Chapter 4: Key Takeaways — Bankroll Management Fundamentals

Core Formulas

Kelly Criterion (General Form)

$$f^* = \frac{p \cdot d - 1}{d - 1}$$

Where: - f = optimal fraction of bankroll to wager - p = true probability of winning - d = decimal odds offered - p * d - 1 = expected edge (must be positive for a bet to be worthwhile) - d - 1* = net odds (profit per dollar wagered if you win)

Even-money special case (d = 2.0): f* = 2p - 1

Risk of Ruin (Even-Money Bets, Fixed Staking)

$$R = \left(\frac{1 - p}{p}\right)^{B/u}$$

Where: - R = probability of eventual ruin (going broke) - p = win probability (must be > 0.50 for formula to apply) - B = current bankroll - u = fixed unit (bet) size - B/u = number of units in bankroll

Expected Growth Rate (Per Bet)

$$g(f) = p \cdot \ln(1 + f \cdot (d - 1)) + (1 - p) \cdot \ln(1 - f)$$

Where f is the fraction of bankroll wagered. Maximized when f = f* (the Kelly fraction).

Fractional Kelly Growth Rate Retention

Key Concepts at a Glance

The Kelly Criterion

• Maximizes the geometric growth rate (long-run compounding) of a bankroll.
• Balances the trade-off between betting too much (excessive risk) and too little (slow growth).
• Requires knowing your true probability of winning, which in practice must be estimated.
• A negative Kelly fraction means the bet has negative expected value -- do not bet.

Risk of Ruin

• The probability that a bettor's bankroll will eventually reach zero under fixed staking.
• Exponentially sensitive to edge: doubling your edge reduces risk of ruin dramatically.
• Exponentially sensitive to bankroll depth: doubling your units halves the log of ruin probability.
• Under pure proportional (Kelly) staking, theoretical risk of ruin is zero, but practical constraints (minimum bets, withdrawal needs) reintroduce ruin risk.

Why Fractional Kelly Dominates in Practice

1. Edge estimation error: If your estimated edge is wrong, full Kelly based on overestimated edge is actually over-Kelly, which is destructive.
2. Flat growth curve near optimum: The growth rate curve is flat at the top, so reducing from full to half Kelly sacrifices only ~25% of growth.
3. Drawdown reduction: Half Kelly roughly halves the variance and dramatically reduces maximum drawdown.
4. Asymmetric consequences: Overbetting by X% is far more costly than underbetting by X%.

Unit Sizing Guidelines

• Conservative: 1% of bankroll per unit (100-unit bankroll)
• Standard: 2% of bankroll per unit (50-unit bankroll)
• Aggressive: 3-5% of bankroll per unit (20-33 unit bankroll)
• Maximum single-bet exposure: Rarely exceed 5% of bankroll regardless of perceived edge

Decision Framework

Before Placing Any Bet

Step 1: Estimate your true win probability (p)
         |
Step 2: Calculate the implied probability from the odds
         |
Step 3: Determine your edge = p * d - 1
         |
         +--> If edge <= 0: DO NOT BET
         |
         +--> If edge > 0: Continue to Step 4
         |
Step 4: Calculate Kelly fraction: f* = edge / (d - 1)
         |
Step 5: Apply fractional Kelly (recommended: 0.25 to 0.50 of f*)
         |
Step 6: Check against maximum exposure limits
         |
         +--> Single bet: max 5% of bankroll
         +--> Daily total: max 10-15% of bankroll
         +--> Correlated bets: reduce further
         |
Step 7: Place the bet at the smaller of Kelly-recommended and exposure limit

When to Adjust Your Staking

Emergency Rules

1. Never chase losses by increasing bet sizes after a losing streak.
2. Never exceed your maximum single-bet limit regardless of how confident you feel.
3. Stop betting if your bankroll drops below 50% of its starting value -- reassess your edge.
4. Keep separate records for each sport/bet type to identify where your edge truly exists.
5. Reassess your edge estimate every 200-300 bets using confidence intervals.

Common Mistakes to Avoid

Quick Reference: Formulas in Python

def kelly_fraction(p: float, d: float) -> float:
    """Kelly fraction for win probability p at decimal odds d."""
    return (p * d - 1) / (d - 1)

def risk_of_ruin(p: float, bankroll_units: float) -> float:
    """Risk of ruin for even-money bets with fixed staking."""
    return ((1 - p) / p) ** bankroll_units

def expected_growth(p: float, f: float, d: float) -> float:
    """Expected log growth rate per bet."""
    import math
    return p * math.log(1 + f * (d - 1)) + (1 - p) * math.log(1 - f)

def bet_size(bankroll: float, p: float, d: float, kelly_mult: float = 0.5) -> float:
    """Recommended bet size using fractional Kelly."""
    f = kelly_fraction(p, d) * kelly_mult
    return max(0, bankroll * f)

One-Sentence Summary

Bankroll management is the discipline of sizing your bets so that your edge compounds over time without exposing you to catastrophic losses -- and the Kelly Criterion, tempered with conservative fractional adjustments, is the mathematical foundation for doing this optimally.

Review this summary card regularly. The concepts here form the foundation for every betting decision you will make.