Never Go Broke: The Kelly Formula Explained
Never Go Broke: The Kelly Formula Explained
By:Kai Zeng
Science beats gut feelings when it comes to determining how much to bet
- The Kelly criterion provides a mathematical framework for position sizing that balances growth potential with capital preservation.
- Fractional Kelly approaches help accommodate real-world uncertainties and individual tolerance for risk.
- Kelly applications are particularly well-suited for options strategies because probability and payoff can both be estimated before execution.
- Consistently applying appropriate Kelly percentages to uncorrelated strategies improves long-term returns while reducing drawdown risk.
Imagine three gamblers, each with $10,000, playing a game with a 60% chance of winning and even-money payouts. After 100 bets:
- Gambler A bets 50% of the bankroll each time.
- Gambler B bets 20% of the bankroll each time.
- Gambler C bets 10% of the bankroll each time.
Which strategy leads to the highest ending bankroll? The answer is B with the 20% allocation.
This optimal betting size wasn't discovered through trial and error but through the Kelly criterion, a mathematical formula developed by John Kelly at Bell Labs in 1956. The formula determines exactly how much to invest when probability favors you, balancing aggressive betting with protection against ruin.
The Kelly formula is expressed as:
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Where:
- f* = optimal bankroll percentage to allocate
- p = winning probability
- q = losing probability (1-p)
- b = net odds received (amount won per dollar bet)
In our scenario: p = 0.60, q = 0.40, b = 1
Plugging these values into the formula: f* = (0.60 × 1 - 0.40) ÷ 1 = 0.2 or 20%
This approach applies perfectly to options strategies, where both probability and odds are known before trade entry. Let's examine iron condors as an example.
With a $100,000 portfolio trading only iron condors with these parameters:
- p: 85% success rate
- q: 15% losing rate
- b: 0.2 ($3 premium over $1,500 buying power reduction)
The Kelly formula gives us: f* = (0.85 × 0.2 - 0.15) ÷ 0.2 = 0.1 or 10%
The theoretical optimal capital allocation is 10%. Monte Carlo simulations show this portfolio should generate approximately 25% profit after 100 trades with zero chance of ruin.
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However, profitability varies based on success rate and odds. If we maintain the 85% success rate but increase the odds to 0.25, the optimal Kelly percentage rises to 25%. Conversely, if odds decrease to 0.15, Kelly produces a negative number—signaling the portfolio will eventually lose money.
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Most professionals don't use the full Kelly percentage but instead use a fraction of it—typically between 25-75% of the calculated value. This conservative approach accommodates real-world uncertainties and personal risk tolerance. Over or under-allocating relative to the optimal Kelly threshold significantly impacts both performance and risk profile.
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Kai Zeng, director of the research team and head of Chinese content at tastylive, has 20 years of experience in markets and derivatives trading. He cohosts several live shows, including From Theory to Practice and Building Blocks. @kai_zeng1
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