Position Sizing in Python: Kelly Criterion + Fixed-Fractional

Your signal quality is half the game. The other half is how much capital you risk when the signal fires. A positive-expectancy strategy can still blow up an account if position sizing is wrong — and a mediocre strategy can survive if sizing is conservative enough. This walkthrough implements Kelly Criterion and fixed-fractional sizing in ~25 lines of numpy and pandas, using the same worked example as the Kelly Criterion guide so the two pages agree on every number.

"""AlgoDrill — Kelly fraction and fixed-fractional position sizing.

~25 lines of numpy/pandas to compute full Kelly, half-Kelly,
quarter-Kelly, and fixed-fractional risk from a trade series.
Pure math — no broker, no execution.

Versions: numpy 2.4.6 · pandas 2.3.3 (run 2026-06-05)
Install:  pip install numpy pandas
"""
import numpy as np
import pandas as pd


def kelly_fraction(win_rate: float, avg_win: float, avg_loss: float) -> dict:
    """Kelly fraction from win rate and average payoffs."""
    loss_rate  = 1.0 - win_rate
    expectancy = win_rate * avg_win - loss_rate * avg_loss  # E[trade]
    kelly_f    = expectancy / avg_win                       # f* = E/W
    return {
        "expectancy":   round(expectancy, 4),
        "payoff_ratio": round(avg_win / avg_loss, 4),
        "kelly_f":      round(kelly_f, 4),
        "half_kelly":   round(kelly_f / 2, 4),
        "qtr_kelly":    round(kelly_f / 4, 4),
    }


# ── Example: 55% WR, $150 avg win, $100 avg loss (matches /kelly-criterion) ──
r = kelly_fraction(0.55, 150.0, 100.0)
print("=== Kelly sizing: 55% WR, $150 avg win, $100 avg loss ===")
print(f"Expectancy      ${r['expectancy']:.2f} per trade")
print(f"Payoff ratio      {r['payoff_ratio']:.4f}")
print(f"Full Kelly        {r['kelly_f']:.1%}")
print(f"Half-Kelly        {r['half_kelly']:.1%}")
print(f"Quarter-Kelly     {r['qtr_kelly']:.1%}")
print()

# ── Fixed-fractional comparison on a $10,000 account ─────────────────────────
ACCOUNT = 10_000
print(f"=== Fixed-fractional risk on ${ACCOUNT:,} account ===")
for pct in (0.01, 0.02, r["half_kelly"], r["kelly_f"]):
    risk = ACCOUNT * pct
    print(f"  {pct:.1%}  → ${risk:>7.2f} at risk per trade")
print()

# ── Batch Kelly from a real trade P&L series (pandas path) ───────────────────
pnl = pd.Series([150, -100, 200, -90, 175, -110, 300, -95, 225, -100])
wins   = pnl[pnl > 0]
losses = pnl[pnl < 0]
wr     = len(wins) / len(pnl)
r2 = kelly_fraction(wr, float(wins.mean()), float(abs(losses.mean())))
print(f"=== Batch Kelly: 10-trade P&L series ===")
print(f"Trades {len(pnl)}  WR {wr:.0%}  Avg win ${wins.mean():.2f}  Avg loss ${abs(losses.mean()):.2f}")
print(f"Full Kelly        {r2['kelly_f']:.1%}")
print(f"Half-Kelly        {r2['half_kelly']:.1%}")

Script Output

=== Kelly sizing: 55% WR, $150 avg win, $100 avg loss ===
Expectancy      $37.50 per trade
Payoff ratio      1.5000
Full Kelly        25.0%
Half-Kelly        12.5%
Quarter-Kelly     6.2%

=== Fixed-fractional risk on $10,000 account ===
  1.0%  -> $  100.00 at risk per trade
  2.0%  -> $  200.00 at risk per trade
  12.5%  -> $1250.00 at risk per trade
  25.0%  -> $2500.00 at risk per trade

=== Batch Kelly: 10-trade P&L series ===
Trades 10  WR 50%  Avg win $210.00  Avg loss $99.00
Full Kelly        26.4%
Half-Kelly        13.2%

Line-by-Line Walkthrough

Lines 14–24: The kelly_fraction function

The function takes three inputs: win_rate (fraction of trades that win, e.g., 0.55), avg_win (average dollar profit on winning trades), and avg_loss (average dollar loss magnitude, positive). It computes:

• Expectancy: the average dollar profit per trade across both wins and losses. expectancy = win_rate * avg_win - loss_rate * avg_loss. For (0.55, 150, 100): 0.55 × 150 − 0.45 × 100 = 82.50 − 45.00 = $37.50. A positive expectancy is necessary but not sufficient — you also need enough trades for it to compound reliably.
• Payoff ratio: avg_win / avg_loss = 150 / 100 = 1.50. A ratio above 1.0 means wins are larger than losses on average. Combined with a >50% win rate, this system has clear positive expectancy. Many trend-following strategies operate with win rates below 50% but payoff ratios above 2.0 — they lose more often but win much larger.
• Kelly fraction: kelly_f = expectancy / avg_win = 37.50 / 150 = 0.25 = 25%. This is the fraction of capital to risk per trade that maximizes long-run geometric growth. The formula is derived from maximizing the expected logarithm of portfolio value — the correct objective for multi-period compounding.

Lines 27–36: Reading the sizing output

For the worked example (55% WR, $150/$100):

• Full Kelly (25.0%): risk $2,500 of a $10,000 account per trade. This maximizes long-run growth but implies drawdowns of 30–50% during bad streaks. Most professional traders consider full Kelly too aggressive.
• Half-Kelly (12.5%): risk $1,250 per trade. Captures roughly 75% of full Kelly growth with approximately half the drawdown. This is the standard practitioner starting point for a well-understood edge.
• Quarter-Kelly (6.2%): risk $625 per trade. Appropriate when your trade history is small (<50 trades), the edge is unverified out-of-sample, or you want a large margin of safety against estimation error.

Lines 39–44: Fixed-fractional comparison

Fixed-fractional sizing risks a flat percentage of account equity regardless of Kelly. For a $10,000 account:

• 1% ($100): the conservative default for retail systematic traders. At this level, you need 25× the Kelly-optimal edge to deploy the same capital — growth is slow but the risk of blowing up is minimal.
• 2% ($200): the common upper bound for beginning algorithmic traders. Still far below Kelly-optimal for this example (25%), so growth is suboptimal but protected.
• 12.5% ($1,250) = half-Kelly and 25.0% ($2,500) = full Kelly are shown for comparison.

Lines 47–54: Batch Kelly from a P&L series

The batch section demonstrates how to compute Kelly from a raw list of trade P&L values using pandas. Separate wins (positive P&L) from losses (negative P&L), compute win rate, average win, and average loss magnitude, then apply the formula. For the 10-trade series [150, −100, 200, −90, 175, −110, 300, −95, 225, −100]:

• Win rate: 5/10 = 50%
• Avg win: ($150+$200+$175+$300+$225)/5 = $210.00
• Avg loss: ($100+$90+$110+$95+$100)/5 = $99.00
• Expectancy: 0.50 × 210 − 0.50 × 99 = $55.50 per trade
• Full Kelly: $55.50 / $210.00 = 26.4%

At only 10 trades, these estimates carry wide confidence intervals. The true win rate for a 50% observed rate at n=10 has a 95% confidence interval of approximately 19–81%. Use Kelly at this sample size as a rough directional check, not a precise sizing target.

When to Use Each Approach