Kelly Criterion

The Kelly formula determines the fraction of your capital to risk on a single trade:

f = (bp - q) / b*

Where:

• f* = fraction of capital to bet
• b = the ratio of amount won per dollar risked (reward/risk ratio)
• p = probability of winning
• q = probability of losing (1 - p)

Example 1: You win 55% of trades with a 1:1 risk/reward (risk $1 to make $1). f* = (1 x 0.55 - 0.45) / 1 = 0.10 → Kelly says bet 10% of capital.

Example 2: You win 45% of trades but winners are 2x losers (2:1 R/R). f* = (2 x 0.45 - 0.55) / 2 = 0.175 → Kelly says bet 17.5% of capital.

Example 3: You win 40% of trades with a 3:1 R/R ratio. f* = (3 x 0.40 - 0.60) / 3 = 0.133 → Kelly says bet 13.3% of capital.

If the Kelly fraction is zero or negative, the formula says do not take the trade — you have no edge. This is one of the formula's most valuable uses: it tells you when a strategy is not worth trading at all.

Full Kelly sizing maximizes long-term growth rate mathematically, but it produces violent account swings that most traders cannot stomach. A full-Kelly bettor with 55% win rate and 1:1 payoff can experience 40-50% drawdowns — even though the strategy is profitable long-term.

The core problem: the Kelly formula assumes you know your exact edge. In trading, win rates and payoff ratios are estimated from historical data, which is noisy and changes over time. If you overestimate your edge by even a small amount and use full Kelly, you over-bet and can blow up.

Fractional Kelly solves this by using 25-50% of the full Kelly fraction:

• Half Kelly (50%): Captures 75% of the growth rate with dramatically less volatility. The standard recommendation for most traders.
• Quarter Kelly (25%): Very conservative. Captures about 50% of the growth rate but with drawdowns that feel manageable. Good for beginners or uncertain edges.

The mathematical insight: the growth rate curve vs bet size is parabolic. Betting slightly below Kelly costs very little growth, but betting slightly above Kelly costs a lot. This asymmetry means it is always better to err on the side of underbetting.

In practice, most professional trading firms use 10-25% of full Kelly, not because they lack confidence in their edge, but because they value consistency and survival over maximum theoretical growth.

Fixed percentage (e.g., risk 1% of capital per trade) is simple and widely used. You risk the same fraction regardless of edge quality. A trade with 70% win rate and 2:1 payoff gets the same 1% allocation as a marginal trade with 52% win rate and 1:1 payoff.

Kelly sizing allocates more capital to higher-edge trades and less to lower-edge ones. This is mathematically optimal because it compounds winners faster while limiting damage from low-edge trades. However, it requires accurate estimates of win probability and payoff ratio — which are hard to measure precisely.

Practical hybrid approach: Use a fixed 1-2% risk cap as a ceiling, and Kelly (or fractional Kelly) to determine the actual size within that ceiling. If Kelly says 0.5%, you risk 0.5%. If Kelly says 3%, you cap at 2%. This combines the mathematical optimization of Kelly with the safety of a hard limit.

Portfolio-level Kelly: For multiple simultaneous positions, the total portfolio Kelly fraction should not exceed your risk tolerance. If you have 5 open positions each sized at half-Kelly, your combined allocation might exceed safe levels. Track total portfolio heat (sum of all position risk) and scale individual positions down proportionally if needed.