
"CashBack" Credit Cards
I recently received an offer for a new credit card. To
protect the innocent, let's assume the offer came from
the "HF" company. Here was part of the offer
(and I quote):
 Automatic enrollment in HF CashAward$
 Earn up to 3% Cash Back on
qualifying purchases! Some credit cards offer up
to 1% cash back, but with HF CashAward$ you get
more cash back...up to 3%.
(Cash back programs rebate someusually smallamount
of money to the card holder. The amount rebated is tied
to the amount charged in a oneyear period.) Forgetting
for a second about what qualifications might be hidden in
"qualifying purchases," let's examine the claim
"you get more cash back...up to 3%." Naturally
one must read the fine print. It's right in the middle on
the back side of the offer; right where it's least
likely to be read.
IMPORTANT INFORMATION REGARDING HF
CashAward$ REBATE:
Wait! Stop right there. If it's "IMPORTANT"
why isn't it on the front page in big letters? OK,
continue...
Rebate rates are based on qualifying purchases
posted to your account during a rebate calculation
period of 12 month billing cycles following
enrollment in CashAward$. The rebate rate on
qualifying purchases over $14,000.01 and not
exceeding $17,000 is 3%; between $10,000.01 and
$14,000.00 is 2%; ; between $5,000.01 and $10,000.00
is 1%; between $2,000.01 and $5,000.00 is .5%; and on
the first $2000.00 is .25%. Purchases posted to the
account that exceed $17,000 in the aggreagate during
the rebate calculation period are not eligible for
the rebate.
Get that? First of all, any charges over $17,000 in
the oneyear rebate accumulation period do not generate any
cash back. Second...it's only the charges you make after
having already charged $14,000 that generate a 3%
rebate.
Let's assume you charge an amount A where A is no more
than $2,000. Here you're subject to the rate of
.25%that's 1/4 of 1%. Your rebate will then be .0025A.
If you charge exactly $2,000 you will earn a rebate of
$5. Whoopee.
You begin to do better by charging more. Suppose A is
between $2,000 and $5,000. You'll make the $5 on the
first $2,000 charged, for the remainder you'll earn a
rebate of .5% (1/2 of 1%). The remainder is A2,000;
you'll earn .005(A2,000) on the remainder, for a total
of 5+.005(A2,000). At the top of this range, $5,000, you
earn a rebate of 5+.005(5,0002,000) = 5+15 = 20 bucks. A
little better? Your total rebate of $20 is 20/5000 = .004
= .4% of the total amount charged.
Continue on. Assume A is between 5000 and 10000.
You'll make the first $20; on the remaining (A5,000) you
make 1%, for a total of 20+.01(A5,000). At $10,000 your
rebate is $70; that's .7% of your purchases.
And so forth...until you get beyond $17,000, where you
can't earn any cash back. Sostop charging at $17,000.
here's a table that illustrates.


At
the Upper Limit

Amount A 
Rebate 
Purchase
Amount

Rebate

% of Purchase
Amount

Under $2,000 
.0025*A 
$2000

$5

0.25%

$2,000 to $5,000 
5+.005(A2,000) 
$5,000 
$20 
0.40% 
$5,000 to $10,000 
20+.01(A5,000) 
$10,000 
$70 
0.70% 
$10,000 to $14,000 
70+.02(A10,000) 
$14,000 
$150 
1.07% 
$14,000 to $17,000 
150+.03(A14,000) 
$17,000 
$240 
1.41% 
Above $17,000 
$240 
$Infinity 
$240 
0.00% 
Here's a plot of the situation.
It is impossible to earn more than
1.41%and that occurs only if you're rich enough to
charge $17,000 in one year. Also note that the cash back
percentage dwindles to 0 as you charge more and more...
Questions:
How much must you charge to earn
better than 1% cash back? Provide a range of
values.
(Hard...requires background in
mathematical statistics.) Suppose HF knows that
the people who have this card charge an amount A
that is normally distributed with mean $10,000
and standard deviation $3500. (Assume that the
small % of folks who'd purchase negative amounts
actually purchase nothing.) What is the average
rebate? If there are 100,000 such customers, how
much can HF plan on paying back through rebates?

