Why is the Minimum Payment on a Credit Card Bill so Small?
by Jason Schattman
Sir John A. Macdonald Secondary School, Waterloo, Canada
jschattman@rogers.com
Introduction
On a monthly credit card balance of $1000, a typical credit card company will only ask for a minimum payment of $20. Why do credit card companies do that? Let's see if Maple can lead us to some insights.
Mathematics of Credit Card Debt
Suppose we do what the company wants and make only the minimum payment p every month against an initial balance of b. If the company charges monthly interest rate r, what is the balance after n months?
Let's make a quick table and see if we can notice a pattern.
Balance After n Months
|
One Month:
 = 
|
| Two Months:
|
Three Months:
|
Four Months:
|
Looking at the pattern we have developed, the balance after "n" months would be:
Example 1
Problem Statement:
If your credit card company charges a monthly interest rate of 2% (annually 24%) on an initial balance of $1000, and you make a monthly payment of $30, what is your balance after one year?
Solution:
Let's use the formula that we just derived, where 
and 
.
= 
Your new balance after one year would be 
This means you've paid the credit card company 
= 
over one year, but only reduced your balance by 
= 
. You've paid 
= 
in interest charges...and this on a balance of just $1000! Yikes.
How Long Does it Take to Get Out of Debt?
We would also like to know how many payments we would need to pay off our debt. Let's set our balance formula to zero, solve for 
and assign this expression to 
= 
Example 2
Problem Statement:
How many months would it take to pay off a balance of $1000 if we made $30 monthly payments while being charged 2% monthly interest?
Solution:
Let's use our formula for 
:
= 
months
It would take 
months (4.5 years!) to pay off our debt with a monthly payment of $30. We've made total payments of 
= 
, of which $620 is interest!
What if we double the payment to $60? Do we cut the time in half?
Compare this to 
= 
months.
The time is much less than half! And this time we've only made total payments of 
= 
, saving $420 in interest payments.
Visualization
Let's visualize the formula for N. Below we plot the number of months N needed to pay down an initial debt of $1000 as a function of the monthly payment p, assuming a monthly interest rate of 2%.
Close to p = $20, notice that the time to pay off our debt increases to infinity!
Why?
Recall the formula for time 
needed to get out of debt:
= 
The time becomes infinite as the value 
approaches 
.
Solving 
for p, we have
(Note that when we select "Isolate for 
", Maple assumes the expression is equal to zero unless otherwise specified.)
That is, if we pay less than $19.61 per month, it will take an infinitely long time to get out of debt. This is where we saw the asymptote above! (Note also that 
= 
. That is, the payment is just enough to cover the monthly interest charge.)
Let's see this graphically by plotting our balance formula for p
, and 
over a five year period. (Notice the green curve crosses the n axis at 
.)
![plot([Balance(n, 0.2e-1, 19.61, 1000), Balance(n, 0.2e-1, 30, 1000), Balance(n, 0.2e-1, 60, 1000)], n = 0 .. 60, view = [default, 0 .. 1000])](/view.aspx?SI=6647/Credit Card Application Maple 12 - Final_61.gif)
![plot([Balance(n, 0.2e-1, 19.61, 1000), Balance(n, 0.2e-1, 30, 1000), Balance(n, 0.2e-1, 60, 1000)], n = 0 .. 60, view = [default, 0 .. 1000])](/view.aspx?SI=6647/Credit Card Application Maple 12 - Final_62.gif)
The balance virtually remains constant for 
=20! Now we know what the credit card companies are after - they want you to stay in debt forever!
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