Tell me more detail about Kelly formula with real world examples

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The Kelly Formula, also known as the Kelly Criterion, is a mathematical formula used to determine the optimal size of a series of bets or investments. Developed by John Kelly in 1956, this formula helps maximize long-term growth while minimizing the risk of ruin. The basic formula is f-star equals b-p minus q divided by b, where f-star represents the fraction of capital to bet, b is the odds received, p is the probability of winning, and q is the probability of losing. The mathematical foundation of the Kelly formula lies in maximizing the expected logarithm of wealth. This approach ensures geometric growth rather than arithmetic growth. The formula components work together: b represents the net odds or profit per unit stake, p is the probability of winning, and q is simply one minus p. The logarithmic utility function creates a curve that peaks at the optimal betting fraction, preventing both under-betting and dangerous over-betting. This mathematical framework is superior to fixed betting amounts because it automatically adjusts bet size based on the edge and bankroll size. Let's work through a concrete example using a biased coin flip. We have a coin that lands heads 60% of the time, with even money payouts of one to one odds. Using the Kelly formula: f-star equals b-p minus q divided by b. Substituting our values: p equals 0.6, q equals 0.4, and b equals 1. This gives us f-star equals 1 times 0.6 minus 0.4, divided by 1, which equals 0.2 or 20% of our bankroll. The chart shows how Kelly betting at 20% significantly outperforms fixed betting amounts over multiple coin flips, demonstrating the power of optimal position sizing. When applying Kelly to stock market investments, we use a modified formula for continuous returns. Consider a stock with 15% expected annual return and 25% volatility. With a 5% risk-free rate, the excess return is 10%. The continuous Kelly formula is f-star equals mu divided by sigma-squared. Substituting our values: mu equals 10% and sigma-squared equals 0.0625, giving us f-star equals 1.6 or 160% allocation. This seems impossibly high because it assumes unlimited leverage, zero transaction costs, and perfect parameter estimates. In practice, most professionals use fractional Kelly betting, typically 25% to 50% of the full Kelly recommendation, to account for real-world constraints and parameter uncertainty. Professional blackjack card counters provide an excellent real-world example of Kelly betting. When the count is favorable, a skilled counter might have a 2% edge over the house. With a 51% win probability and even money payouts, the Kelly calculation gives us: f-star equals 1 times 0.51 minus 0.49, divided by 1, which equals 0.02 or 2% of bankroll. The table shows how bet sizes adjust with different count values - higher counts mean bigger edges and larger Kelly bets. Card counters must constantly recalculate their optimal bet size as the deck composition changes, making this a dynamic application of the Kelly criterion that requires both mathematical precision and disciplined bankroll management.