"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.

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 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

Rebate

% of Purchase Amount
Under $2,000 .0025*A

$2000

$5

0.25%
$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...

Questions:

  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?