Copyright © 2008, The McGraw-Hill Companies, Inc.
10.1 Credit Cards 427Jerome’s $800 predicted balance is large enough to justify passing up the no annual fee
offering at Bank C and instead pay his $25 annual fee at Bank B. But it’s not large enough
to justify the $80 annual fee at Bank A. We noted before, though, that someone with a
“large” balance would have reason to choose Bank A. So, how large is “large”?Example 10.1.7 How large would my balance need to be for it to be worth paying
the annual fee at Bank A to get the lower interest rate?First let’s compare Bank A to Bank B. The difference between the two annual fees is $80
$25 $55. The difference between the interest rates is 15% 9% 6%. So, the balance
would have to be large enough that a 6% rate applied to it for a year would be at least equal
to $55. Solving, we get:I PRT
$55 P(0.06)(1)
P $916.67So we can conclude that the cutoff for choosing Bank A over Bank B is $916.67.Now let’s compare Bank A to Bank C. Using the same logic as before, we see that the differ-
ence in fees is $80 and the difference in rates is 14.99%. Solving for interest gives:I PRT
$80 P(0.1499)(1)
P $533.69Overall, Bank A beats out both of the other banks as long as the average balance is larger
than either of these two amounts. So as long as the average balance is $916.67 or more,
Bank A is the winner.Note that in the example above it was not really necessary to compare Bank A to Bank C.
We know that as the balance increases, a lower interest rate becomes more important.
Therefore we really only needed to compare Bank A against the bank with the next lowest
interest rate.Choosing the Best Deal—“Reward Cards”
The choice of the best credit card offer can be especially complicated for so-called reward
credit cards. Many credit card deals are available that offer some sort of reward to their
cardholders for using the card. When first launched, the Discover card made quite a splash
in the market by offering the then-remarkable deal of paying cardholders back a portion
of their annual charges in cash. The idea might have sounded crazy at first, but proved to
be very successful. Today, there are a wide range of similar deals, ranging from cards that
offer cash-back rewards, to airline frequent flier miles, to credits that can be applied toward
buying specific products, and so on.
In choosing the “best” offer, these rewards can’t be ignored, but it is often hard—if not
impossible—to apply a mathematical analysis. If the reward offered is cash, then you can
estimate the amount, and then subtract that from your total costs. But if a credit card offers
frequent flyer miles based on the amount of your purchases, what sort of value should you
place on them? If you enjoy traveling, the frequent flyer miles from your credit card may
combine with other miles you earn from travel to get you a free trip. In that case the value
of this reward to you may be great, though it is not entirely clear how to put a specific dollar
value on it. On the other hand, if you hate traveling, this is obviously of no value to you.
Likewise, if you are planning on buying a new Pontiac next year, a credit card that allows
you to earn credits toward purchase of a GM car would be of great value. If you have no
plans to buy a new car and like Toyotas better anyway, such a reward may not be worth
anything.
While such rewards can’t be ignored, judging their value is subjective. A convenience
user with a reward card offering something she values is in the terrific position of actually
getting paid to use her credit card! But someone paying a high annual fee and interest rate