Backtesting

Performance statistics, overfitting detection, strategy risk analysis, bet sizing, and synthetic data generation (AFML Ch. 10, 14–15).

backtesting

avg_active_signals

avg_active_signals(signals, num_bars)

Average active signals at each bar.

Computes the mean signal strength from all active positions at each point.

Parameters:

Name	Type	Description	Default
signals	list[tuple[int, int, float]]	List of (start_idx, end_idx, signal_value) triples.	required
num_bars	int	Total number of bars.	required

Returns:

Type	Description
ndarray	Average signal at each bar.

avg_holding_period

avg_holding_period(entry_exit_pairs)

Average holding period from entry-exit pairs.

Parameters:

Name	Type	Description	Default
entry_exit_pairs	list[tuple[int, int]]	List of (entry_idx, exit_idx) pairs.	required

Returns:

Type	Description
float	Mean holding period in bars.

bonferroni_correction

bonferroni_correction(p_values)

Bonferroni correction for multiple hypothesis testing.

Parameters:

Name	Type	Description	Default
p_values	ndarray	Uncorrected p-values.	required

Returns:

Type	Description
ndarray	Corrected p-values (clamped to [0, 1]).

compute_drawdowns

compute_drawdowns(returns)

Compute drawdown statistics from a return series.

Parameters:

Name	Type	Description	Default
returns	ndarray	Periodic return series.	required

Returns:

Type	Description
DrawdownResult	Contains max drawdown, max duration, drawdown series, and time-under-water series.

cscv

cscv(returns_matrix, num_groups)

Combinatorially Symmetric Cross-Validation (CSCV).

Parameters:

Name	Type	Description	Default
returns_matrix	ndarray	Matrix of shape (n_periods, n_strategies).	required
num_groups	int	Number of groups for combinatorial splitting.	required

Returns:

Type	Description
CscvResult	PBO and rank-logit vector.

deflated_sharpe_ratio

deflated_sharpe_ratio(observed_sr, sr_std, n_observations, n_trials, skewness, kurtosis)

Deflated Sharpe Ratio — adjusts for multiple testing (AFML Ch. 14).

Corrects the observed SR for the number of strategies tested, accounting for higher moments.

Parameters:

Name	Type	Description	Default
observed_sr	float	Best observed Sharpe ratio.	required
sr_std	float	Standard deviation of Sharpe ratios across all trials.	required
n_observations	int	Number of return observations.	required
n_trials	int	Number of strategies/configurations tested.	required
skewness	float	Skewness of returns.	required
kurtosis	float	Excess kurtosis of returns.	required

Returns:

Type	Description
float	Deflated Sharpe ratio (probability-adjusted).

discrete_signal

discrete_signal(signal, step_size)

Discretize a continuous signal to the nearest step.

Parameters:

Name	Type	Description	Default
signal	float	Continuous signal value.	required
step_size	float	Discretization step (e.g. 0.1 for 10% increments).	required

Returns:

Type	Description
float	Rounded signal.

estimate_ou_params

estimate_ou_params(series, dt)

Estimate O-U process parameters from a time series via OLS.

Parameters:

Name	Type	Description	Default
series	ndarray	Observed time series.	required
dt	float	Time step between observations.	required

Returns:

Type	Description
tuple[float, float, float]	(theta, mu, sigma) — mean-reversion speed, long-term mean, volatility.

hhi

hhi(weights)

Herfindahl-Hirschman Index measuring concentration.

Parameters:

Name	Type	Description	Default
weights	ndarray	Weight or share vector (should sum to 1).	required

Returns:

Type	Description
float	HHI value. Ranges from 1/n (perfectly diversified) to 1 (fully concentrated).

hhi_concentration

hhi_concentration(returns)

Concentration of positive and negative returns using HHI.

Parameters:

Name	Type	Description	Default
returns	ndarray	Return series.	required

Returns:

Type	Description
tuple[float, float]	(positive_hhi, negative_hhi) — concentration in gains and losses.

hit_ratio

hit_ratio(returns)

Fraction of positive returns (hit rate / win rate).

Parameters:

Name	Type	Description	Default
returns	ndarray	Return series.	required

Returns:

Type	Description
float	Proportion of returns > 0.

holm_correction

holm_correction(p_values)

Holm-Bonferroni step-down correction for multiple testing.

Less conservative than Bonferroni while still controlling the family-wise error rate.

Parameters:

Name	Type	Description	Default
p_values	ndarray	Uncorrected p-values.	required

Returns:

Type	Description
ndarray	Corrected p-values.

implied_frequency

implied_frequency(target_sr, precision, avg_win_loss_ratio)

Implied trading frequency needed to achieve a target Sharpe ratio.

Parameters:

Name	Type	Description	Default
target_sr	float	Desired annualized Sharpe ratio.	required
precision	float	Hit rate.	required
avg_win_loss_ratio	float	Average win/loss ratio.	required

Returns:

Type	Description
float	Required trades per year.

implied_precision

implied_precision(target_sr, freq, avg_win_loss_ratio)

Implied precision needed to achieve a target Sharpe ratio.

Parameters:

Name	Type	Description	Default
target_sr	float	Desired annualized Sharpe ratio.	required
freq	float	Average trades per year.	required
avg_win_loss_ratio	float	Average win/loss ratio.	required

Returns:

Type	Description
float	Required hit rate (precision).

otr_mesh

otr_mesh(theta, mu, sigma, pt_range, sl_range, n_simulations, seed)

Compute an optimal trading rule (OTR) mesh over profit-take and stop-loss ranges.

Simulates O-U processes and evaluates average returns for each (profit_take, stop_loss) combination.

Parameters:

Name	Type	Description	Default
theta	float	Mean-reversion speed.	required
mu	float	Long-term mean.	required
sigma	float	Volatility.	required
pt_range	ndarray	Profit-take thresholds to evaluate.	required
sl_range	ndarray	Stop-loss thresholds to evaluate.	required
n_simulations	int	Monte Carlo simulations per grid point.	required
seed	int	Random seed.	required

Returns:

Type	Description
ndarray	2-D array of shape (len(pt_range), len(sl_range)) with average returns.

power_bet_size

power_bet_size(prob, num_classes, exponent)

Power-law bet sizing from prediction probability.

Parameters:

Name	Type	Description	Default
prob	float	Predicted probability.	required
num_classes	int	Number of classes.	required
exponent	float	Power exponent (higher = more aggressive sizing).	required

Returns:

Type	Description
float	Bet size in [-1, 1].

probabilistic_sharpe_ratio

probabilistic_sharpe_ratio(observed_sr, benchmark_sr, n_observations, skewness, kurtosis)

Probabilistic Sharpe Ratio (PSR) — probability that the true SR exceeds a benchmark.

Accounts for skewness and kurtosis of the return distribution (AFML Ch. 14).

Parameters:

Name	Type	Description	Default
observed_sr	float	Observed (sample) Sharpe ratio.	required
benchmark_sr	float	Benchmark Sharpe ratio to compare against.	required
n_observations	int	Number of return observations.	required
skewness	float	Skewness of returns.	required
kurtosis	float	Excess kurtosis of returns.	required

Returns:

Type	Description
float	Probability (0 to 1) that the true SR exceeds the benchmark.

probability_of_backtest_overfitting

probability_of_backtest_overfitting(returns_matrix, num_partitions, seed)

Probability of Backtest Overfitting (PBO) via CSCV (AFML Ch. 14).

Estimates the probability that the best in-sample strategy will underperform out-of-sample using combinatorial data splitting.

Parameters:

Name	Type	Description	Default
returns_matrix	ndarray	Matrix of shape (n_periods, n_strategies) with period returns.	required
num_partitions	int	Number of partitions for CSCV (must be even).	required
seed	int	Random seed for partition shuffling.	required

Returns:

Type	Description
float	PBO estimate in [0, 1]. Higher means more overfitting risk.

sharpe_mesh

sharpe_mesh(returns_grid)

Compute Sharpe ratios over a grid of return series.

Parameters:

Name	Type	Description	Default
returns_grid	ndarray	2-D array where each column is a return series.	required

Returns:

Type	Description
ndarray	Sharpe ratio for each column, reshaped to match the input grid.

sharpe_ratio

sharpe_ratio(returns, risk_free_rate=0.0, periods_per_year=252.0)

Annualized Sharpe ratio.

Parameters:

Name	Type	Description	Default
returns	ndarray	Periodic return series.	required
risk_free_rate	float	Risk-free rate per period.	0.0
periods_per_year	float	Annualization factor (252 for daily, 12 for monthly).	252.0

Returns:

Type	Description
float	Annualized Sharpe ratio.

sigmoid_bet_size

sigmoid_bet_size(prob, num_classes)

Sigmoid bet sizing from prediction probability (AFML Ch. 10).

Maps a probability to a position size using a sigmoid function.

Parameters:

Name	Type	Description	Default
prob	float	Predicted probability of the positive class.	required
num_classes	int	Number of discrete classes (usually 2).	required

Returns:

Type	Description
float	Bet size in [-1, 1].

simulate_ou

simulate_ou(theta, mu, sigma, x0, dt, n_steps, seed)

Simulate an Ornstein-Uhlenbeck mean-reverting process.

dX = theta * (mu - X) * dt + sigma * dW

Parameters:

Name	Type	Description	Default
theta	float	Mean-reversion speed.	required
mu	float	Long-term mean.	required
sigma	float	Volatility.	required
x0	float	Initial value.	required
dt	float	Time step.	required
n_steps	int	Number of simulation steps.	required
seed	int	Random seed.	required

Returns:

Type	Description
ndarray	Simulated path of length n_steps + 1 (including x0).

sr_from_precision

sr_from_precision(precision, freq, avg_win_loss_ratio)

Compute Sharpe ratio from precision, frequency, and win/loss ratio.

Parameters:

Name	Type	Description	Default
precision	float	Hit rate (probability of a winning trade).	required
freq	float	Average number of trades per year.	required
avg_win_loss_ratio	float	Average win size divided by average loss size.	required

Returns:

Type	Description
float	Expected annualized Sharpe ratio.

strategy_failure_probability

strategy_failure_probability(estimated_precision, n_observations, break_even_precision)

Probability that a strategy will fail (precision below break-even).

Uses a binomial test to estimate the probability that the true precision is below the break-even threshold.

Parameters:

Name	Type	Description	Default
estimated_precision	float	Observed hit rate.	required
n_observations	int	Number of trades observed.	required
break_even_precision	float	Minimum precision needed to break even.	required

Returns:

Type	Description
float	Failure probability.

turnover

turnover(positions)

Portfolio turnover from a position series.

Parameters:

Name	Type	Description	Default
positions	ndarray	Position sizes over time.	required

Returns:

Type	Description
float	Average absolute change in position per period.