The Kelly Criterion: How to Size Your Bets Optimally

Imagine you've found a coin that lands heads 60% of the time. You have £10,000 to bet on repeated flips, and each bet pays even money (win = double your stake, lose = lose your stake). You clearly have an edge. But how much should you bet on each flip?

Bet too little, and you're leaving money on the table. Bet too much, and a run of bad luck will wipe you out before your edge has time to compound. The extremes are obvious — betting 1% of your bankroll is overly conservative, while betting 100% guarantees eventual ruin (you only need one loss).

This is the bet-sizing problem, and it has an elegant mathematical solution: the Kelly Criterion, developed by John L. Kelly Jr. at Bell Labs in 1956. Originally designed for information theory problems, it quickly found applications in gambling, investing, and anywhere else that decisions must be sized under uncertainty.

For a simple bet with two outcomes (win or lose), the Kelly Criterion says to stake the following fraction of your bankroll:

f = (bp - q) / b*

Where:

• f* = optimal fraction of bankroll to bet
• b = net odds received (e.g., even money = 1, 2-to-1 = 2)
• p = probability of winning
• q = probability of losing (= 1 - p)

For our biased coin at even money:

• b = 1, p = 0.60, q = 0.40
• f* = (1 × 0.60 - 0.40) / 1 = 0.20 (20% of bankroll)

Kelly says to bet exactly 20% of your current bankroll on each flip. This fraction maximises the expected logarithm of wealth — or equivalently, maximises the long-term compound growth rate of your bankroll.

The Kelly Criterion doesn't maximise expected value of your bankroll (that would mean betting everything). Instead, it maximises the expected geometric growth rate. This distinction is crucial.

With a 60% coin at even money, betting 20% each time:

• After a win: bankroll × 1.20
• After a loss: bankroll × 0.80

The expected growth factor per bet is:

• G = (1.20)^0.60 × (0.80)^0.40 ≈ 1.020

That's about 2% growth per bet. After 100 bets, your expected bankroll is £10,000 × 1.020^100 ≈ £72,400.

Now consider what happens if you over-bet at 50% of bankroll:

• After a win: bankroll × 1.50
• After a loss: bankroll × 0.50
• G = (1.50)^0.60 × (0.50)^0.40 ≈ 0.953

Despite having a clear edge, over-betting produces a growth factor below 1.0 — your bankroll will shrink over time! This is the tragedy of over-betting: you can have a genuine advantage and still go broke.

In theory, full Kelly is optimal. In practice, almost everyone uses a fraction of Kelly — typically between one-quarter and one-half. Here's why:

1. Probability estimates are uncertain

Kelly assumes you know the true probability. But in real life, your estimate of p might be wrong. If you think you have a 60% edge but it's actually 52%, full Kelly based on the wrong probability can lead to severe over-betting.

2. Drawdowns are psychologically brutal

Full Kelly can produce drawdowns of 50-80% from peak. Most humans cannot stick to a strategy through such volatility. A smaller stake produces smoother growth, which means you're more likely to actually follow through.

3. Kelly assumes infinite time horizon

If you need the money within a specific timeframe, the variance of full Kelly might be unacceptable. Half-Kelly gives you 75% of the growth rate with significantly reduced variance.

4. Real-world frictions

Transaction costs, taxes, liquidity constraints, and the inability to bet exactly the right fraction all erode the theoretical optimality of full Kelly.

The common recommendation among quantitative traders and professional gamblers is half-Kelly — it captures 75% of the maximum growth rate while dramatically reducing the probability of large drawdowns.

The Kelly framework extends beyond simple win/lose bets to continuous outcomes like stock returns. For an investment with expected return μ and variance σ², the Kelly fraction is approximately:

f = μ / σ²*

This is sometimes called the Merton share in finance. It says you should invest more aggressively when:

• Expected returns are higher
• Volatility is lower

For example, if a stock index has expected excess return of 6% per year and volatility of 16%:

• f* = 0.06 / (0.16)² = 0.06 / 0.0256 ≈ 2.34

Full Kelly suggests 234% allocation — i.e., heavy leverage! This illustrates why fractional Kelly is essential in practice. Half-Kelly gives ~117%, and quarter-Kelly gives ~59%, which aligns much better with conventional investment wisdom for equity allocation.

Warren Buffett, though he likely doesn't use the formula explicitly, has described his approach in Kelly-like terms: "When you have a big edge, bet big. When you don't, don't bet at all." His concentrated portfolio reflects Kelly thinking — large positions where he has high conviction.

Suppose you've built a model that identifies value in football match outcomes. Your model rates Team A to win at 55% probability, but the bookmaker is offering odds of 2.10 (implying only ~48% probability).

Calculating the Kelly stake:

• b = 2.10 - 1 = 1.10 (net profit per £1 staked)
• p = 0.55
• q = 0.45
• f* = (1.10 × 0.55 - 0.45) / 1.10 = (0.605 - 0.45) / 1.10 = 0.155 / 1.10 = 0.141 (14.1%)

Full Kelly says bet 14.1% of your bankroll. Using half-Kelly (a more sensible practical approach), you'd bet 7.05%.

With a £5,000 bankroll, that's:

• Full Kelly: £705
• Half Kelly: £352
• Quarter Kelly: £176

If your model is well-calibrated and you can find several such bets per week, even quarter-Kelly produces substantial long-term growth. The discipline is in never exceeding your Kelly fraction, even when you feel especially confident about a particular bet.

Kelly is powerful but not universal. It breaks down or requires modification in several scenarios:

Correlated bets — If you're making multiple simultaneous bets that are correlated (e.g., several tech stocks), you need a multivariate version of Kelly. Treating them independently will over-allocate.

Uncertain edge — If you're not confident in your probability estimates, Kelly will recommend stakes that are too large. This is the strongest argument for fractional Kelly.

Non-binary outcomes — Real investments have a distribution of returns, not just "win" or "lose". The formula adapts to continuous distributions, but the calculation is more complex.

Finite bankroll concerns — If losing 50% of your bankroll has non-financial consequences (stress, missed rent, relationship strain), you should use a smaller fraction regardless of what Kelly recommends.

Liquidity constraints — You might not be able to actually deploy the Kelly-optimal amount due to position limits, market depth, or capital lock-up periods.