Kelly Criterion and real position sizing

1. The math origin of Kelly

The Kelly Criterion was published by John Kelly Jr at Bell Labs in 1956. Original context: a gambler with a private information edge on a binary bet wants the wagering rule that maximises long-run geometric wealth growth. Kelly's answer is f = (bp − q) / b.

The result extends naturally from binary bets to many-outcome wagers (the so-called "fractional Kelly with continuous distributions"), and it forms the theoretical backbone for a chunk of modern position-sizing in both gambling and trading.

For the original mathematical formulation, Wikipedia's Kelly Criterion entry is the academic reference. For a practitioner-friendly framing, Investopedia's Kelly piece walks through the trading interpretation.

2. Unpacking the formula: f = (bp − q) / b

Definitions: p = win probability, q = 1 − p = loss probability, b = the payoff ratio (how much you win on a winning bet, expressed as a multiple of the amount you risk).

Plain English: the fraction of your bankroll you should risk on a single trade equals (expected return per dollar wagered) divided by (payoff multiple).

Two consequences fall out immediately. (a) If bp − q ≤ 0 (negative expected value), f ≤ 0 — Kelly says do not bet. (b) f scales with bp − q, so a small change in your estimate of p flips the optimal position size substantially.

3. Equity curves at full, half, quarter Kelly

Run a Monte Carlo over 1,000 trades using the same edge (55% win rate, 1.5 payoff) at three fractions. The shapes look like this.

Full Kelly (25%). Highest long-run geometric growth in the theoretical case. Drawdowns over 40% are common. Sequences of bad luck make the equity curve volatile enough that real traders quit.

Half Kelly (12.5%). ~75% of full Kelly's geometric growth, ~50% of its volatility. The standard recommendation in most practitioner literature.

Quarter Kelly (6.25%). ~50% of full Kelly's growth, ~25% of its volatility. Drawdowns top out around 15-20%. Most traders can stomach this through losing streaks.

The trade-off is straight geometric: you sacrifice some return for a much smoother ride. For non-professional traders, the smoother ride is worth more than the optimised geometric average, because nobody trades a strategy through 40% drawdowns without flinching.

4. Worked: 55% win rate × 1.5 payoff → 12.5%

Plug in p = 0.55, q = 0.45, b = 1.5. f = (1.5 × 0.55 − 0.45) / 1.5 = (0.825 − 0.45) / 1.5 = 0.375 / 1.5 = 0.25.

Full Kelly says 25% of bankroll on the next trade. Half Kelly = 12.5%. Quarter Kelly = 6.25%.

Sense check: at 25%, four consecutive losses (probability 0.45^4 = 4.1%) draws the account down by approximately 1 − (1 − 0.25)^4 = 68%. That happens roughly one in 25 trade sequences. Most traders cannot run a strategy with a 1-in-25 chance of a 68% drawdown.

At quarter Kelly (6.25%), the same four-loss streak draws down 1 − (1 − 0.0625)^4 ≈ 23%. That is uncomfortable but recoverable.

5. Why full Kelly fails in financial markets

Kelly assumes you know p and b. In gambling against a known probability table that is fine. In financial markets, p and b are estimates, and small errors in those estimates push the optimal fraction off the cliff.

If your true win rate is 50% but you estimated 55%, full Kelly tells you to bet 25%. The optimal fraction at the true 50% rate would actually be 16.7% (still positive because b = 1.5). Running 25% when the optimum is 16.7% means you are over-betting; the long-run geometric growth is lower than half Kelly's would have been.

If your true rate is 47% (you overestimated by 8 points), the true optimum is around 5.7%. Running 25% in that world is a catastrophic over-bet — the geometric growth actually goes negative even though the strategy is profitable on average.

The mathematics of Kelly is exact. The practical application is bounded by your ability to estimate p and b. Half or quarter Kelly is a robustness adjustment — it accepts a known performance haircut in exchange for tolerance to estimation error.

6. The crypto perpetual adjustment — why 1/4 Kelly

Crypto perpetuals have three things working against full Kelly. First, win rate and payoff are non-stationary — they drift as market regimes change. Your trailing 60-day estimate of p will not match the next 60 days exactly. Second, leverage. Kelly is computed against bankroll, but on a leveraged book the same Kelly fraction is amplified through the leverage multiplier. Third, tail events — a 2024-08-05-style move generates an outsized loss that the historical distribution did not show.

The standard adjustment in practitioner literature is half Kelly. For crypto perps, our desk recommends quarter Kelly as the working ceiling. That gives meaningful buffer against estimation drift, leverage amplification and tail events.

For the textbook take on risk management around sizing, Investopedia's risk-management techniques for active traders covers the qualitative side.

7. Estimating p and b from 60-90 days of records

You need a trade log. Tag every closed trade with the realised R-multiple (how many R you risked, how many R you made — covered in our stop-loss / position-size paradox piece).

From 60-90 days of records, compute. p = number of wins ÷ total trades. avg_win = mean R on winning trades. avg_loss = mean R on losing trades. b = avg_win ÷ avg_loss.

Plug those into Kelly. Multiply the output by 0.25-0.5 depending on your confidence in the estimates. That number is your maximum per-trade fraction.

Re-estimate every 30 days. If the underlying p or b is drifting, your sizing should drift with it.

8. The assumptions Kelly requires

Kelly is rigorous within its assumptions. Outside them it stops being optimal. Two assumptions matter for crypto.

Repeated independent bets. Kelly assumes each trade is independent of the previous one. In crypto, correlated trades (long BTC and long ETH simultaneously) are not independent — a single market move hits both. Treating each as a separate Kelly fraction over-sizes your aggregate exposure. The fix is to compute Kelly on the combined position, or to scale down both individual fractions.

Continuous compounding. Kelly assumes you can resize after every trade. In practice you do — you cap position fraction at a maximum. Use the Kelly output as a ceiling rather than a target.

Within those caveats, Kelly is one of the cleanest sizing tools available. Pair this with the R-multiple framework for the journaling side, and the Kelly position sizer for the in-browser calculator.