Bet Sizing#

This submodule provides powerful tools for calculating optimal bet sizes using the Kelly Criterion, a mathematical formula designed to maximize the long-term growth of a bankroll.

Single Bet Analysis (kelly_criterion)#

Use this function to perform a deep analysis of a single betting opportunity. It calculates the optimal stake and provides a wealth of metrics to help you understand the bet’s risk and reward profile.

penaltyblog.betting.kelly_criterion(
    decimal_odds: Union[float, NDArray, List[float]],
    true_prob: Union[float, NDArray, List[float]],
    fraction: float = 1.0,
) -> KellyResult

Parameters#

  • decimal_odds: The decimal odds for the bet (e.g., 2.5). Can be a single number or a NumPy array for multiple independent bets.

  • true_prob: Your estimated “true” probability of the outcome (from 0 to 1).

  • fraction (default=``1.0``): A fraction of the full Kelly stake to use. Common values are 0.5 for “Half Kelly” or 0.25 for “Quarter Kelly” to adopt a more conservative strategy.

Returns (KellyResult Object)#

The function returns a KellyResult data object containing a complete analysis.

  • stake (float): The recommended fraction of your bankroll to wager (e.g., 0.05 for 5%).

  • expected_growth (float): The expected logarithmic growth rate of your bankroll if you place this bet. This is the metric that the Kelly Criterion optimizes.

  • edge (float): Your mathematical edge on the bet. A positive edge means the bet has a positive expected value.

  • is_favorable (bool): True if the bet has a positive edge.

  • risk_metrics (RiskMetrics): A detailed object containing advanced risk and return metrics. See the Understanding the Risk Metrics section below for a full explanation.

Usage Example#

import penaltyblog as pb

# Analyze a single bet with a 55% chance of winning at odds of 2.1
# We'll use a conservative Half Kelly approach (fraction=0.5)
result = pb.betting.kelly_criterion(2.1, 0.55, fraction=0.5)

print(f"Is the bet favorable? {result.is_favorable}")
print(f"Recommended Stake: {result.stake:.2%} of bankroll")
print(f"Expected Growth Rate: {result.expected_growth:.4%}")
print("-" * 50)
print("Advanced Risk Metrics:")
if result.risk_metrics:
    print(f"  - Sharpe Ratio: {result.risk_metrics.sharpe_ratio:.2f}")
    print(f"  - Volatility of Wealth: {result.risk_metrics.wealth_volatility:.4f}")
    print(f"  - 95% Value at Risk: {result.risk_metrics.value_at_risk_95:.2%} of bankroll")

Is the bet favorable? True
Recommended Stake: 7.05% of bankroll
Expected Growth Rate: 0.8177%
--------------------------------------------------
Advanced Risk Metrics:
  - Sharpe Ratio: 0.11
  - Volatility of Wealth: 0.0736
  - 95% Value at Risk: 7.05% of bankroll

Portfolio Betting (multiple_kelly_criterion)#

This is a powerful function for calculating optimal stakes when you have the opportunity to bet on multiple, mutually exclusive outcomes at the same time (e.g., Home, Draw, and Away in a football match).

Instead of just calculating Kelly for each bet independently, this function uses an optimizer to treat them as a portfolio, finding the allocation that maximizes the overall growth rate of your bankroll.

penaltyblog.betting.multiple_kelly_criterion(
    decimal_odds: Union[List[float], NDArray],
    true_probs: Union[List[float], NDArray],
    fraction: float = 1.0,
    max_total_stake: float = 1.0,
    method: Literal["simultaneous", "independent"] = "simultaneous",
) -> MultipleKellyResult

Parameters#

  • decimal_odds (List[float]): A list of decimal odds for each outcome.

  • true_probs (List[float]): A list of your estimated probabilities for each outcome. The sum must be <= 1.0.

  • fraction (default=``1.0``): A fraction of the optimal Kelly stakes to apply.

  • max_total_stake (default=``1.0``): The maximum total fraction of your bankroll you are willing to stake across all bets combined.

  • method (default=``”simultaneous”``):

    • simultaneous: (Recommended) Uses a numerical optimizer to find the best possible allocation across all bets to maximize portfolio growth.

    • independent: Calculates the Kelly stake for each bet individually and then scales them down if they exceed max_total_stake. Less optimal but faster.

Returns (MultipleKellyResult Object)#

  • stakes (List[float]): The list of recommended stakes for each outcome.

  • total_stake (float): The total fraction of your bankroll to be staked.

  • expected_growth (float): The expected logarithmic growth rate for the entire portfolio.

  • risk_metrics (RiskMetrics): A detailed risk analysis for the entire portfolio.

  • optimization_success (bool): True if the “simultaneous” optimizer found a valid solution.

Usage Example#

import penaltyblog as pb

# A 1X2 football market where we have a very strong edge on the Home team
odds = [2.5, 3.2, 2.8]

# Our probabilities show a much higher chance for a Home win (55%)
# than the odds imply (1 / 2.5 = 40%).
probs = [0.55, 0.25, 0.20]

result = pb.betting.kelly.multiple_kelly_criterion(odds, probs)

print("Optimal Portfolio Stakes:")
print(f"- Home Win (at {odds[0]}): {result.stakes[0]:.2%} of bankroll")
print(f"- Draw (at {odds[1]}): {result.stakes[1]:.2%} of bankroll")
print(f"- Away Win (at {odds[2]}): {result.stakes[2]:.2%} of bankroll")
print("-" * 20)
print(f"Total Stake: {result.total_stake:.2%}")
print(f"Portfolio Expected Growth: {result.expected_growth:.4%}")
print(f"Portfolio Sharpe Ratio: {result.risk_metrics.sharpe_ratio:.2f}")

Optimal Portfolio Stakes:
- Home Win (at 2.5): 27.17% of bankroll
- Draw (at 3.2): 3.26% of bankroll
- Away Win (at 2.8): 0.00% of bankroll
--------------------
Total Stake: 30.43%
Portfolio Expected Growth: 4.6783%
Portfolio Sharpe Ratio: 0.15

Understanding the Risk Metrics#

Both functions return a RiskMetrics object that gives you a deep insight into the risk/reward profile of your strategy.

  • expected_return: Your expected profit as a percentage of your total stake. A 10% expected return means you expect to make £0.10 for every £1 staked.

  • kelly_growth_rate: The expected long-term growth rate of your bankroll, expressed as a percentage. This is the core metric Kelly optimizes. A higher number is better.

  • wealth_volatility: The standard deviation of your final bankroll. This measures how much your bankroll is expected to swing up and down. A lower number indicates a less risky strategy.

  • sharpe_ratio: A measure of risk-adjusted return (growth rate divided by its volatility). It helps you compare strategies with different risk levels. A higher Sharpe Ratio is better.

  • probability_of_ruin: The chance of losing your entire staked capital in this specific round of betting.

  • value_at_risk_95: The maximum you can expect to lose 95% of the time, expressed as a percentage of your bankroll.