Optimizations Reference

Portfolio Optimization Module

This module provides various portfolio optimization methods including Mean-Variance Optimization, Minimum Variance Portfolio, Maximum Sharpe Ratio, Hierarchical Risk Parity, Black-Litterman, Kelly Criterion, and other common portfolio optimization techniques.

`black_litterman(returns, market_caps, views, view_confidences, tau=0.05, risk_free_rate=0.0)`

Implement Black-Litterman portfolio optimization.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required
`market_caps`	`array - like`	Array of market capitalizations for each asset	required
`views`	`dict`	Dictionary mapping asset indices to expected returns	required
`view_confidences`	`dict`	Dictionary mapping asset indices to confidence levels (0-1)	required
`tau`	`float`	Uncertainty in the prior distribution	`0.05`
`risk_free_rate`	`float`	Risk-free rate	`0.0`

Returns:

Type	Description
`tuple`	(optimal_weights, portfolio_return, portfolio_risk)

`equal_weight_portfolio(returns)`

Create an equal-weighted portfolio.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required

Returns:

Type	Description
`tuple`	(weights, portfolio_return, portfolio_risk)

`hierarchical_risk_parity(returns, linkage_method='single', distance_metric='euclidean')`

Implement Hierarchical Risk Parity (HRP) portfolio optimization.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required
`linkage_method`	`str`	Linkage method for hierarchical clustering ('single', 'complete', 'average', 'ward')	`'single'`
`distance_metric`	`str`	Distance metric for clustering ('euclidean', 'correlation')	`'euclidean'`

Returns:

Type	Description
`tuple`	(optimal_weights, portfolio_return, portfolio_risk)

`kelly_criterion_optimization(returns, risk_free_rate=0.0, constraints=None, kelly_fraction=1.0)`

Implement Kelly Criterion portfolio optimization.

The Kelly Criterion determines the optimal fraction of capital to allocate to each investment to maximize long-term growth rate.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required
`risk_free_rate`	`float`	Risk-free rate	`0.0`
`constraints`	`list of dict`	List of constraints for the optimization problem	`None`
`kelly_fraction`	`float`	Fraction of Kelly Criterion to use (e.g., 0.5 for half-Kelly)	`1.0`

Returns:

Type	Description
`tuple`	(optimal_weights, portfolio_return, portfolio_risk)

Notes

The Kelly Criterion maximizes the expected logarithmic growth rate of wealth. The full Kelly Criterion can be aggressive, so practitioners often use a fraction of the Kelly Criterion (e.g., half-Kelly) for more conservative position sizing.

`market_cap_weight_portfolio(market_caps)`

Create a market-cap weighted portfolio.

Parameters:

Name	Type	Description	Default
`market_caps`	`array - like`	Array of market capitalizations for each asset	required

Returns:

Type	Description
`ndarray`	Market-cap weighted portfolio weights

`maximum_diversification_portfolio(returns, constraints=None)`

Find the portfolio that maximizes diversification ratio.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required
`constraints`	`list of dict`	List of constraints for the optimization problem	`None`

Returns:

Type	Description
`tuple`	(optimal_weights, portfolio_return, portfolio_risk)

`maximum_sharpe_ratio(returns, risk_free_rate=0.0, constraints=None)`

Find the portfolio with maximum Sharpe ratio.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required
`risk_free_rate`	`float`	Risk-free rate for Sharpe ratio calculation	`0.0`
`constraints`	`list of dict`	List of constraints for the optimization problem	`None`

Returns:

Type	Description
`tuple`	(optimal_weights, portfolio_return, portfolio_risk)

`mean_variance_optimization(returns, target_return=None, risk_free_rate=0.0, constraints=None)`

Perform Mean-Variance Optimization to find optimal portfolio weights.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required
`target_return`	`float`	Target portfolio return. If None, maximizes Sharpe ratio	`None`
`risk_free_rate`	`float`	Risk-free rate for Sharpe ratio calculation	`0.0`
`constraints`	`list of dict`	List of constraints for the optimization problem	`None`

Returns:

Type	Description
`tuple`	(optimal_weights, portfolio_return, portfolio_risk)

`minimum_variance_portfolio(returns, constraints=None)`

Find the portfolio with minimum variance.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required
`constraints`	`list of dict`	List of constraints for the optimization problem	`None`

Returns:

Type	Description
`tuple`	(optimal_weights, portfolio_return, portfolio_risk)

`risk_parity_portfolio(returns, constraints=None)`

Find the portfolio where risk is equally distributed across assets.

Parameters:

Name	Type	Description	Default
`returns`	`array - like`	Matrix of asset returns where each column represents an asset	required
`constraints`	`list of dict`	List of constraints for the optimization problem	`None`

Returns:

Type	Description
`tuple`	(optimal_weights, portfolio_return, portfolio_risk)