One of the examples in the
Miracle of Compounding
page used a formula to compute the future value
of a single sum using *continuous compounding*.

The formula looked like this:

This formula certainly doesn't look
like a TVOM formula.
Where did it come from?
And why doesn't it look like a 'regular'
TVOM formula?

The math purist (and the anal retentive) will be comforted
to know that the formula for continuous compounding can be
derived directly from the original TVOM formula.

First we start with our original formula for the
future value of a single sum:

It is important to understand that **n**, the number
of compounding periods, is actually a product of two numbers:
the number of years (**Y**) and the number of compounding periods
per year (**m**). Thus, for a two year term with monthly
compounding, **n** is equal to 24 (2 x 12).
So we can redefine **n** as:

Now we can make the original formula more precise:

Note that I have divided **i** by **m**. This is because
by convention **i** is given as an annual interest rate.
If the compounding period is something other than annual, as it usually
is, then **i** must be converted to the period it is being
applied.
E.g., for monthly compounding **i** would be divided
by 12.

Now solely for the sake of making the formula easier to work with,
I introduce a new term I call **x**:

Working **x** into the previous formula, we have:

Now that the equation is properly prepared, we can get
to the real work.

Our objective is to derive a formula for continuous
compounding.
In other words, we want the compounding interval to
be very small, less than minutes, less than seconds;
we want it to be infinitesimal.

As shown above, a small compounding period is related
to a large **m** (**m** being the number of
'slices' the year is divided into).
Thus to make the compounding period
continuous, we need to evaluate the formula when
**m** is very big. Or, to be more specific, when
**m** is infinite.

Since we set up **x** as a proxy for **m**, we
can now re-write our formula as:

The key to the solution is the term in brackets,
the limit of which converges to a single value: 2.71828.
This is the familiar exponential constant *e*, the base of natural
logarithms.
(In fact, this intimate relationship between
continuous compounding and *e* is the
reason why natural logarithms are so popular!)

So now we have:

Now if the PV is 100 and the interest rate
is 8% over a single year (**Y** = 1), then the
solution becomes:

The continuous compounding factor can be used with
any of the TVOM formulas.
In fact, a more generic way to refer to continuous
compounding may be in terms of the
familiar 'discount factor:'

Since continuous compounding is the greatest frequency
of compounding possible, we can calculate
the maximum effective rate (**i**_{max})
for a given nominal annual rate of interest (**i**) as:

Thus, for a nominal annual rate of 12%,
the maximum effective rate is 12.75% calculated
as shown:

There. The mystery of continuous compounding is no
more.