Money and Finance

Ed Thorp, Jack Schwager, and the Kelly criterion

Excerpt from Jack Schwager’s interview with Ed Thorp in the book Hedge Fund Market Wizards:

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 = P_W – (P_L/W)

where

F = Kelly criterion fraction of capital to bet

W = Dollars won per dollar wagered (i.e., win size divided by loss size)

P_W = Probability of winning

P_L = Probability of losing

When win size and loss size are equal, the formula reduces to:

F = P_W – P_L

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.

[Thorp]: 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 chose 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.

…

…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.

[Schwager]: Say I am playing casino blackjack, and I know what the odds are. Do I bet full Kelly?

[Thorp]: 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.

[Schwager]: In practice, did you end up gravitating to half Kelly?

[Thorp]: I was never forced to make that decision because there were so many trade opportunities that I usually couldn’t put on more than a moderate fraction of Kelly on any single trade. Once in a while, there would be an exceptional situation, and I would hit it pretty hard.

…

…there are no zero-risk trades.

[Schwager]: Do you want to expound?

[Thorp]: There was some remote possibility that we overlooked something. There is always the possibility that there is some unknown factor.

………………………

Joe’s comments:

A couple of important things to remember about the Kelly criterion are that it is really only useful when a large enough number of bets can be made (i.e. you need repeatability) and over-betting will eventually lead to ruin. It is hard to apply to investing because you almost never know the exact odds or the exact payoffs. But I do believe it can be generalized to investing.

If you can do enough work so that you have extremely high confidence that P_W is greater than P_L and where the amount you make if you are right is greater than the amount you lose if you are wrong, then the Kelly criterion will say it is a bet worth taking. Yes, there are situations where the payoffs can be high enough where the Kelly criterion would say to make the bet even if P_Wis less than P_L, but I think you can build in an extra margin of safety by only focusing on situations where your diligence leads you to believe that P_W > P_L AND$W > $L.

Then you next need to decide what size to make the position. You want to have a big enough position size to make a difference, but you have to make sure not to make your position sizes too big, which I think may happen as a result of having too much confidence when you put in a lot of work to understand something, or from not doing enough work and thus not understanding the odds and payoffs well enough....so either from overconfidence or lack of effort.

And as all of these decisions still depend on judgment and putting in the work to able to make good judgments, I think this Charlie Munger quote might be most fitting: "It's not supposed to be easy. Anyone who finds it easy is stupid."

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Related link: Understanding Fortune's Formula - by Edward O. Thorp (Column 21)

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