Kelly Criterion: Mathematically Optimal Position Sizing — At 55% Win Rate, Invest Only 10%

What Is the Kelly Criterion?

The Kelly Criterion was proposed by John Kelly in 1956, originally for telephone network optimization, later adopted by gamblers and investors to calculate the optimal fraction to bet each time.

Formula: f = (bp - q) / b

• f = fraction of capital to invest
• b = odds (profit amount / loss amount)
• p = win rate
• q = 1 - p = loss rate

Calculation Examples

Example 1: 55% Win Rate, 1:1 Odds

• b = 1 (win $1 for every $1 risked)
• p = 0.55
• q = 0.45
• f = (1×0.55 - 0.45) / 1 = 0.10

Kelly tells you: at 55% win rate, invest only 10% of your capital per trade.

Example 2: 40% Win Rate, 1:1 Odds

• b = 1, p = 0.40, q = 0.60
• f = (1×0.40 - 0.60) / 1 = -0.20

Kelly result is negative → Don’t trade. When your win rate is below 50% (at 1:1 odds), math tells you not to bet.

Example 3: 40% Win Rate, 2:1 Odds

• b = 2 (win $2 for every $1 risked)
• p = 0.40, q = 0.60
• f = (2×0.40 - 0.60) / 2 = 0.10

With sufficiently high odds, even a 40% win rate allows a 10% position. High odds compensate for low win rate.

Half Kelly: Safer in Practice

Kelly gives the theoretical optimum — at this fraction, long-term growth is fastest. But two problems:

1. Too volatile — Betting at full Kelly can produce 50%+ drawdowns along the way
2. Win rate/odds estimates are imprecise — If you overestimate win rate, full Kelly leads to overbetting

In practice, use Half Kelly: half the Kelly formula result.

Situation	Full Kelly	Half Kelly
55% win rate, 1:1 odds	10%	5%
60% win rate, 2:1 odds	20%	10%
70% win rate, 1:1 odds	40%	20%

Half Kelly’s long-term growth rate is only 25% lower than full Kelly, but drawdowns are dramatically smaller. Sacrifice a little return for a much lower bankruptcy risk.

Crypto Practical Application

1. Futures Position Calculation

Gate.io BTC futures trading:

• Your estimated win rate: 55%
• Odds: 1:1 (1% stop-loss, 1% take-profit)
• Half Kelly = 5% → Use only 5% of total capital per trade

$100,000 capital → $5,000 per opening → 10x leverage → nominal position $50,000 → even if liquidated, you only lose $5,000 (5% of capital)

2. When Not to Trade

Kelly result negative → don’t trade. This means:

• Win rate below 50% and odds 1:1 → don’t trade
• Odds insufficient to compensate for low win rate → don’t trade

Many retail traders enter when they shouldn’t — Kelly math tells you when to skip.

3. Take-Profit/Stop-Loss Ratio and Odds

Take-profit 2%, stop-loss 1% → odds 2:1 Take-profit 3%, stop-loss 1% → odds 3:1 Take-profit 5%, stop-loss 1% → odds 5:1

Higher odds allow larger Kelly positions — but only if your take-profit targets are actually achievable.

Common Misconceptions

1. High win rate = large position — Wrong! At 70% win rate but 0.5 odds (win $0.5 per $1 risked), Kelly position = 5%, because odds are too low
2. Kelly guarantees profit — Wrong! It only guarantees optimal long-term growth rate — short-term drawdowns can be steep
3. Win rate/odds can be known precisely — Wrong! Always estimate conservatively — use Half Kelly
4. Kelly works for all markets — Wrong! Crypto volatility is far higher than traditional markets — use more conservative fractions

Connection to Position Management

Kelly is the mathematical foundation of position management. Demon Trading position management principles:

1. Calculate theoretical optimal position with Kelly
2. Take Half Kelly as actual position
3. Single-trade risk ≤2% of total capital
4. Total exposure ≤30% of total capital

---

Kelly tells you the mathematically optimal position size: at 55% win rate and 1:1 odds, invest only 10%. In practice, Half Kelly is safer (5%). When Kelly is negative, don’t trade. Core: Position sizing isn’t intuition — it’s math. Calculate with Kelly, execute with Half Kelly.