THORP ON OPTIMAL BET SIZE AND THE KELLY CRITERION
The Market Wizards series is a collection of books written by Jack Schwager that captures the philosophies, traits, experiences, and advice of great traders, seeking to draw out lessons that could help all traders from novices to professionals. The following is an excerpt from Jack about Ed Thorp. A new never before seen video recalling this interview will be hosted to FundSeeder later this week.
In Hedge Fund Market Wizards, the fourth book of the series, Schwager interviewed Edward Thorp, a math professor turned hedge fund manager whose track record must certainly stand as one of the best of all time. His original fund, Princeton Newport Partners, achieved an annualized gross return of 19.1 percent (15.1 percent after fees) over a 19-year period. Even more impressive was the extraordinary consistency of return: 227 out of 230 winning months and a worst monthly loss under 1 percent. A second fund, Ridgeline Partners, averaged 21 percent annually over a 10-year period with only a 7 percent annualized volatility.
Before he ever became interested in markets, Edward Thorp’s avocation was devising methods to win at various casino games—an endeavor widely assumed to be impossible. After all, how could anyone possible devise a winning strategy for games in which the player had a negative edge? One might think that a math professor would be the last person to devote time to such a seemingly futile goal. Thorp, however, approached the problem in a completely unconventional manner. For example, in roulette, Thorpe, along with Claude Shannon (known as “the father of information theory”), created a miniature computer that used Newtonian physics to predict the octant of the wheel in which the ball was most likely to land. In blackjack, Thorp’s insight was that by betting more on high-probability hands than on low-probability hands, it was possible to transform a game with a negative edge into a game with a positive edge. His book, Beat the Dealer, was a best seller and changed the way casinos operate.
Thorp’s insight about the importance of bet size has important ramifications for trading as well: varying position size could improve performance. By analogy to blackjack, trading larger for higher-probability trades and smaller, or not at all, for lower-probability trades could even transform a losing strategy into a winning one. Although in trading, probabilities cannot be accurately defined, as they are in blackjack, traders can often still differentiate between higher and lower probability trades. For example, if a trader does better on high-confidence trades, then the degree of confidence can serve as a proxy for the probability of winning. The implication then is that instead of risking an equal amount on each trade, more risk should be allocated to higher-confidence trades and less to lower-confidence trades.
In the excerpt from Hedge Fund Market Wizards below, Schwager and Thorp discuss optimal bet size and the Kelly Criterion.
For purposes of background, the Kelly Criterion is the fraction of capital to wager to maximize compounded growth of capital. Even when there is an edge, beyond some threshold, larger bets will result in lower compounded return because of the adverse impact of volatility. The Kelly Criterion defines this threshold. The Kelly Criterion indicates that the fraction that should be wagered to maximize compounded return over the long run equals:
F = PW – (PL/W)
F = Kelly Criterion fraction of capital to bet
W = Dollars won per dollar wagered (i.e., win size divided by loss size)
PW = Probability of winning
PL = Probability of losing
When win size and loss size are equal, the formula reduces to:
F = PW – PL
For example, if a trader loses $1,000 on losing trades and gains $1,000 on winning trades, and 60 percent of all trades are winning trades, the Kelly Criterion indicates an optimal trade size equal to 20 percent (0.60 – 0.40 = 0.20).
As another example, if a trader wins $2,000 on winning trades and loses $1,000 on losing trades, and the probability of winning and losing are both equal to 50 percent, the Kelly Criterion indicates an optimal trade size equal to 25 percent of capital: 0.50 – (0.50/2) = 0.25).
Proportional overbetting is more harmful than underbetting. For example, betting half the Kelly Criterion will reduce compounded return by 25 percent, while betting double the Kelly Criterion will eliminate 100 percent of the gain. Betting more than double the Kelly Criterion will result in an expected negative compounded return, regardless of the edge on any individual bet.
The Kelly Criterion implicitly assumes that there is no minimum bet size. This assumption prevents the possibility of total loss. If there is a minimum trade size, as is the case in most practical investment and trading situations, then ruin is possible if the amount falls below the minimum possible bet size.
How important was determining the optimal bet size in your trading success? How and why did you decide to use the Kelly Criterion as the method for determining bet size?
I learned about the Kelly Criterion from Claude Shannon back at MIT. Shannon had worked with Kelly at Bell Labs. I guess Shannon was the leading light at Bell Labs and Kelly was perhaps the second most significant scientist there. When Kelly wrote his paper in 1956, Shannon refereed it. When I told Shannon about my blackjack betting system, he told me to look at Kelly’s paper in deciding how much to bet because in favorable situations, you will want to bet more than in unfavorable situations. I read the Kelly paper, and it made a lot of sense to me.
The Kelly Criterion of what fraction of your capital to bet seemed like the best strategy over the long run. When I say long run, a week playing blackjack in Vegas might not sound very long. But long run refers to the number of bets that are placed, and I would be placing thousands of bets in a week. I would get to the long run pretty fast in a casino. In the stock market, it’s not the same thing. A year of placing trades in the stock market will not be a long run. But there are situations in the stock market where you get to the long run pretty fast—for example, statistical arbitrage. In statistical arbitrage, you would place tens or hundreds of thousands of trades in a year. The Kelly Criterion is the bet size that will produce the greatest expected growth rate in the long term. If you can calculate the probability of winning on each bet or trade and the ratio of the average win to average loss, then the Kelly Criterion will give you the optimal fraction to bet so that your long-term growth rate is maximized.
The Kelly Criterion will give you a long-term growth trend. The percentage deviations around that trend will decline as the number of bets increases. It’s like the law of large numbers. For example, if you flip a coin 10 times, the deviation from the expected value of five will by definition be small—it can’t be more than five—but in percentage terms, the deviations can be huge. If you flip a coin 1 million times, the deviation in absolute terms will be much larger, but in percentage terms, it will be very small. The same thing happens with the Kelly Criterion: in percentage terms, the results tend to converge on the long-term growth trend. If you use any other criterion to determine bet size, the long-term growth rate will be smaller than for the Kelly Criterion. For betting in casinos, I choose the Kelly Criterion because I wanted the highest long-term growth rate. There are, however, safer paths that have smaller drawdowns and a lower probability of ruin.
I understand that if you know your edge and it is precisely defined—which of course is not true in the markets—then the Kelly Criterion is the amount you should bet to maximize the compounded return and that betting either a smaller or larger fraction will give you a smaller return. But what I don’t understand is that the Kelly Criterion seems to give all the weight to the return side. The only way the Kelly Criterion reflects volatility is through its impact on return. Besides the fact that people are uncomfortable with high volatility, there is the very practical consideration that your down-and-out point is not zero as the Kelly Criterion implicitly assumes, but rather your maximum tolerable drawdown. It seems to me that the criterion should be what maximizes growth subject to the constraint of minimizing the risk of reaching your cutout point.
Suppose you have a bankroll of $1 million and your maximum tolerable drawdown is $200,000. Then from the Kelly Criterion perspective, you don’t have $1 million in capital, you have $200,000.
So, in your example, you still apply the Kelly Criterion, but you apply it to $200,000. When you played blackjack, did you apply the Kelly Criterion straightforward?
Yes, assuming I was sure the dealer was not cheating, because my objective was to make the most money, in the least time.
What about when you managed the fund?
When I managed the fund, I wasn’t forced to make a Kelly Criterion decision. If you use hedges to theoretically neutralize your risk, then the Kelly Criterion might well imply using leverage. In Princeton Newport Partners where all positions were hedged, I found that I couldn’t leverage up my portfolio as much as the Kelly Criterion said I should.
Because the brokerage firms wouldn’t give me that much borrowing power.
Does that imply that you would have traded the Kelly Criterion if it was feasible in a practical sense?
I probably wouldn’t have because if you bet half the Kelly amount, you get about three-quarters of the return with half the volatility. So it is much more comfortable to trade. I believe that betting half Kelly is psychologically much better.
I think there is a more core reason why betting less than the Kelly amount would always be the rational decision in the case of trading. There is an important distinction between trading and playing a game such as blackjack. In blackjack, theoretically, you can know the precise probabilities, but in trading, the probability of winning is always an estimate—and often a very rough one. Moreover, the amount of extra gain forgone by betting less than the Kelly Criterion is much smaller than the amount that would be lost by betting more than the Kelly Criterion by the same percentage. Given the uncertainty of the probability of winning in trading combined with the inherent asymmetry in returns around the Kelly fraction, it would seem that the rational choice is to always bet less than the Kelly Criterion, even if you can handle the volatility. In addition, there is the argument that for virtually any investor, the marginal utility of an extra gain is smaller than the marginal utility of an equal percentage loss.
That’s true. Say I am playing casino blackjack, and I know what the odds are. Do I bet full Kelly? Probably not quite. Why? Because sometimes the dealer will cheat me. So the probabilities are a little different from what I calculated because there may be something else going on in the game that is outside my calculations. Now go to Wall Street. We are not able to calculate exact probabilities in the first place. In addition, there are things that are going on that are not part of one’s knowledge at the time that affect the probabilities. So you need to scale back to a certain extent because overbetting is really punishing—you get both a lower growth rate and much higher variability. Therefore, something like half Kelly is probably a prudent starting point. Then you might increase from there if you are more certain about the probabilities and decrease if you are less sure about the probabilities.
The Undiscovered Market Wizards Search
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