This page covers the following topics regarding the
calculation of the future value of an annuity:
 Formula and Definition
 FV of Annuity Illustrated
 Solving for Other Variables in the FV Equation
 Compounding Frequency
 Payment and Compounding Periods Do Not Coincide
 Excel
 HP12C
 Programming Languages
1. Formula and Definition
The equation below calculates the future value
of a stream of equal payments made
at regular intervals
over a specified period of time at a given rate.
This value is referred to as the future
value (FV) of an annuity.
In plain terms, the FV of an annuity equation
calculates how much a stream of
payments will be worth at a specified time in the future.
The FV is an accumulated value in that it represents the
accumulation of both payments made or borrowed and interest earned or
charged.
In practice the FV of an annuity equation is used to
calculate the accumulated growth of a series of payments such
as deposits to a savings account or contributions to a retirement plan.
However the calculation applies to liabilities as well as assets.
So in addition to computing the growth of a savings plan, the
equation can also calculate the accumulation of borrowings
against a line of credit.
By rearrangement of the equation above it is possible to solve
for the payment amount (PMT) necessary to
accumulate a desired FV.
While the FV tells us what the ending accumulated
balance will be, the PMT amount tells us how much we must
save or borrow each period in order to achieve the
specified FV.
For example, if we want to accumulate $5,000 in 3 years, the PMT
calculation tells us how much we must set aside each period in
order to achieve this goal.
Note the distinction between the FV of an annuity
and the PV of an annuity.
The FV of an annuity equation answers the question
"What will it be worth then?" while the PV of an annuity
equation answers the question "What is it worth now (or
some time prior to then)?".
The PV of an annuity is discussed separately
here.
The formula above assumes an ordinary annuity,
one in which each payment is made at the end of
the compounding period.
An annuitydue is
one in which the payments are made at the beginning
of the compounding period.
See AnnuityDue
for more information on the distinction between an
annuitydue and an ordinary annuity.
This distinction is further illustrated in
example problems #7
and
#32.
2. Future Value of an Annuity Illustrated
The following simplified example
illustrates the basic operation of the
FV of an annuity formula.
What is the accumulated value of a $25 payment to be
made at the end of each of the next three years
if the prevailing rate of interest is 9% compounded
annually?
Or, put another way, "How much will I
have at the end of three years if I save $25 a year
at 9%?"
This problem is represented graphically in the diagram below...
In this situation the inputs to the FV of an
annuity equation are as follows...
PMT = 25.00
i = 0.09
n = 3
...and plugging these values into the equation...
...tells us that the accumulated value at the end of three years is
$81.95 (rounded).
According to our calculation, the FV ($81.95 rounded) is greater than the
sum of the three $25 payments ($75.00).
The difference between the two values is interest.
Each payment of $25 except the last one starts earning interest
as soon as it is made.
This interest is added to the accumulated balance
each period.
So the first payment earns interest for two years...
..and the second payment earns interest for one year...
...but the third payment is made at the very
end of the term and earns no interest at all.
The mechanics of the FV calculation
are illustrated below...
...so at 9%, three payments of $25 plus accumulated interest
will be equal to $81.95 in three years.
How much interest is in the accumulated balance?
81.95  75.00 = 6.95
Additional problems illustrating the calculation of
the FV of an annuity can be viewed
here
under Application #5 "Find FV Annuity."
3. Solving for Other Variables
While the basic FV of an annuity formula
presented above allows us to calculate
FV, we often need to calculate
one of the other variables in the equation such as the number
of compounding periods (n), the payment
amount (PMT), or the interest rate (i).
These calculations are illustrated below.
Calculating the PV of an annuity (the current value
of a series of periodic payments) is discussed
separately here.
a. Number of compounding periods (n)
Solving for n is a simple matter of algebraic rearrangement
of the basic FV of an annuity formula for
which the following algebraic identity is helpful...
...and rearranging the FV of an annuity
equation to solve for n as we did for "y" above,
we get this...
...and using the values for the other variables from
our earlier example, we calculate the
number of compounding periods (n) as...
b. Payment amount (PMT)
Rearranging the basic FV of an annuity
formula to solve for PMT is a little easier
than it was for n.
What we end up with looks like this..
...and again using the values for the other variables from
our original example, we calculate PMT as...
c. Interest rate (i)
This is the tough one.
Unfortunately there is no easy way to isolate
the interest rate (i) variable in the basic
FV of an annuity equation.
What we end up with is a value
referred to as k_{FV} defined
below...
...and since we know that FV is 81.95 and PMT
is 25.00, then k_{FV} must be equal to 3.2780.
We know that n = 3 so we keep substituting
different values for i
into the right hand side of the equation
until we get close to the value of 3.2780.
This process of iteration (getting successively
closer to the desired value for k_{FV}) continues
until an acceptably accurate value for i
is found.
Alternatively, we can find two reasonably accurate
values that bracket our desired k_{FV}
and then calculate an i based on
interpolation.
This entire process is illustrated more clearly in
example problem #38.
For practical purposes i is
typically computed using a calculator
or computer program rather than through manual iteration.
In Excel the RATE function is used for this purpose.
The builtin TVOM functions of the HP12C
make it easy to calculate i for an annuity.
However, if we have to code the calculation of i
in a financial application then we're basically stuck
with iteration.
4. Compounding Frequency
The equation for the FV of an annuity
presented at the top of this page assumes
annual compounding.
But what if instead of annual payments of $25,
we were dealing with semiannual payments of $12.50.
In total the annual amount of the payment is the
same (12.50 x 2 = 25.00).
However, the FV is not.
The reason the FV changes is that
the compounding frequency (n) has changed;
it has increased from once a year to twice a year.
The net effect of this changes is to increase the value
of FV; payments are now being made earlier
(frontloaded) and since more cash is being made
available sooner, the FV will be larger.
Q. How do we account for the nonannual payments
in our calculation?
A. By a process referred to as synchronization.
The standard formula for the FV of an annuity
assumes annual compounding.
Synchronization involves modifying the values of
i and n to take into account
nonannual compounding.
In order to understand the process, we start by defining n
(the number of compounding periods in the term of the
annuity) as...
...where m is the number of compounding
periods in a year and Y is the number of
years in the term of the annuity.
So when semiannual compounding is used with our example,
n is calculated as...
This makes sense: we are compounding twice a year over three
years so the number of compounding periods (n) is six.
Now that we have adjusted n, we need to take care
of i.
The 9% interest rate we were given in the original
example was an annual nominal rate.
This is typically how rates are specified in TVOM problems.
A nominal rate is also referred to as an "applied" rate
because it is the rate at which interest is
applied to principal.
Since we increased our compounding frequency from
annual to semiannual, the nominal interest rate also needs to
change.
On an annual basis we were charging 9%.
We simply need to adjust this rate to make it
proportional to the new compounding frequency.
So, since we doubled the compounding frequency, we
must halve the interest rate.
More formally, we divide i by
the number of compounding periods in the year (m)
and our applied interest rate becomes...
... and this also makes sense; we were applying
9% annually and now we are applying 4.5%
semiannually.
What is potentially confusing is that even with nonannual
compounding, we still reference an annual rate.
So instead of "4.5% applied semiannually," we
say "9% with semiannual compounding."
It is understood that 9% is an annual rate and that
only 4.5% is applied each semiannual compounding period.
We can now modify the basic FV of an annuity equation
to incorporate the synchronization process as follows...
...and we calculate the FV under semiannual
compounding as...
We can used the same modified equation to calculate the
FV under other compounding frequencies.
For example, under quarterly compounding the FV is...
...and summarizing the effect of
increasing compounding frequencies...
FV_{annual} = 81.95
FV_{semiannual} = 83.96
FV_{quarterly} = 85.01
...we can see that increasing n has the effect of increasing
the FV of an annuity.
This is because we are compounding more frequently
so there is more interest being earned on interest and the
FV grows larger.
5. Payment and Compounding Periods Do Not Coincide
One of the basic assumptions under TVOM
theory is that of a "simple" annuity
(see Assumptions and Definitions).
A simple annuity is one where the payments and compounding
periods coincide.
For example, when we are compounding monthly,
we should also be making payments monthly.
However, in real life it is not uncommon to find a situation where
compounding is occurring more or less frequently than
payments are being made.
There are two approaches
for handling such situations: rate equivalence
and deconstruction.
a. Rate Equivalence Approach
Under this approach we convert the rate used for compounding
into an equivalent rate based on the payment frequency.
For example, consider our original example where we are making
payments annually but compounding monthly.
What annual rate is equivalent to 9% compounded monthly?
The following equation takes the 9% annual rate and
converts it to an annual effective rate under monthly
compounding...
In other words, 9% compounded monthly is equivalent
to 9.38% compounded annually.
Now we can perform our FV of an annuity calculation using
the equivalent annual rate...
...and we find that the FV increases by $0.31 (82.26  81.95)
using monthly rather than annual compounding.
b. Deconstruction Approach
As an alternative to the rate equivalence approach,
we can compute the FV for
each payment and the summation of all of these individual values
will be the FV of the annuity.
Typically this approach is used when the payment amounts
are not equal or the interval between payment dates varies.
However, it can also be applied to standard annuities.
Using our original example, the FV of a series of
three annual payments of $25 at 9% monthly
compounding is computed as the sum of the FVs of
three single sum payments of $25 each with
terms of 2, 1 and 0 years.
Graphically this approach looks like this...
...and crunching the numbers...
..we find that our calculated FV of $82.26 is the same as it was
under the rate equivalence approach.
6. Excel Spreadsheet
Excel provides us with two approaches to solve
for the FV of an annuity.
a. The FV Function.
If all we want to know is the FV of an annuity,
then we can use Excel's builtin FV Function
as shown here...
..where the cell contents look like this...
The FV Function takes the following parameters..
Note that the function can be used for both single sum and annuity calculations
depending on the parameters supplied.
b. The Accumulation Schedule.
A accumulation schedule can be constructed in
Excel to compute the FV of the annuity
without having to use any special functions.
Such a s schedule looks like this...
...and the cell contents look like this...
Note that the schedule above relies entirely upon
basic math operators; no special FV Function
was used.
This approach is typical of how a programmer might solve the
problem.
Absent knowledge of a specific mathematical equation, a common
operation (accruing interest on a cummulative balance in this case) is
simply repeated over and over again to arrive at the solution.
One benefit of the discount schedule that we do not have
with a direct FV calculation is that we can see
the FV of the annuity for all of the payment
frequencies.
For example, we know that the value of the annuity after
two payments is $52.25.
7. HP12C
8. Programming Languages
Practical application of TVOM concepts often involves
using a programming language to code the calculations.
Listed below are some very simple illustrations of how
the standard TVOM equation for the future value of an
annuity can be coded in four different
programming languages:
C#:
JavaScript:
VB Script:
TSQL:
Note on SQL.
Be careful about performing this type of math calculation using SQL
because the code executes on the database server.
As a general rule, the processing power of the database server
is best reserved for performing large scale data modification
and retrieval operations rather than arithmetic calculations.
In a production environment such calculations are typically performed
in a COM object on a middle tier server or perhaps by a VB Script in
an Active Server Page on the web server or even by JavaScript on the
client's browser.
I included this example only to show that such TVOM calculations
are possible using Microsoft's implementation of the SQL
language (called TSQL or TransactSQL).
The reader should be aware that doing so
can make for a very expensive query.
Be sure to consider all of the options before including
such functionality in your production SQL code.


