This page covers the following topics regarding the
calculation of the future value of a single sum:
 Formula and Definition
 FV of a Single Sum Illustrated
 Solving for Other Variables in the FV Equation
 Compounding Frequency
 Excel
 HP12C
 Programming Languages
1, Formula and Definition
The equation below calculates how large
a single sum will become at the end of a specified
period of time.
This value is referred to as the future value (FV)
of a single sum.
Observe from the formula that the future value
(FV) consists of both a present value (PV)
piece  an initial lump sum  and an accumulated interest piece.
Thus, we start with a fixed amount and calculate
how large it will grow (i.e., accumulate or compound)
over the specified period of time and interest rate.
The FV of a single sum formula serves as a means
of valuation.
It tells us what something will be worth at a future
date.
This "something" can be an asset or a
liability.
For example, if we borrow $1,000 today and don't
make any payments until the loan comes due two
years from now, how much will we owe at that time?
Likewise, if we deposit $5,000 into a bank account
today, how much will it be worth in 180 days?
Note the distinction between the FV of a single sum
and the PV of a single sum.
The FV of a single sum answers the question
"How much will it be worth then?" while
the PV of a single sum answers the
question "What is it worth now (or before 'then')?".
The PV of a single sum is discussed separately
here.
Also note that the formula above gives us the FV of
a single sum; in other words, a fixed, lump
sum amount.
The future value of an annuity formula
gives us the FV of a series of periodic
payments.
The FV of an annuity is discussed separately
here.
2. Future Value (FV) of a Single Sum Illustrated
The following simplified example illustrates the basic
operation of the FV of a single sum formula.
How much will I receive at the end of 3 years if
I invest a single sum of $50 today at 8% interest
compounded annually?
In other words, at 8%, how large will my $50
grow in 3 years.
Drawn from the the perspective of the investor,
the problem is illustrated below.
The investor deposits $50 (the PV)
and this amount "accumulates" over 3 years
at 5% to some larger amount (the FV)...
The arrow drawn pointing toward the time line
(labeled "50.00") represents a cash outflow
from the investor.
The arrow drawn pointing from the time line
(labeled "?FV") represents a cash inlow
to the investor, in this case it is the
accumulated amount of the CD which the investor
may withdraw at maturity.
The question mark denotes the fact that this is the unknown
amount whose value is the object of our calculation.
(For additional assistance reading a cash flow diagram, click
here.)
So now that we have identified our initial deposit,
the term of the investment, and the interest rate, we can summarize
our inputs to the FV of a single sum equation as...
FV = 50.00
i = 0.08
n = 3
...and plugging these values into the equation...
...we calculate a FV of $62.99.
The mechanics of the calculation are illustrated below...
So what does it mean when we say that the future
value of $50 in 3 years at 8% is $62.99?
It means that since $50 today will grow (accumulate)
to $62.99 in three years time, we should be
indifferent as to our preference for one
option over the other.
In other words, $50.00 today
and $62.99 in three years are financially equivalent.
Or at least they are according to TVOM principals
and a set of assumptions discussed more fully
here.
In reality, there are other factors that
need to be taken into consideration (taxes,
default risk, cash flow, etc.) before we
can really declare "equivalence."
Still, TVOM theory and its associated calculations
provide a powerful tool for analyzing
financial alternatives by providing a
mechanism for placing cash flows
at different time periods on a comparable basis.
3. Solving for Other Variables
While the equation discussed above allows us to calculate
the FV of a single sum, there are times
when we need to know the value of one
of the other variables (n, i, or PV) .
For a single sum, solving for any of the other TVOM variables
is a simple matter of rearranging the basic
formula to isolate the variable being sought.
a. Compounding periods (n)
Knowledge of the following algebraic identity is
necessary for isolating the exponent n...
Now by rearrangement of the FV
of a single sum equation
we can find the number of compounding
periods (n) in our
original example as...
b. Interest rate (i)
The following algebraic identity is helpful
when solving for i...
Now we can solve for the interest rate (i) in our
original example as...
If the compounding frequency is something
other than annual, the interest rate (i)
determined above would need to be multiplied by
the number of compounding periods per year (m) in
order to return the annual interest rate.
For example, the FV of $50 in 3 years
at 8% under monthly compounding is $63.51.
In this case we would calculate
an annual i as...
See the discussion on "Compounding Frequency"
that follows for more information on
adjustments made to the values of i
and n under nonannual
compounding frequencies.
c. Present Value (PV)
Rearranging to solve for the PV of a single sum
is fairly straight forward...
...and using the values from our original example,
we confirm the PV as...
The PV of a single sum is discussed
in more detail here.
4. Compounding Frequency
The FV of a single sum equation at the
top of the page assumes annual compounding.
But what if in our original example we were compounding
quarterly rather than annually?
In this case we must "synchronize" the values
for i and n in order to accommodate
the nonannual compounding frequency.
We start by defining n, the number of
compounding periods in the term, as being equal
to the product of two numbers:
the number of compounding periods
in the year (m) and
the number of years in the term (Y)...
Thus for a three year term (Y=3) with quarterly compounding
(m=4), the number of compounding periods (n)
is 12 (4 x 3).
Now that we have modified n, we must adjust i.
i is almost always given as a annual nominal rate.
If the compounding frequency is something
other than annual, then i must be made
proportional to the the period in which it is
being applied.
Typically this is accomplished by dividing i
by m.
Since here we are compounding quarterly,
i would be divided by 4.
Taking all of this into account, if we
rewrite the standard future value of a single sum
equation to incorporate the
synchronization process, it looks like this...
...and if in our original example above we had used quarterly
rather than annual compounding, the FV
is calculated as...
Changing the compounding period from annual
to quarterly increases the future value by $0.42
over the 3 year period ($63.41  $62.99).
More frequent compounding means more interest is
being earned on interest resulting in a greater
accumulation (future value).
Under monthly compounding, the FV is even larger...
...and larger still under daily compounding...
...and largest under continuous compounding...
The FV formula used for continuous compounding
looks a little strange.
However, it is derived directly from
the standard FV of a single sum equation.
The concept of continuous compounding and derivation of the
formula are discussed in more detail at
Continuous Compounding.
Additional information on the impact of frequency and term
on TVOM calculations can be found at
Miracle of Compounding.
5.Excel
There are two approaches to solving for the FV
of a single sum in Excel:
a. FV function
If all we want is the FV of a single sum,
we can use Excel's FV function as shown here...
...where the cell formulas look like this...
...and the input parameters to the function are defined as follows...
Note that this function can be used for both single sum and
annuity calculations depending on the parameters supplied.
b. Accumulation Schedule
We can compute the FV of a single sum without the aid of
a special function by creating a accumulation schedule
as shown here...
...where the cell contents look like this...
The schedule simply accumulates the balance
one period at a time.
In other words, the "beg" balance is
the PV and the "end" balance is the
FV.
The "int" amount is equal to the interest
earned on the PV for the current period.
6. HP12C
7. 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 a
single sum 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.


