Time Value of Money Concepts

What is continuous compunding?

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 (imax) 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.