The Kelly criterion offers a mathematical formula to determine the optimal stake size for bets, aiming to maximize the long-term expected growth rate of your capital or bankroll. Developed by John Kelly Jr., bettors, and investors use this Kelly strategy to manage risk and optimize returns over time.
What is the Kelly Criterion?
At its core, the Kelly criterion provides a disciplined approach to bet sizing. John Larry Kelly Jr., working at Bell Labs, first described the formula in 1956. His work focused on maximizing the long-term expected value of the logarithm of wealth. Maximizing this logarithm is mathematically equivalent to maximizing the long-term expected geometric growth rate of capital.
People primarily use the formula in gambling contexts, but its principles extend to investment management, particularly in explaining diversification strategies. Some analyses suggest prominent investors like Warren Buffett employ methods similar to the Kelly bet. According to Arabcasino.vip, the fundamental idea is to bet a specific percentage of your capital on each opportunity, where the percentage depends on the perceived edge or advantage.
The Kelly Criterion Formula
For a simple bet with two outcomes (win or lose), you can calculate the optimal fraction of your bankroll to stake using a basic formula. The formula often appears as follows:
Kelly % = (BP – Q) / B
- Kelly % = The suggested percentage of your bankroll to bet.
- B = The decimal odds received for the bet minus 1 (e.g., if odds are 3.0, B is 2.0).
- P = The probability of winning the bet.
- Q = The probability of losing the bet (calculated as 1 – P).
Example:
Imagine a biased dice roll where landing on 1, 2, or 3 has a 60% probability (P = 0.60), meaning landing on 4, 5, or 6 has a 40% probability (Q = 0.40). Assume you get even money odds (2.0 decimal odds), so B = 1.0.
Applying the formula:
Kelly % = (1.0 * 0.60 – 0.40) / 1.0 = (0.60 – 0.40) / 1.0 = 0.20
The Kelly criterion suggests staking 20% of your current bankroll on the low numbers appearing. If the edge were smaller, say a 53% win probability, the formula would recommend staking only 6%. Calculating the correct percentage helps manage your funds effectively based on the advantage you believe you possess.
Also Read: Two-Odds Betting Techniques
Why Use the Kelly Criterion in Betting?
Bettors face the constant challenge of deciding how much to stake. Staking too little on favorable opportunities means leaving potential profit unrealized. Staking too much increases the risk of ruin, potentially wiping out your bankroll quickly.
An issue arises particularly in value betting, where you identify overpriced selections and bet without laying them off on an exchange. Here, the amount you stake directly impacts both potential gains and losses.
The Kelly criterion addresses this directly. It optimizes stake size to achieve the maximum long-term expected growth rate. Instead of relying on gut feelings or fixed stakes, it provides a mathematical basis for bet sizing, considering your bankroll, the odds, and your assessed win probability. Adhering strictly to the Kelly calculation maximizes your capital’s growth rate over many bets, a key objective for serious bettors.
Practical Application and Challenges
While powerful in theory, applying the Kelly criterion effectively presents challenges.
Estimating Probabilities: The formula’s greatest weakness lies in its reliance on accurate probability estimates. Your calculation for the optimal stake is only as reliable as your assessment of the likelihood of winning. Overestimating your edge leads to over-betting, increasing risk, while underestimation results in under-betting and slower growth. In real-world scenarios like sports betting or stock market investment, determining true probabilities is difficult and often involves significant uncertainty.
Volatility and Risk: The stake calculated by the full Kelly criterion often represents an aggressive approach. While mathematically optimal for maximizing the growth rate, it can lead to substantial fluctuations and sharp downswings in your bankroll. A run of bad luck, even when making +EV (positive expected value) bets, can significantly deplete capital if using full Kelly stakes. Importantly, betting more than the Kelly fraction is counterproductive; it increases risk while decreasing the expected growth rate.
Fractional Kelly Staking
Given the aggressive nature and volatility associated with full Kelly stakes, many bettors and investment managers adopt a more conservative approach known as Fractional Kelly. Instead of betting the full percentage suggested by the formula, you bet a fixed fraction of that amount.
Common fractions include 50% (half-Kelly), 25%, or even 10%. For instance, if the Kelly formula suggests a 20% stake, a half-Kelly approach means betting 10% of your bankroll, while a quarter-Kelly approach involves betting 5%.
Outplayed’s software, for example, defaults to a 0.25 fraction (a quarter of the optimal Kelly stake). Using Fractional Kelly results in a lower theoretical maximum growth rate compared to the full Kelly stake.
However, the significant advantage is substantially reduced risk and much smoother bankroll performance, mitigating the severity of downswings. Bettors achieve a large portion of the potential growth with a much smaller, more sensible, and balanced stake amount, greatly lessening the chance of catastrophic bankroll loss.
Many find this trade-off between slightly slower growth and increased capital preservation favorable. A prudent approach often suggests using a fraction, like half-Kelly, especially when dealing with uncertain estimates like stock market returns.
Kelly Criterion for Multiple Outcomes
The Kelly criterion is not limited to simple win/lose bets. It can be generalized to handle situations with multiple potential outcomes, such as betting on a horse race where various bet types exist for the same event (non-mutually exclusive bets). Applying Kelly to multiple outcomes involves a more complex algorithm:
- Calculate the expected revenue rate for each potential outcome you are considering betting on.
- Compare these expected rates to a calculated ‘reserve rate’.
- Identify the optimal set of outcomes where the expected revenue rate exceeds the reserve rate. Betting only occurs on outcomes within this set.
- Determine the optimal fraction of your budget to allocate to each bet within the optimal set using a specific formula. An outcome is included only if its expected revenue rate is greater than the reserve rate.
If no outcome meets the criteria (the optimal set is empty), the strategy advises placing no bets at all. The method helps bettors systematically decide which horses (or outcomes) offer value and how much capital to allocate across those bets.
Kelly Criterion in Investment and Finance
The principles of the Kelly criterion extend beyond gambling into economic decision-making and wealth management, particularly investment and portfolio theory. In finance, a portfolio whose security weights maximize the expected geometric growth rate is termed growth optimal.
Formula adaptations exist for applying Kelly to investment scenarios, considering factors like expected returns, volatility (variance), and the risk-free rate. However, using it in financial markets involves significant “garbage in, garbage out” risk.
Expected returns and covariance structures are estimates, often subject to considerable uncertainty and estimation errors. If portfolio weights heavily depend on these potentially inaccurate estimates, the actual performance can deviate wildly from predictions.
Parameter uncertainty is a major concern, often leading practitioners to use Fractional Kelly (like half-Kelly) as a more prudent approach, reducing exposure compared to naive estimates derived from potentially flawed inputs.
Final Words about Kelly Criterion in Betting
The Kelly criterion provides a valuable mathematical framework for optimizing bet sizing to maximize long-term capital growth. Whether in sports betting or financial investment, it offers a disciplined alternative to arbitrary staking by linking bet size to perceived advantage and bankroll.
Success hinges on accurately assessing probabilities, a significant practical challenge. Due to the inherent volatility of full Kelly stakes, Fractional Kelly strategies are widely adopted, balancing growth potential with crucial risk management and capital preservation.
