The Kelly Formula and the Financial Crisis

The Kelly Formula (also known as the Kelly System or Kelly Criterion) is a proportional betting system that optimizes a gambler’s bankroll.  In other words, it is a money management system that enables the maximum rate of compound wealth concurrent with zero chance of ruin.

Brought to life by William Poundstone in Fortune’s Formula, the Kelly Formula is used by some of the most successful investment managers in the world.  The system also sheds some interesting light on the financial crisis.

The Kelly System boils-down to the following: Edge / Odds.

Odds are set by market forces (non private information) and public beliefs: think of the horse track or stock market.

Edge basically boils down to private information about the outcome of an event, and it is the amount one would expect to win, on average, assuming that the same bet could be placed over and over at the same probabilities.

Consider the example of a horse track that has posted odds saying the horse Blue Bell has a 1 in 8 chance of winning. 

Now assume that Bob the Arb has private information that leads him to the conclusion that Blue Bell actually has a 1 in 4 chance of winning the race.  Bob can wager $100 and have a 1 in 4 chance of winning $800.  The expected value of this wager is $200, which would lead to a net profit of $100 based on his $100 wager.

In the example above, Bob the Arb’s Edge is the $100 profit / the $100 wager, or 1.  The Kelly Formula in this case is 1 (Edge) / 8 (Odds), or 12.5 percent.  As a result, Bob the Arb should wager 12.5 percent of his bankroll on Blue Bell and feel confident (assuming he has full confidence in the accuracy of his private information) that he has zero chance of ruin.

So what, exactly, does the Kelly Formula tell us about the financial crisis?

Well, one critical consideration is that the Kelly System tells us that in a fixed race, Edge is equal to Odds and one can therefore wager/invest one’s entire pool of capital with little or no concern about ruin (depleting one’s bankroll).

Now imagine Bob the Arb running the risk arbitrage desk at Goldman Sachs in the mid 1970s. His entire day consists of trying to find an Edge that will enable him to put the Goldman partners’ capital to work in ways that result in positive expected value outcomes related to, say, merger and acquisition activity.  Bob needs to be very careful about determining his Edge because it is the partners’ capital that serves as his bankroll.  Bob would likely never work on Wall Street again if he simply punted and had an unfortunate “event” with the partners’ money.

Skip ahead twenty or so years.  Goldman is now a public company rather than a private partnership and is operating with an implicit guarantee or preferential treatment from the federal government (think “systemic risk,” “too important to fail,” etc). 

Bob the Arb’s world now looks very different.  In fact, one might argue that the combination of implicit guarantees and other people’s money (shareholders, depositors, taxpayers) result in a race that is fixed.

In other words, Edge is approaching Odds so it makes perfect sense for Bob the Arb to bet the house.  The irony is that the source of Bob’s private information or edge the combination of implicit guarantees and other people’s money that ensure Bob’s bankroll will never go to zero.  The taxpayer is primary source of private information and the sucker at the table.