(and L'Hopital's rule -- Sometimes limits are kind of
Derivation of the Compound Interest Formula
In financial mathematics we tend to use an unusual base for an exponential:
Let's see where this model comes from: it's the compound interest model -- that is, interest on top of interest. If you receive simple interest at a rate on an initial investment , then at the end of the year you'd have
If the interest is truly simple, then you simply accumulate at the end of each year, for a total (after years) of .
If, on the other hand, you keep re-investing the money accrued, then you have what is known as "annual compounding": the formular for it would be
Now, in compound interest, you take your interest rate, split it up several (say 12) ways, and then do the computation several (e.g. 12) times:
And if you did it for years, you'd have
Hence, more generally,
You may have encountered a few laws such as The rule of 72 (70, 71,
69.3,...) to calculate doubling time: forget them! Just do the
math. The question is this: for what value of t is
for t, we get
where n is the number of compoundings per year, and r is
given as a decimal (e.g. 9% is represented by .09). This is the
When compounding is continuous (i.e. ), this reduces
to the very lovely rule
Now, how do we know that
That is, how do we know that
The answer, of course, is L'Hopital's Rule, which is useful in solving certain indeterminate limits. Let's rewrite it a little: we want to show that
Here's the general situation: http://www.nku.edu/~longa/classes/mat122/days/day35/thm7-7-1.gif
And if the limit of the quotient of the derivatives is indeterminate, then iterate -- that is, do it again! Then work with the ratio of the second (or higher) derivatives.
Motivation of L'Hopital's Rule
(requires the limit definition of the derivative).
Let's suppose that we're looking at an indeterminate limit
of the form 0/0. We're assuming that both functions are differentiable at
, which means that both functions are continuous at
(i.e. ). Let's
also assume that .
We start with the limit of the derivatives, and make our way over to the