Bankstocks.com: Bankstocks Booklist: Sizing Your Bets, Made Easy

Bankstocks.com: Bankstocks Booklist: Sizing Your Bets, Made EasyBankstocks Booklist: Sizing Your Bets, Made Easy
Monday, December 05, 2005
Matt Stichnoth
bankstocks.com
mstichnoth@bankstocks.com

Solve the following problem. You’re at the track with $1,000 in your pocket, and see that the posted odds on a certain horse winning an upcoming race are 5 to 1. You (and only you) have a secret line of communication to the horse’s trainer, and learn that the horse’s chances of winning are meaningfully higher than the posted odds—say, 1 in 3. Which is to say, you have a material information advantage over other bettors. How much of your $1,000 do you bet?

That, in a nutshell, is one of the most crucial and least discussed dilemmas in the capital allocation process. While CAPM types preach about the virtues of diversification, the Warren Buffetts of the world know better. Diversification only assures mediocre returns, they point out; the real money is made when you put a lot of capital to work in those rare opportunities when you have a true edge. Like, say the 1-in-3 shot above that’s going off at 5-to-1.

William Poundstone gets at this issue in Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. The book is a history of a formula called the “Kelly Criterion” that allows gamblers (and other capital allocators) to maximize their profits on a series of bets where they have an information edge, but without betting so much that they risk going broke. Take the horse-racing example, above. Yes, you’ll want to bet more than you normally would, to make the most of your insider knowledge. But you don’t want to bet everything: even by your own reckoning, the horse has just a 33% chance of winning. Once you’re bankrupt, you can’t get back in the game. The optimal bet size is somewhere in between.

The namesake and inventor of the Kelly formula is a man named John Kelly, a mathematician at Bell Labs in the 1950s and 1960s. Kelly developed his formula by building on the work of another Bell Labs mathematician, Claude Shannon. Poundstone says Shannon is considered by many to be the second-most-brilliant individual of the twentieth century, after Einstein. In particular, Shannon is the father of “information theory,” which serves as the broad mathematical foundation for essentially the entire electronics and digital revolutions. Everything from integrated circuits to fiber-optic cable to DNA sequencers rely at rock-bottom on Shannon’s work. His models apply to any kind information conduit, electronic or otherwise. They allow communications engineers to minimize the amount of noise—static, gossip, whatever--in a given conduit, and maximize the amount of information the conduit can carry. Which is to say, Shannon essentially developed a mathematical way to convert uncertainty into certainty.

Communications engineers aren’t the only ones with an interest in separating information from noise, of course. Bettors and investors could use some help there, too. So it’s perhaps not coincidental that some of Shannon’s math can be put to use at the race track, the blackjack table, and on Wall Street. One of the first to apply Kelly’s formula was a young physics grad student, Edward Thorp, who used it in conjunction with a card counting system he developed for blackjack. (Thorp later wrote a book on card counting called Beat the Dealer that’s now considered a classic among blackjack aficionados. Later on he ran a hugely successful quant fund, Princeton-Newport Partners, that eventually got tangled up in Rudolph Giuliani’s pursuit of Michael Milken in the 1980s. But that’s another story.)

How does the Kelly formula work, you ask? It’s pretty simple. The formula says that the optimal wager size is determined according to the following fraction:

Edge/Odds

The denominator, odds, is the public odds posted on the track’s tote board. The numerator, edge, is the amount you stand to profit, on average, if you could make this same bet over and over and over. Let’s go back to the horse racing hypothetical in the first paragraph, and see how it works. The posted odds are 5 to 1. So we’ll put a 5 in the denominator. But recall that you believe the true odds are 1 in 3, not 5 to 1. If you bet $1,000, then, you’ll have a 33% chance of winning $6,000 ($5,000 plus your original $1,000 wager), or $2,000, on average. On a $1,000 bet, your profit is thus $1,000. That’s your edge. For the formula’s purposes, the $1,000 becomes a 1.

So according to Kelly, the edge is 1 and the odds are 5. Plug in the numbers and you get 1/5. You should bet 20% of your bankroll.

A few comments are in order. First off, this only works in instances when you have a true, material information advantage. If you don’t, your edge is zero, so you shouldn’t bet. Second, the only time the formula will tell you to bet all you’ve got is when you’re absolutely, positively sure you’ll win. In the real world, that hardly ever happens. Thus Kelly prevents bettors avoid being wiped out completely, so that they’ll have capital to put to work when the next opportunity rolls around. This is no small advantage. Other capital allocation strategies gamblers use, most notably “martingale,” in which the player doubles down after a losing bet in order to quickly recoup losses, a can be quick trips to bankruptcy. Finally, using Kelly on a series of bets is the most efficient way to compound your winnings. Models show that, say, a more aggressive “Kelly times 2” strategy actually leads to lower long-term returns.

The chart below, from the book, illustrates Kelly’s advantages. It shows the results of various strategies for betting on a series of hypothetical coin flips where the bettor has a 55% chance of winning.

It scarcely needs to be added, of course, that the economics profession has roughly zero use for all this. First off, the formula was developed by a mathematician, not an economist, which naturally makes economists skeptical. Second, the notion that an investor can have a true edge is anathema to the efficient-market dogma that still dominates most economics departments. Paul Samuelson is particularly scornful of Kelly (or “g,” as it’s referred to in economics circles), calling it a “fallacy.”

The Kelly criterion’s virtual absence in economics and M.B.A. curricula explains why the formula is not well known on Wall Street. It shouldn’t be. It is hard enough to find ideas where an information advantage is even possible. When those do occur, investors can use all the help they can get in figuring out how much capital to apply. Kelly may not be as ideally suited to Wall Street as it is to blackjack, but it sure seems like a good place to start.


What do you think? Let me know!



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