This page covers the following topics regarding the
calculation of the present value of a single sum:
 Formula and Definition
 PV of a Single Sum Illustrated
 Solving for Other Variables in the PV Equation
 Compounding Frequency
 Excel
 HP12C
 Programming Languages
1. Formula and Definition
The equation below calculates the current value
of a single sum to be paid at a specified date
in the future.
This value is referred to as the present value (PV)
of a single sum.
If we remember that 1/x^{n} can be written
as x^{n}, then a more compact form of the
equation can be written as:
The PV of a single sum formula is used as
a valuation mechanism.
It tells us how much an amount to be transacted in
the future is worth today (or some date prior to
the receipt or payment date).
For example, if we are want to buy a $1,000 zero coupon
bond that matures in 10 years, how much is it worth
right now?
In other words, how much should we pay for that bond
today?
The PV of a single sum formula can value
liabilities as well as assets.
For example, if we owe a debt that obligates us
to pay $5,000 in 3 years time, what is a fair
amount we could offer to settle that debt today?
Note the distinction between the PV of a single sum
and the future value (FV) of a single sum.
The PV of a single sum answers the
question "What is it worth now (or before some future
date)?" while
the FV of a single sum answers the question
"How much will it be worth then?"
The FV of a single sum is discussed separately
here.
Also note that the formula above gives us the PV of
a single sum; in other words, a fixed, lump
sum amount.
The present value of an annuity formula
gives us the PV of a series of periodic
payments.
The PV of an annuity is discussed separately
here.
2. Present Value (PV) of a Single Sum Illustrated
The following simplified example illustrates the basic
operation of the PV of a single sum formula.
What is the current value (PV) of a CD that will
pay $100 in 3 years if the prevailing interest rate is 5%
compounded annually?
In other words, how much do I need to deposit to have
$100 in 3 years?
Drawn from the the perspective of the investor,
the problem is illustrated below.
The investor will receive $100 in three years time (the FV)
and this amont is "discounted" back to today at 5% in order to
calculate the required deposit (the PV)...
The arrow drawn pointing away from the time line
(labeled "100.00") represents a cash inflow to the investor.
The arrow drawn pointing to the time line
(labeled "?PV") represents a cash outflow from the investor,
in this case it is the amount invested in the CD.
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 how much we will receive at maturity,
the term of the investment, and the interest rate, we can summarize
our inputs to the PV of a single sum equation as...
FV = 100.00
i = 0.05
n = 3
...and plugging these values into the equation...
...we calculate a PV of $86.38.
The mechanics of the calculation are illustrated below...
So what does it mean when we say that the present
value of $100 in 3 years at 5% is $86.38?
In essence it means that the receipt of $100
in three years is worth the same as the receipt of $86.38 today.
The logic behind this assertion is that if we deposited
$86.38 into an investment account paying 5% annually,
it would grow to $100 in three years.
In this case we should be indifferent as to our preference for one
option over the other because $86.38 today
or $100 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 PV of a single sum, there are times
when we need to know the value of one
of the other variables (n, i, or FV) .
For single sums, solving for any of the other TVOM variables
is simply a 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 PV
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 identities are 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 PV of $100 in 3 years
at 5% under monthly compounding is $86.10.
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. Future Value (FV)
Rearranging to solve for the FV of a single sum
is fairly straight forward...
...and using the values from our original example,
we confirm the FV as...
The FV of a single sum is discussed
in more detail here.
4. Compounding Frequency
The PV 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 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 present 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 present value PV
is calculated as...
Changing the compounding period from annual
to quarterly reduces the present value by $0.23
over the 3 year period ($86.38  $86.15).
More frequent compounding means less money
is required up front (i.e., at "present")
in order to grow to a specified amount in the future.
Under monthly compounding, the PV is even smaller...
...and smaller still under daily compounding...
...and smallest under continuous compounding...
The PV formula used for continuous compounding
looks a little strange.
However, it is derived directly from
the standard PV of a single sum equation.
The concept of continuous compounding and derivation of the
formula is 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 PV
of a single sum in Excel:
a. PV function
If all we want is the PV of a single sum,
we can use Excel's PV function as shown here...
...where the cell formulas look like this...
...and the input parameters to the function are defined as follows ...
b. Discount Schedule
We can compute the PV without the aid of
a special function by creating a discount schedule
as shown here...
...where the cell contents look like this...
The schedule simply discounts the FV balance
one period at a time.
In other words, the "beg" balance is
the FV and the "end" balance is the
PV.
The "int" amount is equal to the difference
between the two and is calculated as...
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 present 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.


