Continuous Compounding

We continue the discussion that was started in the previous section: Simple versus Compound Interest.

In a nutshell:

• You invest some money (the principal) in a bank which pays interest.
• After a specified time (the compounding period), you add in the interest that has been earned on the principal.
• After another compounding period, you add in the interest earned on the principal plus accumulated interest.
• Repeat.

Compound interest, by definition, is interest calculated on the principal amount together with accumulated interest. Interest can be added in at different fixed intervals: annually, monthly, weekly, daily, and so on.

The following are equivalent:

• Add in interest $\,n\,$ equally-spaced times per year.
• Add in interest every $\,\frac{365}{n}\,$ days (called the compounding period). (Ignoring leap years for simplicity, there are $\,365\,$ days in a year.)

Adding in interest more and more often produces a slightly better accumulation. (Most banks add in interest daily.)

What happens as you increase the number of times that interest is added in each year? Your earnings will get bigger. How much bigger? Will your earnings increase without bound?

Equivalently, what happens as you add in interest over shorter and shorter periods of time? In other words, what happens as you let the compounding period approach zero? You'll end up with more money. How much more? Will you get rich?

The amount of money produced under these situations does not increase without bound! (Wishful thinking, but doesn't happen.) Instead, the irrational number ‘$\,\text{e}\,$’ emerges to describe the limiting amount earned. The resulting formula is called the Continuous Compounding Formula, and is the subject of this section.

A Quick Review of the Compound Interest Formula

• Put $\,P\,$ dollars (the principal) in a bank.

• Assume the bank offers an annual interest rate $\,r\,.$

  For example, $\,3\%\,$ per year corresponds to $\,r = 0.03\,.$

• Add in the earned interest $\,n\,$ equally-spaced times each year. In other words, assume there are $\,n\,$ compounding periods per year.

  For example, adding in interest monthly corresponds to $\,n = 12\,.$ Adding in interest daily corresponds to $\,n = 365\,.$

• Continue this process for $\,t\,$ years.
• The accumulation (principal plus interest) is given by the variable $\,A\,$ in the Compound Interest Formula: $$ \cssId{s39}{A=P{(1+\frac{r}{n})}^{nt}} $$

Letting $\,n\,$ go to Infinity in the Compound Interest Formula

As $\,n\,$ gets large, $\,\frac{r}{n}\,$ gets small. That is, as $\,n\rightarrow\infty\,,$ we have $\,\frac{r}{n}\rightarrow 0\,.$

As $\,\frac{r}{n}\,$ approaches zero, $\,1 + \frac{r}{n}\,$ approaches $\,1\,.$ That is, as $\,\frac rn\rightarrow 0\,,$ we have $\,1 + \frac rn \rightarrow 1\,.$

As $\,n\,$ gets big, $\,nt\,$ gets big. That is, as $\,n\rightarrow\infty\,,$ we have $\,nt\rightarrow\infty\,.$

So, as we add interest in more and more frequently (let $\,n\rightarrow\infty\,$), here's what happens in the Compound Interest Formula:

We're looking at a ‘$\,1^\infty\,$’ form—which is just a shorthand for saying that the base approaches $\,1\,,$ and the exponent approaches $\,\infty\,.$

But, here's the important question:

As the base and exponent travel together along their journeys to $\,1\,$ and $\,\infty\,,$ respectively, what number (if any) does the entire expression approach?

A Discussion of the ‘$\,1^\infty\,$’ Form

There are lots of things that make the ‘$\,1^\infty\,$’ form both interesting and difficult to analyze. All of the following are true:

• $\,1\,$ to any finite power equals $\,1\,.$   For example, $\,1^{1,000,000,000} = 1\,.$

• When a number a bit less than $\,1\,$ is raised to higher and higher powers, it approaches zero.

  For example, $\,(\frac{99}{100})^n\,$ approaches $\,0\,$ as $\,n\,$ approaches infinity. (This is an exponential function with base between zero and one.)

• When a number a bit more than $\,1\,$ is raised to higher and higher powers, it approaches infinity.

  For example, $\,(\frac{101}{100})^n\,$ approaches $\,\infty\,$ as $\,n\,$ approaches infinity. (This is an exponential function with base greater than one.)

Depending on precisely how the base approaches $\,1\,,$ and how the exponent approaches infinity, the form $\,1^\infty\,$ can approach different numbers!

These three different $\,1^\infty\,$ forms approach (respectively) $\,\text{e}\,,$ $\,\infty\,,$ and $\,1\,$:

• Here's a familiar one:   $\displaystyle\lim_{n\rightarrow\infty} (1 + \frac 1n)^n = \text{e}$

  Go to wolframalpha.com and enter:

  the limit, as n goes to infinity, of (1 + 1/n)^n

• $\displaystyle\lim_{n\rightarrow\infty} (1 + \frac 1n)^{n^2} = \infty$

  Go to wolframalpha.com and enter:

  the limit, as n goes to infinity, of (1 + 1/n)^(n^2)

• $\displaystyle\lim_{n\rightarrow\infty} (1 + \frac 1n)^{\sqrt{n}} = 1$

  Go to wolframalpha.com and enter:

  the limit, as n goes to infinity, of (1 + 1/n)^(sqrt(n))

‘$\,1^\infty\,$’ is an example of what is called (in calculus) an indeterminate form. An indeterminate form is when you can't figure out what is happening without further analysis . That is, each occurence of ‘$\,1^\infty\,$’ must be investigated separately.

For us, the investigation will involve noting the similarity of our situation to the definition of the irrational number $\,\text{e}\,.$ Compare these:

There are some pesky differences:

• $\displaystyle\,\frac rn\,$ instead of just $\displaystyle\,\frac 1n$
• $\,nt\,$ instead of just $\,n$
• the factor of $\,P\,$ in front

We can deal with these differences—keep reading!

Derivation of the Continuous Compounding Formula

So, you want $\displaystyle\,\frac rn\,$ to ‘look like’ $\displaystyle\,\frac 1n\,$?   Make it happen!

Define a new variable $\,m\,$ by this relationship:

Recall that ‘ $:=$ ’ means ‘equal, by definition’. (By the way, this is the tricky part of the derivation.)

Cross-multiplication in equation (*) gives $\,n = mr\,.$

Now, rewrite the Compound Interest Formula in terms of $\,m\,$ instead of $\,n\,,$ and do a bit of algebra:

Taking reciprocals in equation (*), we have $\,m = \frac nr\,.$ So, as $\,n\,$ goes to infinity, $\,m\,$ also goes to infinity.

And (here's the punch line!) as $\,m\,$ goes to infinity:

Example

Suppose that $\,\$1000\,$ is invested at $\,3\%\,$ annual interest. What is the accumulation after ten years if compounded monthly, daily, and continuously?

Compounded monthly:

Compounded daily:

Compounded continuously:

Not much difference! You won't get rich if your bank decides to compound continuously!