In Bill Ackman's Q3 letter to investors, he laid out his approach to investing in high-risk, high-reward scenarios:

"We occasionally are willing to invest a small amount of fund capital in situations which offer the potential for a many-fold profit at the risk of a large or near-total loss of capital invested. I typically call these investments mispriced options."

Not long ago, while discussing credit default swaps in an interview, Ackman explained how an investment in these instruments presented such an opportunity before the credit crisis hit, where 2% of your portfolio could return "ten times — *the portfolio.*" Ackman went on to say that John Paulson "made a fortune this way," with his historic bet against subprime MBS and other weak credits.

What Ackman is really saying is to make investments where you have a high expected value. This means analyzing the size of potential gains and losses, adjusted for the respective probabilities that such gains or losses will be realized. Here is how to define it, in the language of elementary statistics:

In words, it is the sum of all possible gains and losses multiplied by their relative probabilities. This type of reasoning is commonly used by skilled gamblers, and you can look over some basic examples on this gambling webpage

This is the essence of what Ackman is talking about with "mispriced options"; a small position that will either expire worthless or be worth a fortune.

One current example might be General Growth Properties (NYSE: GGP), a beleaguered REIT facing bankruptcy. But whatever the case is with General Growth, it seems that another Ackman holding, Borders Group (NYSE: BGP), might be a safer bet.

This month,

And while the probabilities inherent in a potential General Growth bankruptcy filing are anyone's guess,

Borders' current share price is roughly 53 cents. This is down from over $20 before the credit crisis began. Given that Ackman started buying at the latter price, we might assume that he believes the company is worth much more.

Indeed, going by what Ackman said at the time, in late 2006, the shares might be worth about $36 each. But since then, we must also assume that value has been destroyed, from the economic malaise as well as a dilutive emergency financing, which granted Ackman warrants to purchase additional shares at $7 each.

Let us assume that Borders' intrinsic value is $10.60 for our example, or twenty times the current market price. This is less than a third of Ackman's original estimate, but a price that puts his warrants in the money. Thus, we have a possible gain of $10.07 and a possible loss of $0.53, on a per share basis. But what is the probability of such a gain or loss?

By Ackman's definition of a "mispriced option," as set forth above, Borders fits the bill. This is an opportunity that could provide a tremendously outsize return, if one is willing to accept the chance of a total loss of capital invested.

Let me be clear: this is an illustration, *not* a recommendation. Borders' stock does not trade for 53 cents because it is a risk-free proposition. The company faces an onerous debt burden and, depending what happens with the economy, still might not make it through these hard times.

Moreover, such a situation is clearly inappropriate for investment of more than even a few percent of one's portfolio. This is *not* a situation analogous to Warren Buffett's famous 1964 investment in American Express, where he bet roughly 40% of his partnership's capital on the stock. In that case, Buffett had high confidence that his chance of loss was near zero, with a company that is immensely stronger than Borders is.

This illustrates a current opportunity cost that investors must weigh: Should you accept a relatively lower gain, although still handsome by most standards, with near certainty? Or should you shoot for truly eye-popping gains, but with a risk of permanent loss?

Moreover, how do you gauge your confidence level and accuracy in assessing such risks? Should you attempt to create a portfolio across the risk spectrum, with a small amount of capital in Borders-type stocks and a larger amount in American Express-type stocks?

These are questions that only you can answer.