The Mathematics of Money

426 Chapter 10 Consumer Mathematics

calculations behind your decision. There is something to be said for simplicity—keeping

all your financial dealings at one institution is more appealing that having accounts in a

dozen different places. However, given that interest rates and fees can vary dramatically

from one issuer to another, it is usually worth shopping around to find the best deal, or to

at least make sure that you have a competitive one. The absolute lowest costs may not be

the most important thing to you, but you probably want to make sure that the deal you are

getting is at least reasonable compared to what’s available elsewhere.

Ideally, as a consumer you would want to choose the card that has both the lowest

interest rate and the lowest annual fee. If one of your options is the lowest cost for both of

these, no calculations are necessary. The choice is obvious. What if, though, the card with

the lowest annual fee carries a higher interest rate, while the card with the lowest interest

rate has a high annual fee. How can we balance the interest rate against the annual fee to

determine which is the best deal?

For example, suppose you have the choice of a Visa card issued by any of three different

banks. The banks’ offers are shown in the table below. (Note that in this table the abbrevia-

tion APR, short for annual percentage rate, is used; it is common practice on credit card

offers to label the interest rate in this way.)

Card Issuer APR Annual Fee

Bank A 9% $80

Bank B 15% $25

Bank C 23.99% None

Which deal is best? The answer really depends on how you use the card. If you are a

convenience user, paying your bill within the grace period each month, the interest rate

is irrelevant. You’re not going to pay any interest anyway, and so the interest rate doesn’t

matter to you in the slightest. In that case, your best choice would be Bank C, because with

no annual fee you will be able to use this card for free. On the other hand, for someone who

carries a very large balance, the savings from a low interest rate would more than make up

for a high annual fee, and so Bank A would be the obvious winner.

The question is more challenging for someone in the middle, who may carry a balance,

but not a large enough one that a lower interest rate would obviously compensate for a higher

annual fee. In such a case, we must make a reasonable estimate of how much of a balance

will be carried, and then crunch the numbers. The following example will illustrate:

Example 10.1.6 Jerome expects that he will normally carry a credit card balance of

around $800. Which of the three options in the table given above would be the lowest

cost option for him?

We can calculate the total annual cost of each card. Since he will be carrying a balance of

around $800, at Bank A his interest charges over the course of a year would be:

I 
 PRT

I 
 ($800)(0.09)(1)

I 
 $72

Combining this with the $80 annual fee means that his total annual cost at Bank A would be

approximately $80  $72 
 $152.

For Bank B, the interest would be I 
 PRT 
 ($800)(0.15)(1) 
 $120. Added to the $25

annual fee would total $145.

For Bank C, interest would be I 
 PRT 
 ($800)(0.2399)(1) 
 $191.92. There is no annual

fee, so this it total cost.

In Jerome’s case, clearly it is worth paying an annual fee and going with either Bank A or B.

Bank B has the lowest overall cost. If Jerome has some other reasons (convenience, existing

banking relationship, etc.) to prefer Bank A, though, he might choose that option, since the

difference is only a few dollars and is only an estimate anyway. Bank B, however, has the low-

est projected cost, and so barring any other factors it is the best choice.