Credit Card Calculations

Date: 1/24/96 at 10:53:17
From: Anonymous
Subject: credit card calculations

I have textbooks and programs, none of which cover the calculation
typical of credit card balance payoffs - how long will it take?  All 
that I have assumes a constant monthly payment, while the credit card 
payment is calculated as a percentage (usually in the 2% - 4% range) 
of the outstanding balance. I would like to be able to determine the 
effect of varying conditions such as APR, the above percentage 
variation, etc.  

Can you help me with a formula or such?   (While it has been 50 years, 
I did graduate with a EE degree, so don't worry about the algebra 
involved....)

Thanks in advance!
Ted Jarmie

Date: 7/29/96 at 13:20:6
From: Doctor Jerry
Subject: Re: credit card calculations

Suppose at t=0 months, P dollars is owed to the credit card company.  
We assume that for all t >= 0 there are no new charges, the APR is 
r percent, and the company requires that s percent of the unpaid 
balance be paid each month.

Letting B(t) denote the balance owed the company at time t months, 

B(0) = P,
B(1) = B(0) + B(0)*(r/100)*(1/12) - B(0)*(s/100),
B(2) = B(1) + B(1)*(r/100)*(1/12) - B(1)*(s/100),
  .
  .
  .

In each line after the first, the new balance is the old balance plus 
the interest the company earned on that balance minus the amount you 
have been asked to pay.  By factoring, for t >= 1,

B(t) = B(t-1)*z

where z= 1 + r/1200 - s/100.  It follows that

B(t) = P*z^t    (read z^t as z to the t power).

I'll work out one case.  Suppose P = 1000, r = 18, and s = 3.  Then

B(0) = 1000.00
B(1) =  985.00
B(2) =  970.22 
B(3) =  955.67
.
.
.
B(24) = 695.78.

Since, in this case, z = 0.985, the graph of B(t) against t will be a 
curve that falls slowly, approaching the t-axis asymptotically.  After 
5 years, the person will still owe 403.81.

I hope this answers your question.

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