Which has better rewards: BankAmericard vs Amex Blue Preferred rewards card?

This is an interesting question that could be asked in a consumer ed class.

Suppose Alice is shopping for a cash-back rewards credit card. She is considering between the BankAmericard and the American Express Blue Preferred credit cards. Here are the details of those cards, at the time of this writing.

SUMMARY OF REWARDS
Category	BankAmericard	Amex Blue Preferred
Supermarkets	3%	6%
Gas	2%	3%
Department stores	1%	3%
Everywhere else	1%	1%
Annual Fee	0	$75

Which card will give Alice more cash back? Under which conditions is one card better than the other?

You may assume the following: ars a responsible consumer, Alice never carries a balance and always pays her credit card every month in full. She spends around $1000 a month, $200 in gas, $200 in groceries, and $50 in department stores.

(Also, there are no affiliate links in this article as I am interested in math and have no ulterior motive).

A concrete calculation

Part of the setup was how much Alice spends: she spends around $1000 a month, $200 in gas, $200 in groceries, and $50 in department stores.

To get to annual spending, we multiply everything by 12 months in a year. Her annual spending is $12,000 total, $2,400 in gas, $2,400 in groceries, and $600 in department stores.

For BankAmericard, this means she spends $2,400 in bonus categories of supermarkets and gas, and she spends $7,200 in everything else.

So now let’s compute. First the BankAmericard rewards:

BankAmericard = 0.01 (everything else) + 0.03 (supermarket) + 0.02 (gas)

BankAmericard = (0.01)(7,200) + (0.03)(2,400) + (0.02)(2,400)

BankAmericard  = $192

For the American Express card, this means she spends $2,400 in bonus categories of supermarkets and gas, $600 in department stores, and $6,600 in everything else.

Now we compute the American Express Blue Preferred card rewards.

Amex = 0.01 (everything else) + 0.05 (supermarket) + 0.02 (gas + department) – 75

Amex = (0.01)(6,600) + (0.06)(2,400) + (0.03)(3,000) – 75

Amex = $300 – 75 = $225

It is a bit surprising to me, but the American Express card actually wins out, even with the $75 annual fee.

The extra rewards for the groceries and department stores more than compensate for the annual fee.

But how much does Alice have to spend for the American Express to be worth it?

Let’s dig a bit deeper into the math.

A useful notation

Above we did the calculation by finding out spending in “everything else” and then adding on the bonus for specific categories.

That can become cumbersome as we are constantly figuring out what spending in “everything else” is.

Another way to calculate is as follows. Both cards offer a base 1 percent on everything. This means we can write the bonus cash back rates in terms of the additional amount ON TOP of 1 percent.

For example, the American Express card offers 3 percent cash back on gas. We can think about this as a 1 percent base bonus, and then a 2 percent bonus cash back for gas. Similarly, the supermarket category is a 2 percent cash back, which consists of a 1 percent base bonus, and a 1 percent category bonus rate.

A cashback formula

Using the notion of bonus cash back rates, we can write slightly simplified cash back formulas.

Let’s use the symbols T for total spending, S for supermarkets, G for gas, and D for department stores.

The BankAmericard cash back rewards can be expressed in the following formula:

BankAmericard = 0.01 T + 0.01 S + 0.02 G

Notice how we don’t have to compute spending in “everything else.” We just get a 1 percent cash back for everything, then we add on the bonus cash back rates for special categories.

Analogously, we can write out the American Express Blue Preferred card rewards in terms of bonus rates. So the 6 percent on supermarkets becomes a 5 percent bonus rate, and the 3 percent bonus on gas and department stores is a 2 percent bonus rate.

The only thing we have to add is the annual fee of 75 dollars which subtracts from the earned rewards.

Amex = 0.01 T + 0.05 S + 0.02 (G + D) – 75

So these are the formulas for cashback. For the Amex card, we want to make sure the spending is in annual terms.

What’s the breakeven point?

Now that we have simplified formulas for both cards, we can figure out where the breakeven point is.

We have:

BankAmericard = Amex

0.01 T + 0.01 S + 0.02 G =  0.01 T + 0.05 S + 0.02 (G + D) – 75

75 = 0.04 S + .02 D

Notice how much the equation simplifies!

The BankAmericard only wins out when your spending on shopping and department stores would earn less than $75, at cash back bonus rates of 4 percent and 2 percent, respectively.

Conversely, the American Express Blue Preferred card is worth it if you earn more than $75 with those bonus rates.

This implies you will be better off with the American Express if you spend $1875 a year in groceries alone ($156 monthly), or $3,750 ($312 monthly) in department stores alone.

Or it will be worth it if you spend some blended amount between the two: let’s say $1500 in groceries ($125 monthly) and $750 in department stores ($62.5 monthly).

In conclusion

The American Express Blue Preferred card is mathematically a better deal than the BankAmericard rewards so long as you spend enough on groceries and department stores. You’ll have to make sure you earn enough to merit the $75 annual fee, but this is probably a reasonable assumption since people often do spend $150 per month in groceries.

And you’ll also have to watch out when cards pull back on their cash back rates–this happens all the time.

(On a closing point, this article is not saying this is the best cash back card. In upcoming weeks we’ll consider the popular Chase Freedom card to see how its quarterly rewards stack up).