Kelly Criterion

The Kelly Criterion (also known as the Kelly formula, Kelly strategy, or Kelly bet) is a mathematical formula used for bet sizing that seeks to maximize the long-term growth rate of capital. Originally developed by scientist John Kelly Jr. at Bell Labs in 1956 to analyze long-distance telephone signal noise, it was quickly adopted by gamblers and, later, by some of the world's most successful investors. The formula tells an investor what percentage of their capital they should allocate to a particular opportunity based on the perceived odds of success and the potential payoff. Rather than just picking winners, the Kelly Criterion addresses a critical, often overlooked question: How much should you invest? While its direct application can be tricky, its underlying logic provides a powerful framework for thinking about risk, reward, and position sizing, a philosophy that resonates deeply with investors like Warren Buffett and Charlie Munger.

At its heart, the Kelly Criterion is a surprisingly simple equation that helps you size your bet when you believe you have an “edge”—an advantage over the prevailing odds. It calculates the optimal fraction of your bankroll to wager to ensure your wealth grows fastest over the long run. The most common version of the formula is: Kelly % = W - [(1 - W) / R] Let's break down the variables:

• Kelly %: The result of the formula. This is the recommended percentage of your capital to allocate to the investment.
• W: The probability of winning. This is your estimated chance that the investment will be successful.
• R: The win/loss ratio. This is your average gain on a winning trade divided by your average loss on a losing trade.

The formula essentially weighs your probability of being right against how much you stand to gain if you are right, compared to how much you stand to lose if you are wrong.

Imagine you're offered a bet on a biased coin that lands on heads 60% of the time. If it's heads, you win the amount you bet (a 1-to-1 payout). If it's tails, you lose your bet. You have an edge, but how much should you wager on each flip? Let's plug this into the Kelly formula:

• W (Probability of winning): 0.60 (or 60%)
• R (Win/Loss Ratio): You win $1 for every $1 you bet, so R = 1.

Kelly % = 0.60 - [(1 - 0.60) / 1] Kelly % = 0.60 - [0.40 / 1] Kelly % = 0.60 - 0.40 = 0.20 The formula suggests you should bet 20% of your capital on each coin flip. Betting more would expose you to excessive risk and could wipe you out, while betting less would slow your long-term growth.

For a value investing practitioner, the Kelly Criterion isn't about coin flips; it's about sizing investments based on conviction and the margin of safety.

• W (Probability of winning): This becomes the investor's confidence that their analysis is correct. For instance, what is the probability that the market price of a stock will eventually rise to meet your calculation of its intrinsic value? This is highly subjective and depends on the quality of your research.
• R (Win/Loss Ratio): This is the classic risk/reward analysis. If you believe a stock worth $100 is trading at $50, your potential upside is $50. If you estimate that the worst-case scenario (your thesis being wrong) is the stock falling to $40, your potential downside is $10. Your R would be $50 / $10 = 5.

Using these inputs, the Kelly formula helps an investor move from a vague feeling of “this is a good investment” to a concrete allocation decision. It's the mathematical expression of Warren Buffett's famous advice to make large, concentrated bets on your best ideas—the ones with the highest probability of success and the most favorable risk/reward profiles. Investors like Bill Miller have also famously applied its principles.

While intellectually elegant, applying the Kelly Criterion directly to investing is fraught with challenges.

• Garbage In, Garbage Out (GIGO): The formula's output is entirely dependent on your inputs for W and R. Unlike a casino game, these probabilities are not known in the world of investing. Overestimating your analytical prowess can lead to dangerously aggressive allocations.
• Extreme Volatility: Following the “Full Kelly” recommendation can lead to terrifying portfolio swings and severe drawdowns. The path to maximum long-term growth is not a smooth one; it's a bumpy ride that few investors have the stomach to endure. This is why many practitioners advocate for a Fractional Kelly (e.g., using half or a quarter of the recommended percentage) to temper volatility and reduce the risk of ruin.
• Black Swans: The model assumes you can reasonably estimate your maximum loss. It doesn't account for black swan events—unforeseeable disasters that can cause losses far exceeding your worst-case scenarios.

Ultimately, the Kelly Criterion is best viewed as a powerful conceptual tool rather than a rigid command. It forces you to rigorously quantify your conviction and the attractiveness of each opportunity, fostering the discipline and rational thinking that are the cornerstones of successful long-term investing.