The Mathematics of Money

106 Chapter 3 Compound Interest

marketing gimmick) it is seldom considered seriously. For any practical purposes, there is

seldom any point in compounding more frequently than daily.

Continuous Compounding (Optional)

But what if we really push the envelope? While compounding every minute didn’t produce

much of a gain, it still did earn more than just daily compounding. What if we compound

interest every second? Or every thousandth of a second? How about compounding every

nanosecond?^3 Compounding interest a billion times each second must result in some pretty

staggering gains! Even if the quantity of interest earned each nanosecond is vanishingly

small, surely the power of interest building upon itself so mind-bogglingly often can be

expected to result in a staggering future value!

Yet the astounding answer to this question is that even incomprehensibly fast com-

pounding produces little extra beyond dowdy, boring old daily compounding. In fact,

mathematicians discovered hundreds of years ago that there is a limit to just how much

the interest can grow to, no matter how fast it compounds. Assuming that the interest

compounds every fraction of a fraction of a fraction of a second is known as continuous

compounding. Continuous compounding can be a tricky idea to wrap your mind around.

How fast is it? Continuous compounding is so fast that no matter how fast you want your

interest compounded, continuous compounding is faster. Continuous compounding is the

“limiting case,” the result of assuming that interest compounds the fastest it even theoreti-

cally could.

If we go ahead and pretend that our interest is compounding infinitely often, our com-

pound interest formula becomes useless. We would have to divide our interest rate by

infinity (whatever that means) and then use an infinite exponent (whatever that means).

However, the following remarkable formula gets the job done:

FORMULA 3.2.1

The Continuous Compound Interest Formula

FV 
 PVert

where

FV represents the FUTURE VALUE (the ending amount)

PV represents the PRESENT VALUE (the starting amount)

e is a mathematical constant (approximately 2.71828)

r represents the ANNUAL INTEREST RATE

and

t represents the NUMBER OF YEARS

The e used in the formula is a mathematical constant. Like its better known cousin , e

is an irrational number, meaning that when we try to represent it with a decimal, the

decimal never stops or forms any repeating pattern. This of course makes the precise value

of e impossible to work with. However, for all practical purposes, we only need to use an

approximate value for e that is accurate to the first few decimal places ( just as the value of

 is often approximated by 3.14 or 22/7.) For us, the approximation e 
 2.71828 will be

close enough for all practical purposes.

Example 3.2.7 Find the future value of $5,000 at 8% interest for 5 years, assuming

that the interest compounds continuously.

This problem is yet another variation to add to the table from Example 3.2.5. This time, we

use the continuous compounding formula to fi nd the result:

FV 
 PV(ert)

FV 
 $5,000(2.71828)(0.08)(5)

(^3) A nanosecond is one billionth of a second.