The Kelly Criterion: The Math of Optimal Bet Size (and Why You Shouldn't Use It)

In 1956, a Bell Labs physicist named John Kelly published a paper titled "A New Interpretation of Information Rate." It contained a formula that would become legendary in gambling, investing, and trading circles — and misunderstood by almost everyone who encountered it.

The Kelly Criterion tells you the mathematically optimal fraction of your bankroll to bet, given your edge. It is the answer to the position sizing question — and it is also a fantastic way to bankrupt yourself if you use it naively.

This article explains both halves of that sentence.

The Formula

For a simple bet with two outcomes (win or lose), the Kelly formula is:

f* = (bp - q) / b

Where:

• f* is the fraction of your bankroll to bet
• b is the net odds received (e.g., 2 means you win 2× your stake)
• p is the probability of winning
• q is the probability of losing (= 1 - p)

For trading, we typically restate this in terms of win rate and payoff ratio:

f* = (W × R - (1 - W)) / R
   = W - (1 - W) / R

Where:

• W is your win rate (e.g., 0.45 for 45%)
• R is your win/loss ratio (average win / average loss, e.g., 2.5)

Worked Example

Suppose your trading strategy has:

• Win rate: 45% (W = 0.45)
• Average win: $300
• Average loss: $100
• Win/loss ratio (R): 3.0

f* = 0.45 - (1 - 0.45) / 3.0
   = 0.45 - 0.55 / 3.0
   = 0.45 - 0.183
   = 0.267

Kelly says: bet 26.7% of your account on each trade.

Read that again. Twenty-six point seven percent. On a strategy with a 45% win rate.

This is where most retail traders get excited, plug in their numbers, and blow their accounts within a week.

Why Full Kelly Is Insane

The Kelly Criterion is mathematically optimal — under specific assumptions that almost never hold in real trading:

Assumption 1: You Know Your True Edge

Kelly requires exact knowledge of p and b. In trading, you have estimates based on historical backtests — and backtests are notoriously overfitted (see Backtesting Without Overfitting).

If your true win rate is 40% but your backtest says 45%, Kelly tells you to bet 26.7% when the correct answer is 16.7%. That's a 60% over-bet — catastrophic when compounded over many trades.

Assumption 2: Bets Are Independent

Kelly assumes each bet is independent of the others. In trading, positions are correlated (see Portfolio Heat and Correlation). Five "1% risk" trades on USD-bloc pairs are not five independent 1% bets — they're closer to one 3.5% bet.

Kelly applied to correlated positions dramatically over-bets.

Assumption 3: Returns Are Normally Distributed

Kelly optimizes for geometric mean return — the long-run compounding rate. This optimization assumes returns follow a roughly normal distribution. Real trading returns have fat tails: 4-6 standard deviation moves happen far more often than the normal distribution predicts.

When fat tails hit, full Kelly produces drawdowns of 50-90% — mathematically optimal in the long run, but psychologically and practically un-survivable in the short run.

Assumption 4: You Can Survive the Drawdowns

Even with perfect knowledge of your edge, full Kelly produces drawdowns of 30-50% with regularity. The math says "keep betting, you'll recover." The human says "I can't sleep, I'm closing the position."

Most traders — even professionals — cannot stomach a 50% drawdown without abandoning their strategy. They sell at the bottom, breaking the geometric compounding that Kelly relies on.

The Solution: Fractional Kelly

The professional compromise is fractional Kelly — betting a fixed fraction (typically 0.25× or 0.5×) of the Kelly-optimal amount.

f_actual = fraction × f*

For our example above:

• Full Kelly: 26.7%
• Half Kelly: 13.4%
• Quarter Kelly: 6.7%
• Eighth Kelly: 3.3%

The mathematical reason this works is elegant: fractional Kelly captures most of the geometric growth while dramatically reducing drawdowns.

| Kelly Fraction | Long-Run Growth | Max Drawdown (typical) | |----------------|-----------------|------------------------| | 1.0× (full) | 100% | 50-90% | | 0.5× | 75% | 30-50% | | 0.25× | 50% | 15-25% | | 0.10× | 30% | 5-10% |

(All values relative to full Kelly's growth rate = 100%.)

A trader using 0.25× Kelly gets half the long-run growth of full Kelly, but with drawdowns that are actually survivable. For most retail traders, even 0.25× Kelly is aggressive.

Practical Recommendations

For Beginner Traders

Use 0.10× to 0.25× Kelly — equivalent to 0.5% to 1.5% risk per trade for most strategies. This matches the conventional "1% rule" advice, but with mathematical justification instead of folklore.

For Experienced Traders with Verified Edge

Use 0.25× to 0.50× Kelly (1.5% to 5% risk per trade). This requires:

• A track record of at least 200+ trades
• Out-of-sample validation of your edge
• Explicit correlation management across positions
• The psychological capacity to tolerate 25%+ drawdowns

For Nobody, Ever

Don't use full Kelly. The long-run math is correct, but the short-run reality will put you out of business before the long run arrives.

For Position Sizing Across Multiple Trades

When you have multiple open positions, compute Kelly for each trade individually, then scale down by a diversification factor based on correlation. The simplified formula:

f_adjusted_i = f_i × (1 / max(1, n_correlated_positions))

Where n_correlated_positions is the count of currently-open positions correlated with trade i (correlation > 0.5).

This is approximate but far better than ignoring the correlation problem.

The Kelly Paradox

The Kelly Criterion is one of the most beautiful results in probability theory. It is also one of the most dangerous — not because the math is wrong, but because the conditions under which it works perfectly almost never exist in real trading.

Kelly requires:

• Perfect knowledge of your edge (you have an estimate)
• Independent bets (yours are correlated)
• Normal distributions (yours have fat tails)
• Infinite time horizon (you have a finite career)
• Perfect execution (you have slippage, emotions, and gaps)

When you relax any of these assumptions, full Kelly becomes a recipe for disaster. Fractional Kelly is the principled compromise — it acknowledges that you don't know your edge perfectly, that your bets are correlated, and that you need to survive drawdowns to keep playing.

Use Kelly. Just don't use it at full strength.

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Related: Position Sizing Math covers the mechanical computation of lot size from your chosen risk percentage. Kelly is how you choose that percentage in the first place.