The Kelly Criterion, developed by John Kelly in 1956, is a powerful mathematical formula used in gambling, investing, and portfolio management. It determines the optimal fraction of your capital to bet or invest to maximize long-term wealth growth. The formula considers both the probability of winning and the potential payoff to find the sweet spot that balances risk and reward.
The Kelly formula is expressed as f-star equals b times p minus q, all divided by b. Here, f-star is the optimal fraction of your capital to bet, b represents the net odds you receive, p is the probability of winning, and q is the probability of losing, which equals one minus p. Let's look at a simple example: imagine a coin flip game where you win two dollars for every dollar bet, but lose one dollar if wrong. If the probability of winning is sixty percent, then b equals 2, p equals 0.6, and q equals 0.4. Plugging into the formula gives us f-star equals 0.2, meaning you should bet twenty percent of your capital.
One of the most critical insights from the Kelly Criterion is understanding the severe consequences of overbetting. When you bet more than the optimal Kelly fraction, several negative effects occur. First, your long-term growth rate decreases significantly. Second, the volatility of your returns increases dramatically, making your wealth fluctuate wildly. Third, you face a real risk of ruin, where you could lose everything. The graph clearly shows how the expected growth rate drops as you move beyond the optimal point, entering what we call the danger zone. This is why disciplined adherence to the Kelly fraction is so important.
To truly understand the power of the Kelly Criterion, let's examine a wealth growth simulation over one hundred bets. We compare three different strategies: the Kelly optimal strategy betting twenty percent, a conservative approach betting only ten percent, and an aggressive strategy betting forty percent. The simulation uses our coin flip example with sixty percent win probability and two-to-one odds. As we can see from the results, the Kelly optimal strategy significantly outperforms both alternatives. The conservative approach grows steadily but slowly, while the aggressive strategy shows high volatility and ultimately underperforms due to the compounding effect of losses from overbetting.