Facebook and the St. Petersburg Paradox

FEBRUARY 4, 2012   THE WALL STREET JOURNAL

If you are considering buying into Facebook's initial public offering of stock, take a moment to ponder the St. Petersburg Paradox, an old riddle still relevant to investing today.

Proposed in the 18th century, the paradox works like this: I will toss a coin until it comes up heads, at which point you get paid and the game ends. You get $1 if the coin comes up heads on the first toss, $2 if the coin comes up heads on the second, $4 if it is heads on the third, $8 on the fourth and so on. Your prospective payoff doubles with each successive flip until the coin finally lands on heads and the game is over.

Investors considering Facebook should think first about the St. Petersburg Paradox, the 18th-century mathematical riddle that helps explain why growth stocks can get overvalued so easily. Jason Zweig has details on The News Hub. Photo: Getty Images

How much would you pay to play? The paradox is that the potential payoff is infinite; after the first flip, you have a 50% chance of winning $2, plus a 25% chance of winning $4, plus a 12.5% chance of winning $8 and so forth. You will win $537 million if you get heads on the 30th toss; on the 50th, $563 trillion.

However, psychologists have found, most people won't pay more than $20 or so for a chance to play the game. After all, you could get heads as early as the first toss, leaving you with only a $1 payoff and no chance to earn those giant later gains. With that risk of quick, sharp loss looming at the outset, the infinite value of playing the game feels finite to most people.

A high-growth stock like Facebook is a lot like that St. Petersburg coin. The potential payoffs are enormous, although not infinite—and the game might peter out all too soon. At the end of 2010, 608 million people actively used Facebook every month; by this past Dec. 31, 845 million people did. If Facebook keeps growing that fast, more than 22 billion people will be using it 10 years from now, or three times the estimated population of the planet today.

The main difference is that the warm glow Facebook users get from its services may blind them to the St. Petersburg Paradox. Because Facebook connects people so powerfully, the people who use it may feel powerfully connected to the company, too—and to its stock.

Christophe Vorlet for The Wall Street Journal

Nevertheless, "it's an incredibly difficult thing to forecast the future cash flows of this kind of company, even for quantitative investors," says Charles Lee, a professor of accounting at Stanford University's business school and a former head of equity research at Barclays Global Investors (since acquired byBlackRock). Thus, says Mr. Lee, "once your projections go out beyond two or three years, you're in very murky waters."

The slightest stumble in high growth rates can lead to enormous changes in value at fast-moving companies, since today's stock prices are highly sensitive to projections of long-term future profits.

Just as the coin-flipping game in the St. Petersburg Paradox can end on any toss, even the fastest-growing company's upward trajectory can flatten in a flash. Just ask shareholders in Amazon.com, who this week lost 8% in one day on the company's warning that it might lose money next quarter.

If Facebook comes out at the high end of the valuation range proposed for the stock in its first sale to the public, the company would have a total market value of around $100 billion.

Now, let's say Facebook will be as successful in the future as Google already has been, suggests Jay Ritter, a finance professor at the University of Florida and an expert on initial public offerings. "Facebook is basically on Google's trajectory, so I think that's a very reasonable scenario," he says.

Facebook, at $3.7 billion in revenues and $1 billion in profits in 2011, already has nearly three times the sales and 10 times the profits that Google had when that company first listed in 2004.

Now, imagine that Facebook continues its torrid growth and expands over the next 10 years until it has grown as big as Google is today—with annual revenue of nearly $40 billion and net income of almost $10 billion. That would imply that Facebook will grow roughly tenfold over the coming decade—an average annual growth rate of about 26%, which is seldom sustained by big companies.

In this bullish scenario, what would happen to Facebook's stock?

At today's valuations, Google's shares trade at a total market value of just over $190 billion. If Facebook's shares rose from a total initial value of $100 billion to $190 billion 10 years from now, they would deliver a 90% cumulative gain, for an average annual return of 6.8%. Bottom line: "The valuation is so high today that the upside potential is limited," Mr. Ritter says.

Like most companies planning a public offering, Facebook declined to comment.

There certainly is a chance that Facebook's first public shareholders will be richly rewarded over the years to come—the same way someone taking the bet in the St. Petersburg Paradox could become fabulously wealthy. But, with Facebook, the odds would be a lot better if the price to make this particular bet were a lot lower.

— intelligentinvestor@wsj.com; twitter.com/jasonzweigwsj