Interest Continuously Compounded or Compounded Daily

Date: 10/11/2002 at 00:08:24
From: Mike Foroobar
Subject: Continuously Compounded Interest or Compounded Daily

I lent a friend $40 on the condition that he pay 3% interest per day 
until he decided to pay me back, and he agreed. What formula should I 
use to calculate the amount he owes me at any day x? Additionally, 
would I have made more money if I had told him that it would be 
compounded continuously? (And if so, what is the formula for this at 
at any day x?)

Thanks in advance.

Date: 10/11/2002 at 05:23:05
From: Doctor Mitteldorf
Subject: Re: Continuously Compounded Interest or Compounded Daily

Mike -

You can multiply the amount by 1.03 for every day that he has the loan 
out. For example, if he pays you after 1 week, he owes $40 times
1.03^7, where ^7 means "raised to the seventh power."

The more frequently you compound the interest, the more interest you
accumulate. But the process doesn't go on without limit. Suppose you
were owed a total of 10%, and let's make the original principal $100.

If you didn't compound at all, then the payback amount would be $110.

Suppose you compounded twice during that time; the way to do that
calculation would be to divide the 10% into two equal portions; but
instead of adding 5% and 5%, you'd multiply 1.05 times 1.05, as we did
above. The result is 1.1025, so the payback amount would be $110.25 -
slightly more than without compounding.  

To continue this process, imagine compounding 10 times over that 
period. You'd divide the 10% into 10 equal portions and multiply 1.01
by itself ten times: 1.01^10=1.104622. The payback amount would be
$110.4622.  

Is there a limit? Suppose you divided the 10% into N equal portions, 
then raised the result to the Nth power.

           (1 + 0.10/N) ^ N

We know the results for N=1 and N=2 and N=10. Suppose we let N get 
very large? The results will continue to increase, but not forever;
they will approach a limit, which just means there's a maximal amount
that the interest can't get to, no matter how big it is. But by making 
N larger and larger, you can get arbitrarily close to that amount.

That amount is the number e raised to the 0.10 power = e^0.10 =
1.105171. The payback amount would be $110.5171.  

In fact, the most common way to define the number e is as e raised to 
the power 1 from the above analysis. In other words, suppose you had, 
not 10% interest, but a full 100%. Divide this up into N packages, and 
raise the interest to the Nth power.  

   (1 + 1/N) ^ N

The limit of this quantity as N gets very large is e=2.7182818...

So I leave it to you to finish your example: suppose your friend kept 
the $40 for one week at 3% a day, and you had agreed on continuous
compounding: how much would he owe you in that case?

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/