Interest: Definitions

Date: 9/25/96 at 23:12:48
From: Jake Taylor
Subject: math help

What does the term compound interest mean? What does continuous 
compounding mean?

Date: 10/29/96 at 20:15:14
From: Doctor Mason
Subject: Re: math help

Hi Jake!

Let's begin with simple interest.  Simple interest is computed only 
once for the duration of the loan or investment.  Therefore, if you 
plan to invest $1000 for 3 years, the bank will figure the interest 
only once at the end of the three years and then give it to you.  
Let's say the account pays 5 percent interest.  At the end of three 
years, the bank will calculate  $1000 x .05 x 3 = $150 and pay you 
$150 in interest.  If you add that to what you started with, you will 
end up with $1150.

Compound interest is calculated more often, and as soon as it is 
calculated, the interest in added to your account. That way the next 
time the interest is computed, you have MORE in your account than the 
money you put there. Let's suppose your $1000 is in an account paying 
5 percent compounded semi-annually (twice a year) for three years.  
We can use the Simple Interest formula over and over to figure your 
interest. This formula is I = PRT, where I = the interest you get,  
P = the amount you invest, R = the interest rate written as a decimal, 
and T = the time in years. (For us our time is 1/2 since we're 
figuring the interest twice each year.):

$1000.00 x .05 x (1/2) = $25 for the first 6 months.  
   Add it to your account.
$1025.00 x .05 x (1/2) = $25.63 for the second 6 months. 
   Add it to your account.
$1050.63 x .05 x (1/2) = $26.27 for the third 6 months. 
   Add it to your account.
$1076.90 x .05 x (1/2) = $26.92 for the fourth 6 months. 
   Add it to your account.
$1103.82 x .05 x (1/2) = $27.60 for the fifth 6 months. 
   Add it to your account.
$1131.42 x .05 x (1/2) = $28.29  for the last 6 months. 
   Add it to your account.
                                
This makes for a total of $1159.71 in your account!

As you can see, you end up with $9.71 more if they calculate the 
interest multiple times during the time the money is invested rather 
than just at the end. You make more money because they are paying 
you interest on the money they paid you earlier! Pretty neat, isn't 
it? The more often the interest is compounded, the more interest you 
earn. (By the way, banks don't do it by writing it all out the way we 
did in our example above. They have charts and formulas that help 
them calculate the compound interest quickly.)

Interest compounded continuously uses a formula that involves the 
number "e".  "e" is like "pi" in that it's an irrational number that 
people just found as they calculated things like growth and decay.  
Continuous compounding is growth of money, so it fits the "e" 
situation. Anyway, if you find an account that compounds continuously, 
you would earn even more interest.  The formula for continuous 
compounding is A = Pe^(rt) where the rate x time is the 
exponent on the constant "e" (which is about 2.7).

Hope this helps.

-Doctor Mason,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/