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What's American Express (AXP 1.60%) without its deal with Costco (COST 0.77%)? That's the billion-dollar question shareholders of the credit card company are seeking answers to.

The company's conference calls tell the tale. Executives now believe that American Express will be able to retain "at least 20% of the out-of-store spending of the former Costco co-brand card members," thanks to its efforts to market new cards to these card members prior to turning over its Costco book to Citigroup.

The upper end of the range -- really, a pie-in-the-sky estimate -- was that the card company could retain 50% of out-of-store Costco spending, consistent with its experience in Canada. To be sure, the card company always warned that its ability to retain customers in the United States would be materially lower than its experience in Canada, because it retained its portfolio in Canada rather than selling it off.

The lower end of the range for retaining Costco spend has always been...well, just about zero. The new Costco card issued by Citi and run over the Visa network has substantially better rewards than the old American Express card. I've previously estimated that American Express managed to put new cards in the hands of approximately 20% of its former Costco card members, based on commentary from previous calls.

But issuing new cards is only half the battle; getting its customers to use them is another matter. The promotions were rich with rewards -- as much as $250 for spending just $1,000 in the first three months. After that period ends, newly issued cards will have to compete on a level playing field with the new Costco card.

Estimating the actual amount of Costco spend that American Express can retain has always been difficult. Keep in mind that American Express's projections come just one month after Citi and Visa took over the Costco deal on June 20.

Rebuilding the business

American Express indicated several times on its conference call that it plans to increase spending to acquire new customers. In his prepared remarks, Jeffrey Campbell, CFO and executive VP of American Express, stated that "while we had previously expected our total spending on growth initiatives during full-year 2016 to be similar to 2015, we now expect to spend at a somewhat higher level."

He went on to add that "when you look at the back half of 2016, it's certainly not news to anyone on this call; it's a very competitive environment right now," and that the company expects to spend at least $200 million more on marketing and promotion in 2016, versus its 2015 spending of $3.1 billion.

That implies marketing and promotion expenses should ramp to more than $890 million per quarter for the remainder of 2016, significantly higher than the first-half average of about $760 million per quarter. The difference adds up to at least $0.29 of drag on the company's pre-tax earnings per share for the second half of 2016.

If you're looking for a reason for the market's disappointed reaction to American Express's second-quarter earnings, this is it. A return to growth is seemingly coming at a much higher price than anyone had expected.