The Kelly Criterion: Optimal Bet Sizing (And Why You Should Use a Fraction of It)

The Kelly criterion is a mathematical formula for the bet size that maximizes long-run growth of capital. It tells you precisely what fraction of your account to risk on any bet given the bet's edge and odds. The math is well-established. The practical application requires understanding why "full Kelly" is too aggressive for almost everyone and how to use a fraction of it instead.

The basic Kelly formula

For a bet with two outcomes (win or lose):

f* = (bp - q) / b

Where:

• f* = fraction of capital to bet
• b = net odds received on win (e.g., 2:1 = 2)
• p = probability of winning
• q = probability of losing (1-p)

For trading, often expressed as:

f* = (W × R - L) / R

Where:

• W = win rate (e.g., 0.55)
• L = loss rate (1 - W = 0.45)
• R = ratio of average win to average loss

A worked example: 55% win rate, 1.5R average winner vs 1R average loser:

• f* = (0.55 × 1.5 - 0.45) / 1.5
• f* = (0.825 - 0.45) / 1.5
• f* = 0.375 / 1.5
• f* = 0.25 = 25%

Full Kelly says risk 25% of your account per trade.

That's a lot. And exactly why full Kelly is rarely used in practice.

Why full Kelly is too aggressive

Full Kelly maximizes expected log growth in the long run. But:

1. Inputs are uncertain. The formula assumes you know W and R precisely. You don't. Your "55% win rate" might really be 50% (or 60%). The formula's optimal answer is wrong if the inputs are wrong.

2. Drawdowns are extreme. Full Kelly produces enormous drawdowns. A strategy at 25% Kelly might routinely have 50%+ drawdowns. Most traders psychologically can't survive these.

3. Behavioral failure under stress. Even if you mathematically should hold through a 50% drawdown, the emotional reality is that you'll deviate from strategy under that pressure. The deviations destroy the math's expected value.

4. Path dependency. Full Kelly's expected value doesn't account for the path the capital takes. A path through a 80% drawdown might mathematically work but practically can mean ruin if the drawdown forces capital withdrawal.

5. Estimation error compounds. Small overestimation of edge produces dramatic overbetting. The cost of being wrong about edge is asymmetrically large with full Kelly.

The result: full Kelly is optimal only under unrealistic assumptions (perfect knowledge of edge, infinite patience, no emotional response to drawdown). Practical Kelly is much smaller.

Fractional Kelly, the practical version

Most professional traders use fractional Kelly: risk some fraction of full Kelly.

Half-Kelly. Risk 50% of what full Kelly suggests. For the example above, this would be 12.5% per trade.

• Captures most of the long-run growth (~75% of full Kelly's expected log return)
• Substantially reduces drawdowns (about half full Kelly's drawdown)
• More forgiving of input error (a 10% error in edge estimate is much less catastrophic)

Quarter-Kelly. Risk 25% of full Kelly.

• ~50% of full Kelly's expected log return
• ~25% of full Kelly's drawdown
• Very forgiving of input uncertainty

Tenth-Kelly or smaller.

• ~20% of full Kelly's expected log return
• Very small drawdowns
• Robust to substantial input error
• Closest to the "1% rule" framing

For most retail traders, quarter-Kelly to tenth- Kelly is appropriate. Even tenth-Kelly produces ~20% of full Kelly's growth, which, compounded over years, is substantial.

Comparison with the 1% rule

The 1% rule from earlier chapters is roughly tenth-Kelly for typical retail strategies.

Take a typical retail strategy: 50% win rate, 1.5R average winner vs 1R average loser:

• Full Kelly = (0.5 × 1.5 - 0.5) / 1.5 = 16.7%
• Tenth-Kelly = 1.67% per trade
• Fifteenth-Kelly ≈ 1.1%

The 1% rule lands close to fifteenth-Kelly for this example, appropriately conservative for retail traders with uncertain edge estimates and typical psychological tolerance.

For pros with verified edge across thousands of trades, the rule might be more like quarter to half-Kelly. But for most retail, the 1% (or even 0.5%) rule is the right zone.

Why Kelly isn't directly useful for most retail

Beyond the over-aggression issue, Kelly has practical limitations for retail:

1. Requires accurate W and R estimates. Retail typically doesn't have these. Sample sizes are too small; estimates are too noisy.

2. Assumes binary outcomes. Real trading has continuous outcomes. The Kelly formula needs adaptation for continuous outcomes (it still works, but the math is more complex).

3. Assumes independent bets. Real trades are correlated (multiple positions in the same regime, same asset class). Independence assumption breaks down.

4. Assumes single strategy. Real traders have multiple strategies with different W and R. Optimal sizing across strategies requires more advanced methods.

These limitations don't mean Kelly is useless. They mean it's a framework and upper bound, not a precise recipe.

How to use Kelly thinking practically

Even without precise application:

1. Know roughly where full Kelly is. Compute it for your strategy. Your actual sizing should be a small fraction of this. If your actual sizing is close to full Kelly, you're almost certainly oversized.

2. Use Kelly as an upper bound. "Don't risk more than X per trade" where X is set based on your Kelly calculation with appropriate fractional discount.

3. Adjust as your edge estimates change. If your strategy's W and R deteriorate (per journal), recompute Kelly and adjust your sizing fraction accordingly. Sizing should follow edge.

4. Account for correlation. Multiple correlated positions are functionally one big position. Size accordingly. If you're long 5 correlated alts at 1% each, you have 5% effective exposure to "alt season ending."

5. Use full Kelly only as a thought experiment. "If I knew my edge perfectly and had infinite patience, full Kelly would be optimal. I don't, so I use fractional."

A common mistake: applying full Kelly literally

A trader computes full Kelly = 20%. They size 20% per trade. The first 5-trade losing streak (which happens) drops their account 70%. They can't recover psychologically; they over-trade trying to recover; they end up at zero.

The fix: never use full Kelly in practice. Use fractional. The over-conservative bias of fractional Kelly is the right direction of error.

A common mistake: using Kelly with bad input estimates

A trader has 30 trades. They compute "60% win rate, 2R average winner." They Kelly-size based on these. The estimates are unstable (small sample); the actual long-run rates are different; their sizing is wrong.

The fix: Kelly requires reliable inputs. Without 100+ trades of consistent expectancy, treat Kelly outputs as suggestions with wide error bars. Default to conservative sizing.

A common mistake: ignoring drawdown tolerance

A trader's mathematical Kelly might be 15%. Their psychological tolerance for drawdowns might cap their effective sizing at 1%. The mathematical optimum doesn't matter if the trader can't execute through the drawdowns the math requires.

The fix: size for the larger of: math says it's optimal, OR psychological tolerance for the resulting drawdowns. Whichever is smaller is your real ceiling.

A common mistake: not adjusting for correlation

A trader has 10 alt positions, each at 1% risk. "Diversified, total risk 10%." But all 10 alts are highly correlated; in a downturn they all hit stops together. Real risk on a coordinated bad move: 10% of account, plus correlation-amplified slippage.

The fix: correlated positions should be considered as one. Either size them smaller individually or cap total correlated exposure.

Mental model, Kelly as the recipe with adjustable serving size

A recipe specifies how much of each ingredient. You can scale the recipe up or down. Doubling gives you twice the dish; halving gives you half.

Full Kelly is the recipe for "maximum long-run growth." Fractional Kelly is scaling the recipe down. Half-Kelly is half the recipe; tenth-Kelly is a tenth.

You don't have to follow the recipe at full scale. For most traders, smaller scale is more sustainable. The dish is still good (still positive long-run growth); the cooking process (trading psychology) is much more manageable.

Why this matters for trading

Kelly provides the mathematical framework for position sizing. Even if you don't apply it literally, knowing where full Kelly is for your strategy tells you whether your actual sizing is appropriate, oversized, or potentially undersized. Hex37's position sizer lets you specify percent- of-account risk; a Kelly-aware trader would set this fraction based on their strategy's metrics and their preferred Kelly fraction.

Takeaway

Kelly criterion gives the mathematically optimal bet size for maximum long-run growth: f* = (W × R

• L) / R. Full Kelly is too aggressive in practice (extreme drawdowns, sensitive to input errors, psychological failure under stress). Fractional Kelly (half, quarter, tenth) is what practitioners actually use. For most retail, quarter to tenth-Kelly is appropriate, close to the 1% rule. Use Kelly as a framework and upper bound, not a precise recipe. Account for input uncertainty, correlation, and psychological tolerance. Conservative sizing has the right direction of error.