How the financial markets fell for a 400-year-old sucker bet.

Here's how to make money flipping a coin. Bet 100 bucks on heads. If you win, you walk away $100 richer. If you lose, no problem; on the next flip, bet $200 on heads, and if you win this time, take your $100 profit and quit. If you lose, you're down $300 on the day; so you double down again and bet $400. The coin can't come up tails forever! Eventually, you've got to win your $100 back.

This doubling game, sometimes called "the martingale," offers something for nothing—certain profits, with no risk. You can see why it's so appealing to gamblers. But five more minutes of thought reveals that the martingale can lead to disaster. The coin will come up heads eventually—but "eventually" might be too late. Most of the time, one of the first few flips will land heads and you'll come out on top. But suppose you get 11 tails in a row. Just like that, you're out $204,700. * The next step is to bet $204,800—if you've got it. If you're out of cash, the game is over, and you're going home 200 grand lighter.

But wait a minute, maybe somebody will loan you the $200,000 you need to stay in the game. After all, you've got a great track record; up until this moment, you've always ended up ahead! If people keep staking you money, you can just keep betting until, eventually, you win big time.

See where I'm going with this?

The carefully synthesized financial instruments now seeping toxically from the hulls of Lehman Bros. and Washington Mutual are vastly more complicated than the martingale. But they suffer the same fundamental flaw: They claim to create returns out of nothing, with no attendant risk. That's not just suspicious. In many cases, it's mathematically impossible.

To explain why, I need to introduce the mathematical notion that underlies every price computation in finance, from options to insurance to credit default swaps: expected value. Suppose somebody approaches you and says, "I propose a game of chance. I flip this coin, and if it comes up heads you get $100. If it comes up tails, you get nothing. How much will you pay me for the right to play this game?" In other words: What is the value of a 50 percent chance of winning $100?

If you played this game all day, you'd probably win about half of the time. Most people, then, would value the coin flip game at $50, which is just the probability of success (50 percent, or 0.5) times the value of a successful outcome ($100). In general, to compute the expected value of a game you need to add up the values of all possible outcomes multiplied by their respective probabilities. Consider, for instance, the riskier game where you win $100 if the coin lands heads but lose $100 otherwise.  Each of those outcomes happens 50 percent of the time; so the value of this game is

(0.5) x ($100) + (0.5) x (-$100) = $0

The equation just records the obvious fact that this game favors neither you nor your opponent. It's a wash.

What's the expected value of the martingale? Like the game above, it's no more than a bunch of coin flips, each one of which has a value of 0. So the whole game has a value of 0.