????
I
5
N
50
PMT
10000
2,093,479.995
Again, in this type of question, you enter a zero here. Try to keep in
mind what you are telling your calculator. By entering that N=50 and
PMT=10,000, you are telling the calculator that there are fifty
payments of $10,000. And, in this case, that is all of the payments;
thus, the PV=0. Simply, there is no payment at t=0.
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Example: Future Value of an Annuity - Annual Payments
As another example, suppose your goal is to save $1,000,000 at the end of thirty five years.
To do this you plan to make annual deposits over the next thirty five years, with the first
deposit at the end of the first year. You assume that you can earn a compound return of 6%,
compounded annually, on your investments over this period. How much do these annual
investments need to be to reach your $1,000,000 goal?
Buttons
Numbers to
Enter
PV
0
FV
1,000,000
I
6
N
35
PMT
?????
This problem is a classic problem in which you tell it four variables
and it will tell you the fifth.
-8,973.858
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Example: Future Value of an Annuity - Annual Payments
Continuing with this example: Suppose your goal is to save $1,000,000 at the end of
thirty five years. To do this you plan to make annual deposits over the next thirty five
years, with the first deposit at the end of the first year. You assume that you can earn a
compound return of 6% on your investments over this period. How much do these
annual investments need to be to reach your $1,000,000 goal?
One very useful way to help your understanding of TVM problems is to build a simple
spreadsheet that models the cash flows and balance across time.
The next slide describes one version of a simple spreadsheet. Warning: Building this
type of model is not as bad as the next slide makes it look; it is actually very easy to
construct.
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Assumed compounding rate…
Assumed annual investment…
This column is the balance moving across time. Shown
below is the calculation for D5.
Subject to some rounding, by the time we make the final
payment at t=35, the balance is $1,000,000.
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