Kelly Criterion

The Kelly Criterion is a mathematical formula used to determine the optimal size of a bet in order to maximize the long-term growth of capital while minimizing the risk of ruin. Developed by mathematician John L. Kelly Jr. in the 1950s, the Kelly Criterion is widely used in various fields, including gambling, investing, and finance. Here’s how the Kelly Criterion works:

1. Basic Principle:
   • The Kelly Criterion provides a method for determining the proportion of a bankroll that should be wagered on a particular bet or investment opportunity.
   • The goal of the Kelly Criterion is to find the optimal bet size that maximizes the expected logarithm of wealth over time, balancing the trade-off between risk and return.
2. Calculation:
   • The Kelly Criterion formula is given by:�∗=(��−�)�f∗=b(bp−q)​Where:
     • �∗f∗ is the fraction of the bankroll to bet,
     • �b is the odds received on the bet (expressed in decimal odds, where 1.0 represents even money),
     • �p is the probability of winning the bet,
     • �q is the probability of losing the bet (1 – �p).
3. Interpretation:
   • The result of the Kelly Criterion formula, �∗f∗, represents the proportion of the bankroll that should be wagered on the bet.
   • If �∗f∗ is positive, it indicates that the bet offers positive expected value, and the optimal strategy is to bet a fraction of the bankroll proportional to the size of the edge.
   • If �∗f∗ is negative, it suggests that the bet has negative expected value, and the optimal strategy is to abstain from betting.
4. Example:
   • Suppose a bettor has determined that a particular bet has odds of 2.5 (implied probability of winning = 1/2.5 = 0.4) and a probability of winning of 0.6 (p = 0.6) based on their analysis.
   • Using the Kelly Criterion formula, the optimal fraction of the bankroll to bet would be:�∗=(2.5×0.6−0.4)2.5=(1.5−0.4)2.5=1.12.5=0.44f∗=2.5(2.5×0.6−0.4)​=2.5(1.5−0.4)​=2.51.1​=0.44
   • This means that the bettor should wager approximately 44% of their bankroll on this particular bet.
5. Risk Management:
   • While the Kelly Criterion can help optimize bet sizes, it’s essential to consider the potential risks and uncertainties associated with any betting or investment decision.
   • The Kelly Criterion does not take into account factors such as variability of returns, transaction costs, and changing market conditions, so it should be used in conjunction with other risk management techniques.
6. Practical Considerations:
   • In practice, bettors often use a fractional Kelly strategy by betting a fraction (e.g., half or one-quarter) of the optimal bet size calculated by the Kelly Criterion to reduce the impact of variability and mitigate the risk of large losses.

Overall, the Kelly Criterion provides a valuable framework for determining optimal bet sizes based on probabilities and expected returns, but it’s important to exercise caution and consider other factors when making betting or investment decisions.