Formula for Compound Interest

Date: 31 Mar 1995 13:20:35 -0500
From: Anonymous
Subject: interest calculations

Please send the the formula(s) for calculating interest compounded annually,
daily, and continuously.

Date: 2 Apr 1995 20:55:00 -0400
From: Dr. Ken
Subject: Re: interest calculations

Hello there!

Well, there's really not much difference in interest compounded yearly and
daily, it's just a question of how much interest is accrued in a certain
length of time.  They both amount to a finite geometric sequence, and we can
deal with those:  

The interest rate, p, is the ratio of successive terms in the sequence
(where p is a number like 1.05 or thereabouts).  Then if you start with a
certain amount, a, in your account, after n pay periods you have a*p^n in
your account.

To find out how interest compounded continuously works, you use limits: say
we've got "p" interest compounded continuously, where p is a number like .05
or thereabouts.  Then if we subdivide the year into m sections, each section
will compound p/m interest, and we'll compound it m times.  So after the
first section of the year we'll have 

a + a*p/m = a(m+p)/m, after the second section we'll have

a(m+p)/m     + a(m+p)/m *p/m     = a(m+p)^2/m^2, and then

a(m+p)^2/m^2 + a(m+p)^2/m^2 *p/m = a(m+p)^3/m^3, and then

a(m+p)^2/m^3 + a(m+p)^3*m^3 *p/m = a(m+p)^4/m^4, .......

So at the end of the year (i.e. after m sections) we'll have

a(m+p)^m/m^m = a*((m+p)/m)^m in our account.

If we take the limit as m goes to infinity, this will go to a*e^p, where e
is the base of the natural logarithm.  So it's similar to the non-continuous
case.

I hope this helps you.

-Ken "Dr." Math