“The fundamental law of investing is the uncertainty of the future.” — Peter Bernstein
What do money management and data transmission over phone lines have in common? Uncertainty.
Problems associated with data transmission are very similar to issues a gambler or trader faces in determining the optimal amount of money to trade at any given time. Believe it or not, the concept of how much money you must trade is related to work of Bell engineers going back decades. In 1956 J. L. Kelly published a paper while working at Bell Labs. The paper sought to solve issues associated with noise over phone lines, noise that was random and very unpredictable.
We found an excerpt from Kelly, J. L., Jr., titled “A New Interpretation of Information Rate”, taken out of the July, 1956 edition of Bell System Technical Journal:
If a gambler places bets on the input symbol to a communication channel and bets his money in the same proportion each time a particular symbol is received, his capital will grow (or shrink) exponentially. If the odds are consistent with the probabilities of occurrence of the transmitted symbols (i.e., equal to their reciprocals), the maximum value of this exponential rate of growth will be equal to the rate of transmission of information. If the odds are not fair, i.e., not consistent with the transmitted symbol probabilities but consistent with some other set of probabilities, the maximum exponential rate of growth will be larger than it would have been with no channel by an amount equal to the rate of transmission of information.
The formula has come to be simply known as the Kelly formula and it was great inspiration to great trend traders and systems traders. They were inspired directly by Bell Labs research to develop systems centered around determining the optimal bet size, otherwise known as money management.
The connection from telephone noise to trading position sizing is not as strange as it first appears. Both problems involve making optimal decisions under uncertainty with incomplete information. Kelly’s insight was that the correct bet size is not the one that maximizes the single-period expected return, but the one that maximizes the long-run rate of capital growth. That is precisely the problem a trend follower faces on every trade. More on Kelly, Bell Labs, and risk management here.
Kelly Formula
A web post from one of our readers gave a great description:
What is the Kelly criterion (or formula)? It is a formula for calculating how much to bet. It assumes that your objective is long term capital growth (getting rich). The handicapper’s choice of money management strategy is similar to the stock market choice between growth stocks and income stocks. Growth stocks tend to be more volatile, but in the long term return more profit. That is because the profits from growth stocks are reinvested rather than skimmed off. Every reinvestment is a calculated risk. Therefore, income stocks tend to fluctuate in value less, but also return less profit in the long term. Kelly betting is for growth. It reinvests profits, and thus puts them at risk. If your objective is to make small but consistent profits, it may be too aggressive a money management scheme.
The Kelly formula is: Kelly % = W – (1-W)/R where:
- Kelly % = percentage of capital to be put into a single trade.
- W = Historical winning percentage of a trading system.
- R = Historical Average Win/Loss ratio.
The formula produces the position size that maximizes the long-term growth rate of capital given a system’s historical win rate and payoff ratio. Two inputs. One output. The simplicity is deceptive. The implications are profound. A system with a 40% win rate and an average win three times the size of the average loss produces a positive Kelly percentage, meaning the math says bet. A system with a 60% win rate but average wins smaller than average losses produces a negative Kelly percentage, meaning the math says do not bet at all, regardless of how often it wins. This is why win rate alone is meaningless as a measure of a system’s quality. The Kelly formula combines both inputs into a single number that tells you exactly how much to bet to grow capital as fast as possible over time, without risking ruin. For the connection between Kelly position sizing and the volatility-based sizing used in the TurtleTrader rules, see the rules page.
Edward O. Thorp: Beat the Dealer
Thorp is famous for his blackjack paperback, Beat the Dealer, where he explores Kelly for gambling:
The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, i.e. how much to bet. In the stock market (more inclusively, the securities markets) the problem is similar but more complex. The gambler, who is now an investor, looks for excess risk adjusted return. In both these settings, we explore the use of the Kelly criterion, which is to maximize the expected value of the logarithm of wealth (maximize expected logarithmic utility).
Thorp continues:
The criterion is known to economists and financial theorists by names such as the geometric mean maximizing portfolio strategy, maximizing logarithmic utility, the growth-optimal strategy, the capital growth criterion, etc.
Thorp’s path from blackjack card counting to financial markets represents one of the cleanest intellectual translations in trading history. The problem is identical: you have a positive-expectation process, you know the win rate and the payoff ratio, and you need to determine the correct bet size to maximize long-term capital growth without risking ruin. Kelly’s formula solves this in both settings. The connection to blackjack and trend following runs deeper than analogy. Both are probability management problems in disguise, and Kelly provides the mathematical bridge between them.
The broader lesson is that money management is not a secondary concern in trading. It is the primary one. Finding a positive-expectation system is the first problem. Determining the optimal bet size to maximize the growth of that edge over time is the second problem, and Kelly showed it is at least as important as the first. A trader with a slight edge who bets too much will go broke. A trader with a strong edge who bets correctly will compound it into significant wealth. This is the mathematical foundation of risk management in trend following, and it traces directly back to a Bell Labs engineer solving a problem about telephone noise in 1956.
Frequently Asked Questions
What is the Kelly formula and where does it come from?
The Kelly formula is a mathematical criterion for determining the optimal fraction of capital to bet in a positive-expectation game or trading system. It was published by J. L. Kelly Jr. in 1956 while working at Bell Labs, originally as a solution to problems of noise and information transmission over telephone lines. Its application to gambling and investing was recognized quickly, and it became the foundation of optimal position sizing theory.
How do you calculate the Kelly percentage for a trading system?
Kelly % = W minus (1-W) divided by R, where W is the historical winning percentage of the system and R is the historical average win to loss ratio. The result is the fraction of capital that should be risked on each trade to maximize long-term capital growth. Most practitioners use a fraction of the full Kelly amount to reduce volatility while preserving most of the growth advantage.
Why is win rate alone insufficient to evaluate a trading system?
Because profitability depends on both the win rate and the average win to loss ratio. A system that wins 40% of the time but makes three times as much on winners as it loses on losers has a positive Kelly percentage and is highly profitable. A system that wins 70% of the time but loses more on average than it wins has a negative Kelly percentage and will lose money over time. The Kelly formula correctly combines both inputs.
What is the connection between Bell Labs and trend following?
Kelly’s insight that the correct bet size is the one maximizing long-run capital growth rather than single-period expected return directly inspired trend traders and systems developers to build position sizing into their systems. The volatility-based position sizing used by the TurtleTraders and other systematic trend followers is a practical implementation of the same principle: size each position to reflect the current risk level and the long-run growth objective, not the maximum possible single-trade return.
What did Edward Thorp contribute to trading through his work on Kelly?
Thorp showed that the Kelly criterion applied equally to blackjack card counting and securities markets. The central problem in both is finding a positive-expectation process and then managing money correctly to exploit it. His work in Beat the Dealer and subsequent financial research demonstrated that the mathematical framework for optimal betting in a casino is the same framework for optimal position sizing in a trading system, connecting two fields that appeared unrelated.
Trend Following Systems
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