Placing a value on a company is always a tricky business. History is filled with examples of disastrous valuations that are hard to credit in retrospect. The dotcom bubble of the late 90s is one of the best known examples.

And yet crazy valuations continue apace. One current bubble involves social media companies such as LinkedIn, Twitter, Groupon and, of course, Facebook. In July, the latter announced that it had 750 million users, an astronomical number that is dwarfed only by the company’s valuation which stands at anything from $65 billion to north of $100 billion.

By that measure, the company’s current and future users will each have to generate a remarkable amount of income for the company, numbers that reek of the boom and bust economics of the dotcom era.

So how much is Facebook really worth? Today, Peter Cauwels and Didier Sornette, econophysicists at the Swiss Federal Institute of Technology in Zurich, inject a little sanity into the debate. They argue that it is actually easier to value social media companies than other firmS because their revenue is so obviously based on a singe simple metric: the number of users.

All that is required is a reasonable model of user growth and a good understanding of the profit each user can generate.

For Facebook, user growth is pretty straightforward. Cauwels and Sornette argue that although Facebook’s growth has been exponential in the past, this cannot continue if only because of the finite number of people on the planet. Instead, Facebook user numbers will eventually level off, following a classic s-shaped curve.

Indeed, they say Facebook’s growth has already changed. In 2010, they say it switched from exponential to s-shaped.

The only question now is how high it will reach. Cauwels and Sornette offer three scenarios in which Facebook eventually plateaus at a base case of 840 million, a high growth case of 1.1 billion or a case of extreme growth reaching 1.8 billion users within a few years. (The graph above shows that Facebook’s growth will probably fall somewhere between the base and high growth scenarios.)

Cauwels and Sornette then calculate a value for the company based on the prospect of each user generating $1 profit per year, the approximate average over the last five years.

This gives a value in the base case of $15 billion, in the high growth case of $20 billion and in the extreme growth case of $33 billion. All these numbers are significantly less than those that are bandied around in the press.

It’s worth pointing out some of the assumptions behind this calculation. It generously assumes that real interest rates are essentially 0% for the next 50 years, that Facebook’s profit margins remain as high in future as they are now and that its revenue per user remains constant in future.

This last one is particularly generous. Cauwels and Sornette have worked out the average revenue per user over the last five years. But the truth is that Facebook’s revenue per user appears to be halving every 3.5 years, a fall that is entirely masked by taking an average.

So these valuations are at the top end of what could be called reasonable calculations.

They imply that if the current valuations are to be achieved, Facebook will somehow have to improve its profit per user by between 1.5 and 6 times.

That’s obviously a tall order.

Facebook might argue otherwise, however. It might say that its real value will only become apparent as it grows and its boffins become better at exploiting ever more clever data mining techniques to provide advertisers and marketeers with value. When this happens, current valuations will seem laughably small.

Maybe. More likely, however, is that the value of Facebook and other social media companies is being driven by the kind of group think that has characterised so many bubbles in the past. In these, it doesn’t matter to a trader what Facebook is really worth as long as he or she thinks some other idiot will pay more.

That kind of thinking always works until the number of idiots, like Facebook users, runs out.

There is a bigger picture here too. Sornette is director of the Financial Crisis Observatory at the Swiss Federal Institute of Technology. His job is to study bubbles and crashes. We’ve looked at some of his work here, here and here.

Why is he interested in the value of Facebook? His motive here appears to be this: to head off yet another boom and bust in the stock market at a time when the world can least afford it. A worthy, if futile, goal.

Ref: arxiv.org/abs/1110.1319: Quis Pendit Ipsa Pretia: Facebook Valuation And Diagnostic Of A Bubble Based On Nonlinear Demographic Dynamics