"Cash-Back" 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 some--usually small--amount of money to the card holder. The amount rebated is tied to the amount charged in a one-year 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.


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 one-year 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 A-2,000; you'll earn .005(A-2,000) on the remainder, for a total of 5+.005(A-2,000). At the top of this range, $5,000, you earn a rebate of 5+.005(5,000-2,000) = 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 (A-5,000) you make 1%, for a total of 20+.01(A-5,000). 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. So--stop charging at $17,000. here's a table that illustrates.


At the Upper Limit

Amount A Rebate

Purchase Amount


% of Purchase Amount

Under $2,000 .0025*A




$2,000 to $5,000 5+.005(A-2,000) $5,000 $20 0.40%
$5,000 to $10,000 20+.01(A-5,000) $10,000 $70 0.70%
$10,000 to $14,000 70+.02(A-10,000) $14,000 $150 1.07%
$14,000 to $17,000 150+.03(A-14,000) $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...


  1. How much must you charge to earn better than 1% cash back? Provide a range of values.

  2. (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?