Do the math: Facebook is not a buy

February 5, 2013, 9:17 PM UTC

FORTUNE — Facebook’s stature with investors, severely diminished following its botched IPO in May, is enjoying a remarkable revival. Since sliding to less than half its offering price in September, the shares have surged 60% to over $28. Its fourth quarter earnings, announced on January 30, garnered upbeat reviews. The perception is spreading once again that the social networking phenomenon, boasting a billion users, will prove a fabulous growth stock. So is it time to buy Facebook?

To decide, let’s first determine how much Facebook’s value needs to rise over the next decade to give investors an acceptable return. Then, it’s straightforward to calculate how fast its earnings must increase to support that future market cap. How Facebook will fare getting friends to share photos, videos, and status updates is uncertain, but determining the profits it will need to generate is mainly math.

Let’s assume that investors demand a 10% annual return on their Facebook shares. That number sounds high, but Facebook is a pricey stock that needs to attract adventurous buyers seeking a big score. Facebook (FB) doesn’t pay a dividend, and we’ll assume that it continues to retain all of its earnings during its rapid phase of expansion, the usual course for shooting stars in tech. Hence, the entire return needs to come in the form of capital gains.

MORE: Exclusive: Twitter nears $10 billion valuation

Today, Facebook’s market cap stands at $69 billion. That’s a big number, in the same with league with Goldman Sachs ($71 billion), 3M ($69 billion) and American Express ($66 billion), proven players that make far more money. Getting to that 10% return through 2022 isn’t as simple as raising the $69 billion valuation by 10% a year. It’s a good bet that Facebook will issue a lot more shares over the next decade. Tech companies are highly active in selling stock to pay employees and make acquisitions, and by its actions so far, Facebook hews enthusiastically to that tradition. We’ll be extremely conservative and project that Facebook will grow its shares outstanding by 2% annually, increasing the “float” 22% by 2022.

So for investors to gain 10% a year for the next 10 years, the stock price must rise 12% annually, keeping in mind that today’s stockholders will be diluted an average of 2 percentage points a year. Hence, Facebook’s market cap will need to grow 200%, to $207 billion.

How high do profits need to grow by 2022? If Facebook is to keep promising 10% returns in 2022, when it should be a reasonably mature company, its price-to-earnings ratio needs to fall to around 14. So profits must reach almost $15 billion. Its PE multiple in this frothy market now exceeds 50.

What’s worrying is the distance between what Facebook earns now and the summit it needs to attain. Facebook guides investors to follow what it calls “non-GAAP” figures that it presents alongside the official GAAP numbers in its earnings releases. The non-GAAP numbers are akin to “cash earnings.” For 2012, the non-GAAP earnings eliminate giant restricted stock awards, much of them made in prior years but expensed in 2012.

MORE: Go ahead, bash your boss on Facebook

So let’s use Facebook’s preferred measure. Its non-GAAP earnings for 2012 were $1.3 billion. To get from $1.3 billion to $15 billion, total profits would need to grow at 31% annually for a full decade. How likely is that? From 2011 to 2012, Facebook’s earnings rose 13%. Interestingly, its EPS, again by its own non-GAAP measure, expanded just 6% because its share count swelled by over 6%. Facebook, for example, used stock, as well as cash, last year to buy photo-sharing site Instagram for $1 billion.

Even if Facebook maintained its current lofty margins, its sales would need to approach $70 billion by 2022. Its current revenue is almost all from advertising. It’s highly questionable whether the world ad market is big enough, or growing fast enough (it’s not), for a relative upstart to mushroom to that size. Two of the world’s biggest sellers of ads are News Corp and Time Warner. To get to $70 billion in sales, Facebook would need to capture the equivalent of today’s combined revenues of those two companies.

Fat margins are also bound to attract challengers, and Facebook has plenty. It faces growing competition in virtually every new area it’s targeted for expansion. Yahoo’s recently revived Flickr is a rival to Instagram in photo-sharing. Twitter is a formidable foe for Facebook’s News Feed.

Facebook’s future as a business is still highly promising. But the odds that it will grow earnings 31% a year are extremely low. Even to get to an 8% return, earnings would need to expand 8-fold in ten years.

MORE: Cracking the mobile ad market

It’s comforting that Facebook is financially solid, bolstered by around $10 billion in cash. That hoard is partially the legacy of its notorious IPO. In effect, Facebook turned the tables on Wall Street. The investment banks typically lobby their IPO clients to substantially underprice their shares so that they “pop” on the first day of trading, using the ludicrous argument that the publicity of the pop is more valuable than more cash in the corporate treasury. It usually works. But Facebook apparently didn’t buy that argument. It received proceeds at $38 a share, far more than the stock was actually worth, judging from how the shares plunged right after the offering.

Expensive stocks typically bring puny rewards for investors. Facebook may turn out to be a great company, but the market’s expectations of its future greatness are just too lofty to make it anything approaching a good buy.