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feat: add Kelly Criterion and Sharpe Ratio to financial algorithms
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‎financial/kelly_criterion.py‎

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"""
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Kelly Criterion for optimal position sizing in betting and trading.
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The Kelly Criterion is a formula used to determine the optimal size of a series of bets
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or investments to maximize logarithmic wealth over time. It was developed by John L.
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Kelly Jr. in 1956.
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Wikipedia Reference: https://en.wikipedia.org/wiki/Kelly_criterion
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Investopedia: https://www.investopedia.com/articles/trading/04/091504.asp
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The Kelly Criterion is widely used in:
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- Sports betting and gambling to determine optimal bet sizes
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- Investment portfolio management to size positions
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- Trading strategies to manage risk and maximize growth
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"""
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from __future__ import annotations
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def kelly_criterion(win_probability: float, win_loss_ratio: float) -> float:
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"""
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Calculate the optimal fraction of bankroll to bet using the Kelly Criterion.
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The Kelly Criterion formula:
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f* = (p * b - q) / b
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Where:
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f* = fraction of bankroll to bet (Kelly fraction)
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p = probability of winning
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q = probability of losing (1 - p)
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b = win/loss ratio (amount won per unit staked / amount lost per unit staked)
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:param win_probability: Probability of winning (0 < p < 1)
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:param win_loss_ratio: Ratio of win amount to loss amount (b > 0)
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:return: Optimal fraction of bankroll to bet
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>>> round(kelly_criterion(0.6, 2.0), 4)
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0.4
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>>> round(kelly_criterion(0.55, 1.0), 4)
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0.1
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>>> kelly_criterion(0.5, 1.0)
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0.0
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>>> round(kelly_criterion(0.7, 3.0), 4)
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0.6
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>>> round(kelly_criterion(0.3, 2.0), 4)
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-0.05
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>>> kelly_criterion(0.0, 1.0)
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Traceback (most recent call last):
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...
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ValueError: win_probability must be between 0 and 1 (exclusive)
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>>> kelly_criterion(1.0, 1.0)
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Traceback (most recent call last):
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...
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ValueError: win_probability must be between 0 and 1 (exclusive)
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>>> kelly_criterion(0.5, 0.0)
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Traceback (most recent call last):
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...
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ValueError: win_loss_ratio must be > 0
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>>> kelly_criterion(0.5, -1.0)
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Traceback (most recent call last):
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...
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ValueError: win_loss_ratio must be > 0
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"""
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if win_probability <= 0 or win_probability >= 1:
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raise ValueError("win_probability must be between 0 and 1 (exclusive)")
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if win_loss_ratio <= 0:
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raise ValueError("win_loss_ratio must be > 0")
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loss_probability = 1 - win_probability
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kelly_fraction = (win_probability * win_loss_ratio - loss_probability) / (
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win_loss_ratio
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)
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return kelly_fraction
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def kelly_criterion_extended(
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win_probability: float, win_amount: float, loss_amount: float
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) -> float:
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"""
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Calculate the Kelly fraction using explicit win and loss amounts.
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This is a more general form of the Kelly Criterion that accepts
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absolute win and loss amounts rather than a ratio.
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Formula:
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f* = (p * W - q * L) / (W * L)
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Where:
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p = probability of winning
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q = probability of losing (1 - p)
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W = amount won per unit bet
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L = amount lost per unit bet (positive value)
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:param win_probability: Probability of winning (0 < p < 1)
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:param win_amount: Amount won per unit bet (W > 0)
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:param loss_amount: Amount lost per unit bet (L > 0)
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:return: Optimal fraction of bankroll to bet
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>>> round(kelly_criterion_extended(0.6, 2.0, 1.0), 4)
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0.4
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>>> round(kelly_criterion_extended(0.55, 1.5, 1.5), 4)
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0.1
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>>> kelly_criterion_extended(0.5, 1.0, 1.0)
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0.0
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>>> round(kelly_criterion_extended(0.7, 3.0, 1.0), 4)
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0.6
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>>> kelly_criterion_extended(0.0, 1.0, 1.0)
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Traceback (most recent call last):
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...
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ValueError: win_probability must be between 0 and 1 (exclusive)
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>>> kelly_criterion_extended(0.5, 0.0, 1.0)
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Traceback (most recent call last):
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...
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ValueError: win_amount must be > 0
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>>> kelly_criterion_extended(0.5, 1.0, 0.0)
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Traceback (most recent call last):
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...
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ValueError: loss_amount must be > 0
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"""
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if win_probability <= 0 or win_probability >= 1:
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raise ValueError("win_probability must be between 0 and 1 (exclusive)")
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if win_amount <= 0:
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raise ValueError("win_amount must be > 0")
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if loss_amount <= 0:
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raise ValueError("loss_amount must be > 0")
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loss_probability = 1 - win_probability
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# Convert to win/loss ratio format: b = win_amount / loss_amount
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# Then apply Kelly formula: (p * b - q) / b
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win_loss_ratio = win_amount / loss_amount
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kelly_fraction = (win_probability * win_loss_ratio - loss_probability) / (
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win_loss_ratio
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)
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return kelly_fraction
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def fractional_kelly(
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win_probability: float, win_loss_ratio: float, fraction: float = 0.5
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) -> float:
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"""
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Calculate a fractional Kelly bet size to reduce volatility.
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Many practitioners use a fraction of the Kelly Criterion (e.g., half-Kelly)
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to reduce risk and volatility while still achieving good growth. This is
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because the full Kelly can lead to large drawdowns.
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Formula:
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f*_fractional = fraction * f*
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Where f* is the Kelly Criterion optimal fraction.
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:param win_probability: Probability of winning (0 < p < 1)
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:param win_loss_ratio: Ratio of win amount to loss amount (b > 0)
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:param fraction: Fraction of Kelly to use (0 < fraction <= 1), default 0.5
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:return: Fractional Kelly bet size
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>>> round(fractional_kelly(0.6, 2.0, 0.5), 4)
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0.2
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>>> round(fractional_kelly(0.55, 1.0, 0.25), 4)
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0.025
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>>> round(fractional_kelly(0.7, 3.0, 1.0), 4)
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0.6
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>>> fractional_kelly(0.6, 2.0, 0.0)
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Traceback (most recent call last):
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...
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ValueError: fraction must be between 0 and 1 (exclusive for 0, inclusive for 1)
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>>> fractional_kelly(0.6, 2.0, 1.5)
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Traceback (most recent call last):
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...
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ValueError: fraction must be between 0 and 1 (exclusive for 0, inclusive for 1)
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>>> fractional_kelly(0.0, 2.0, 0.5)
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Traceback (most recent call last):
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...
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ValueError: win_probability must be between 0 and 1 (exclusive)
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"""
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if fraction <= 0 or fraction > 1:
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raise ValueError(
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"fraction must be between 0 and 1 (exclusive for 0, inclusive for 1)"
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)
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full_kelly = kelly_criterion(win_probability, win_loss_ratio)
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return fraction * full_kelly
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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# Example: A bet with 60% win probability and 2:1 odds
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win_prob = 0.6
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odds = 2.0
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full_kelly = kelly_criterion(win_prob, odds)
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half_kelly = fractional_kelly(win_prob, odds, 0.5)
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print(f"Win probability: {win_prob}")
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print(f"Win/loss ratio: {odds}")
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print(f"Full Kelly fraction: {full_kelly:.2%}")
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print(f"Half Kelly fraction: {half_kelly:.2%}")

‎financial/sharpe_ratio.py‎

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"""
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Sharpe Ratio for measuring risk-adjusted returns in investment portfolios.
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The Sharpe Ratio is a measure of risk-adjusted return developed by Nobel laureate
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William F. Sharpe. It calculates the excess return per unit of risk (standard deviation)
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and is widely used to compare the performance of investment portfolios.
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Wikipedia Reference: https://en.wikipedia.org/wiki/Sharpe_ratio
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Investopedia: https://www.investopedia.com/terms/s/sharperatio.asp
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The Sharpe Ratio is used for:
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- Comparing performance of different investment strategies
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- Evaluating mutual funds and hedge funds
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- Portfolio optimization and risk management
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- Assessing risk-adjusted returns in trading strategies
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"""
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from __future__ import annotations
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def sharpe_ratio(returns: list[float], risk_free_rate: float = 0.0) -> float:
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"""
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Calculate the Sharpe Ratio for a series of returns.
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The Sharpe Ratio formula:
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S = (R - Rf) / σ
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Where:
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S = Sharpe Ratio
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R = Average return of the investment
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Rf = Risk-free rate of return
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σ = Standard deviation of returns (volatility)
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:param returns: List of periodic returns (e.g., daily, monthly)
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:param risk_free_rate: Risk-free rate of return per period, default 0.0
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:return: Sharpe Ratio
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>>> round(sharpe_ratio([0.1, 0.2, 0.15, 0.05, 0.12]), 4)
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2.2164
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>>> sharpe_ratio([0.05, 0.05, 0.05, 0.05, 0.05])
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inf
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>>> round(sharpe_ratio([0.1, 0.2, 0.15, 0.05, 0.12], 0.02), 4)
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1.8589
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>>> sharpe_ratio([0.0, 0.0, 0.0, 0.0, 0.0])
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0.0
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>>> round(sharpe_ratio([-0.05, -0.1, -0.08, -0.12, -0.15]), 4)
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-2.6261
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>>> sharpe_ratio([])
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Traceback (most recent call last):
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...
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ValueError: returns list must not be empty
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>>> sharpe_ratio([0.1])
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Traceback (most recent call last):
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...
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ValueError: returns list must contain at least 2 values
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"""
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if not returns:
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raise ValueError("returns list must not be empty")
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if len(returns) < 2:
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raise ValueError("returns list must contain at least 2 values")
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# Calculate mean return
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mean_return = sum(returns) / len(returns)
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# Calculate excess return
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excess_return = mean_return - risk_free_rate
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# Calculate standard deviation (using sample standard deviation with n-1)
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variance = sum((r - mean_return) ** 2 for r in returns) / (len(returns) - 1)
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std_dev = variance**0.5
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# Handle zero volatility case
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if std_dev == 0:
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return float("inf") if excess_return > 0 else 0.0
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return excess_return / std_dev
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def annualized_sharpe_ratio(
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returns: list[float], risk_free_rate: float = 0.0, periods_per_year: int = 252
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) -> float:
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"""
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Calculate the annualized Sharpe Ratio for a series of periodic returns.
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The annualized Sharpe Ratio accounts for the time period of returns:
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S_annual = S_periodic * sqrt(periods_per_year)
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Common periods_per_year values:
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- Daily returns: 252 (trading days)
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- Weekly returns: 52
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- Monthly returns: 12
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- Quarterly returns: 4
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:param returns: List of periodic returns
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:param risk_free_rate: Risk-free rate per period, default 0.0
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:param periods_per_year: Number of periods in a year, default 252 (daily)
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:return: Annualized Sharpe Ratio
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>>> round(annualized_sharpe_ratio([0.001, 0.002, 0.0015, 0.0005, 0.0012], 0.0, 252), 4)
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35.1844
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>>> round(annualized_sharpe_ratio([0.01, 0.02, 0.015, 0.005, 0.012], 0.0, 12), 4)
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7.6779
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>>> round(annualized_sharpe_ratio([0.05, 0.06, 0.055, 0.045, 0.052], 0.0, 4), 4)
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18.7322
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>>> round(annualized_sharpe_ratio([0.001, 0.002, 0.0015], 0.0001, 252), 4)
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44.4486
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>>> round(annualized_sharpe_ratio([0.001, 0.002], 0.0, 252), 4)
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33.6749
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>>> annualized_sharpe_ratio([0.001, 0.002, 0.0015], 0.0, 0)
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Traceback (most recent call last):
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...
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ValueError: periods_per_year must be > 0
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>>> annualized_sharpe_ratio([0.001, 0.002, 0.0015], 0.0, -252)
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Traceback (most recent call last):
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...
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ValueError: periods_per_year must be > 0
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"""
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if periods_per_year <= 0:
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raise ValueError("periods_per_year must be > 0")
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periodic_sharpe = sharpe_ratio(returns, risk_free_rate)
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# Annualize by multiplying by square root of periods
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if periodic_sharpe == float("inf"):
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return float("inf")
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return periodic_sharpe * (periods_per_year**0.5)
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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# Example: Calculate Sharpe Ratio for a series of monthly returns
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monthly_returns = [0.02, 0.03, -0.01, 0.04, 0.01, 0.02, -0.02, 0.03, 0.02, 0.01]
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risk_free = 0.002 # 0.2% monthly risk-free rate
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sharpe = sharpe_ratio(monthly_returns, risk_free)
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annualized = annualized_sharpe_ratio(monthly_returns, risk_free, 12)
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print(f"Monthly returns: {monthly_returns}")
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print(f"Risk-free rate: {risk_free:.2%}")
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print(f"Sharpe Ratio: {sharpe:.4f}")
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print(f"Annualized Sharpe Ratio: {annualized:.4f}")

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