Kelly Criterion

The Kelly criterion determines the optimal fraction of your bankroll to bet: $$f^* = \frac{bp - q}{b}$$ where $b$ is the odds received (net profit per dollar bet if you win), $p$ is the probability of winning, and $q = 1-p$. Derivation: Maximize the expected log-growth rate $E[\ln(1 + f \cdot \text{outcome})]$. The logarithm makes this equivalent to maximizing the geometric growth rate. Properties: - $f^* = 0$ when edge is zero ($bp = q$) - $f^* < 0$ means don't bet (negative edge) - Overbetting ($f > 2f^*$) loses money in expectation despite positive edge - Half-Kelly ($f = f^*/2$) sacrifices ~25% of growth for ~50% less variance When to use: Sizing bets or positions when you have an edge. Central to trading risk management. Interview question: "You have a coin that's 60% heads. You start with \$100. How much should you bet each flip?" Answer: $f^* = \frac{1 \cdot 0.6 - 0.4}{1} = 0.2$, so bet 20% of current bankroll each round.