skfolio.optimization.MeanRisk#

class skfolio.optimization.MeanRisk(objective_function=MINIMIZE_RISK, risk_measure=Variance, risk_aversion=1.0, efficient_frontier_size=None, prior_estimator=None, min_weights=0.0, max_weights=1.0, budget=1.0, min_budget=None, max_budget=None, max_short=None, max_long=None, transaction_costs=0.0, management_fees=0.0, previous_weights=None, groups=None, linear_constraints=None, left_inequality=None, right_inequality=None, l1_coef=0.0, l2_coef=0.0, mu_uncertainty_set_estimator=None, covariance_uncertainty_set_estimator=None, risk_free_rate=0.0, min_return=None, max_tracking_error=None, max_turnover=None, max_mean_absolute_deviation=None, max_first_lower_partial_moment=None, max_variance=None, max_standard_deviation=None, max_semi_variance=None, max_semi_deviation=None, max_worst_realization=None, max_cvar=None, max_evar=None, max_max_drawdown=None, max_average_drawdown=None, max_cdar=None, max_edar=None, max_ulcer_index=None, max_gini_mean_difference=None, min_acceptable_return=None, cvar_beta=0.95, evar_beta=0.95, cdar_beta=0.95, edar_beta=0.95, solver='CLARABEL', solver_params=None, scale_objective=None, scale_constraints=None, save_problem=False, raise_on_failure=True, add_objective=None, add_constraints=None, overwrite_expected_return=None, portfolio_params=None)[source]#

Mean-Risk Optimization estimator.

The below 4 objective functions can be optimized:

  • Minimize Risk:

\[\begin{split}\begin{cases} \begin{aligned} &\min_{w} & & risk_{i}(w) \\ &\text{s.t.} & & w^T \cdot \mu \ge min\_return \\ & & & A \cdot w \ge b \\ & & & risk_{j}(w) \le max\_risk_{j} \quad \forall \; j \ne i \end{aligned} \end{cases}\end{split}\]

  • Maximize Expected Return:

\[\begin{split}\begin{cases} \begin{aligned} &\max_{w} & & w^T \cdot \mu \\ &\text{s.t.} & & risk_{i}(w) \le max\_risk_{i} \\ & & & A \cdot w \ge b \\ & & & risk_{j}(w) \le max\_risk_{j} \quad \forall \; j \ne i \end{aligned} \end{cases}\end{split}\]

  • Maximize Utility:

\[\begin{split}\begin{cases} \begin{aligned} &\max_{w} & & w^T \cdot \mu - \lambda \times risk_{i}(w)\\ &\text{s.t.} & & risk_{i}(w) \le max\_risk_{i} \\ & & & w^T \cdot \mu \ge min\_return \\ & & & A \cdot w \ge b \\ & & & risk_{j}(w) \le max\_risk_{j} \quad \forall \; j \ne i \end{aligned} \end{cases}\end{split}\]

  • Maximize Ratio:

\[\begin{split}\begin{cases} \begin{aligned} &\max_{w} & & \frac{w^T \cdot \mu - r_{f}}{risk_{i}(w)}\\ &\text{s.t.} & & risk_{i}(w) \le max\_risk_{i} \\ & & & w^T \cdot \mu \ge min\_return \\ & & & A \cdot w \ge b \\ & & & risk_{j}(w) \le max\_risk_{j} \quad \forall \; j \ne i \end{aligned} \end{cases}\end{split}\]

With \(risk_{i}\) a risk measure among:

  • Mean Absolute Deviation

  • First Lower Partial Moment

  • Variance

  • Semi-Variance

  • CVaR (Conditional Value at Risk)

  • EVaR (Entropic Value at Risk)

  • Worst Realization (worst return)

  • CDaR (Conditional Drawdown at Risk)

  • Maximum Drawdown

  • Average Drawdown

  • EDaR (Entropic Drawdown at Risk)

  • Ulcer Index

  • Gini Mean Difference

Cost, regularization, uncertainty set, and additional constraints can also be added to the optimization problem (see the parameters description).

The assets expected returns, covariance matrix and returns are estimated from the prior estimator.

Parameters:
objective_functionObjectiveFunction, default=ObjectiveFunction.MINIMIZE_RISK

ObjectiveFunction of the optimization. Can be any of:

  • MINIMIZE_RISK

  • MAXIMIZE_RETURN

  • MAXIMIZE_UTILITY

  • MAXIMIZE_RATIO

The default is ObjectiveFunction.MINIMIZE_RISK.

risk_measureRiskMeasure, default=RiskMeasure.VARIANCE

RiskMeasure of the optimization. Can be any of:

  • VARIANCE

  • SEMI_VARIANCE

  • STANDARD_DEVIATION

  • SEMI_DEVIATION

  • MEAN_ABSOLUTE_DEVIATION

  • FIRST_LOWER_PARTIAL_MOMENT

  • CVAR

  • EVAR

  • WORST_REALIZATION

  • CDAR

  • MAX_DRAWDOWN

  • AVERAGE_DRAWDOWN

  • EDAR

  • ULCER_INDEX

  • GINI_MEAN_DIFFERENCE_RATIO

The default is RiskMeasure.VARIANCE.

risk_aversionfloat, default=1.0

Risk aversion factor \(\lambda\) of the utility function. Only used for objective_function=ObjectiveFunction.MAXIMIZE_UTILITY. The default value is 1.0.

prior_estimatorBasePrior, optional

Prior estimator. The prior estimator is used to estimate the PriorModel containing the estimation of assets expected returns, covariance matrix, returns and Cholesky decomposition of the covariance. The default (None) is to use EmpiricalPrior.

min_weightsfloat | dict[str, float] | array-like of shape (n_assets, ) | None, default=0.0

Minimum assets weights (weights lower bounds). If a float is provided, it is applied to each asset. None is equivalent to -np.Inf (no lower bound). If a dictionary is provided, its (key/value) pair must be the (asset name/asset minium weight) and the input X of the fit method must be a DataFrame with the assets names in columns. When using a dictionary, assets values that are not provided are assigned a minimum weight of 0.0. The default value is 0.0 (no short selling).

Example:

  • min_weights = 0 –> long only portfolio (no short selling).

  • min_weights = None –> no lower bound (same as -np.Inf).

  • min_weights = -2 –> each weight must be above -200%.

  • min_weights = {"SX5E": 0, "SPX": -2}

  • min_weights = [0, -2]

max_weightsfloat | dict[str, float] | array-like of shape (n_assets, ) | None, default=1.0

Maximum assets weights (weights upper bounds). If a float is provided, it is applied to each asset. None is equivalent to +np.Inf (no upper bound). If a dictionary is provided, its (key/value) pair must be the (asset name/asset maximum weight) and the input X of the fit method must be a DataFrame with the assets names in columns. When using a dictionary, assets values that are not provided are assigned a minimum weight of 1.0. The default value is 1.0 (each asset is below 100%).

Example:

  • max_weights = 0 –> no long position (short only portfolio).

  • max_weights = None –> no upper bound.

  • max_weights = 2 –> each weight must be below 200%.

  • max_weights = {"SX5E": 1, "SPX": 2}

  • max_weights = [1, 2]

budgetfloat | None, default=1.0

Investment budget. It is the sum of long positions and short positions (sum of all weights). None means no budget constraints. The default value is 1.0 (fully invested portfolio).

Examples:

  • budget = 1 –> fully invested portfolio.

  • budget = 0 –> market neutral portfolio.

  • budget = None –> no constraints on the sum of weights.

min_budgetfloat, optional

Minimum budget. It is the lower bound of the sum of long and short positions (sum of all weights). If provided, you must set budget=None. The default (None) means no minimum budget constraint.

max_budgetfloat, optional

Maximum budget. It is the upper bound of the sum of long and short positions (sum of all weights). If provided, you must set budget=None. The default (None) means no maximum budget constraint.

max_shortfloat, optional

Maximum short position. The short position is defined as the sum of negative weights (in absolute term). The default (None) means no maximum short position.

max_longfloat, optional

Maximum long position. The long position is defined as the sum of positive weights. The default (None) means no maximum long position.

transaction_costsfloat | dict[str, float] | array-like of shape (n_assets, ), default=0.0

Transaction costs of the assets. It is used to add linear transaction costs to the optimization problem:

\[total\_cost = \sum_{i=1}^{N} c_{i} \times |w_{i} - w\_prev_{i}|\]

with \(c_{i}\) the transaction cost of asset i, \(w_{i}\) its weight and \(w\_prev_{i}\) its previous weight (defined in previous_weights). The float \(total\_cost\) is impacting the portfolio expected return in the optimization:

\[expected\_return = \mu^{T} \cdot w - total\_cost\]

with \(\mu\) the vector af assets’ expected returns and \(w\) the vector of assets weights.

If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset cost) and the input X of the fit method must be a DataFrame with the assets names in columns. The default value is 0.0.

Warning

Based on the above formula, the periodicity of the transaction costs needs to be homogenous to the periodicity of \(\mu\). For example, if the input X is composed of daily returns, the transaction_costs need to be expressed as daily costs. (See Transaction Costs)

management_feesfloat | dict[str, float] | array-like of shape (n_assets, ), default=0.0

Management fees of the assets. It is used to add linear management fees to the optimization problem:

\[total\_fee = \sum_{i=1}^{N} f_{i} \times w_{i}\]

with \(f_{i}\) the management fee of asset i and \(w_{i}\) its weight. The float \(total\_fee\) is impacting the portfolio expected return in the optimization:

\[expected\_return = \mu^{T} \cdot w - total\_fee\]

with \(\mu\) the vector af assets expected returns and \(w\) the vector of assets weights.

If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset fee) and the input X of the fit method must be a DataFrame with the assets names in columns. The default value is 0.0.

Warning

Based on the above formula, the periodicity of the management fees needs to be homogenous to the periodicity of \(\mu\). For example, if the input X is composed of daily returns, the management_fees need to be expressed in daily fees.

Note

Another approach is to directly impact the management fees to the input X in order to express the returns net of fees. However, when estimating the \(\mu\) parameter using for example Shrinkage estimators, this approach would mix a deterministic value with an uncertain one leading to unwanted bias in the management fees.

previous_weightsfloat | dict[str, float] | array-like of shape (n_assets, ), optional

Previous weights of the assets. Previous weights are used to compute the portfolio cost and the portfolio turnover. If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset previous weight) and the input X of the fit method must be a DataFrame with the assets names in columns. The default (None) means no previous weights.

l1_coeffloat, default=0.0

L1 regularization coefficient. It is used to penalize the objective function by the L1 norm:

\[l1\_coef \times \Vert w \Vert_{1} = l1\_coef \times \sum_{i=1}^{N} |w_{i}|\]

Increasing this coefficient will reduce the number of non-zero weights (cardinality). It tends to increase robustness (out-of-sample stability) but reduces diversification. The default value is 0.0.

l2_coeffloat, default=0.0

L2 regularization coefficient. It is used to penalize the objective function by the L2 norm:

\[l2\_coef \times \Vert w \Vert_{2}^{2} = l2\_coef \times \sum_{i=1}^{N} w_{i}^2\]

It tends to increase robustness (out-of-sample stability). The default value is 0.0.

mu_uncertainty_set_estimatorBaseMuUncertaintySet, optional

Mu Uncertainty set estimator. If provided, the assets expected returns are modelled with an ellipsoidal uncertainty set. It is called worst-case optimization and is a class of robust optimization. It reduces the instability that arises from the estimation errors of the expected returns. The worst case portfolio expect return is:

\[w^T \cdot \hat{\mu} - \kappa_{\mu} \lVert S_{\mu}^\frac{1}{2} \cdot w \rVert_{2}\]

with \(\kappa\) the size of the ellipsoid (confidence region) and \(S\) its shape. The default (None) means that no uncertainty set is used.

covariance_uncertainty_set_estimatorBaseCovarianceUncertaintySet, optional

Covariance Uncertainty set estimator. If provided, the assets covariance matrix is modelled with an ellipsoidal uncertainty set. It is called worst-case optimization and is a class of robust optimization. It reduces the instability that arises from the estimation errors of the covariance matrix. The default (None) means that no uncertainty set is used.

linear_constraintsarray-like of shape (n_constraints,), optional

Linear constraints. The linear constraints must match any of following patterns:

  • “2.5 * ref1 + 0.10 * ref2 + 0.0013 <= 2.5 * ref3”

  • “ref1 >= 2.9 * ref2”

  • “ref1 == ref2”

  • “ref1 >= ref1”

With “ref1”, “ref2” … the assets names or the groups names provided in the parameter groups. Assets names can be referenced without the need of groups if the input X of the fit method is a DataFrame with these assets names in columns.

Examples:

  • “SPX >= 0.10” –> SPX weight must be greater than 10% (note that you can also use min_weights)

  • “SX5E + TLT >= 0.2” –> the sum of SX5E and TLT weights must be greater than 20%

  • “US == 0.7” –> the sum of all US weights must be equal to 70%

  • “Equity == 3 * Bond” –> the sum of all Equity weights must be equal to 3 times the sum of all Bond weights.

  • “2*SPX + 3*Europe <= Bond + 0.05” –> mixing assets and group constraints

groupsdict[str, list[str]] or array-like of shape (n_groups, n_assets), optional

The assets groups referenced in linear_constraints. If a dictionary is provided, its (key/value) pair must be the (asset name/asset groups) and the input X of the fit method must be a DataFrame with the assets names in columns.

Examples:

  • groups = {“SX5E”: [“Equity”, “Europe”], “SPX”: [“Equity”, “US”], “TLT”: [“Bond”, “US”]}

  • groups = [[“Equity”, “Equity”, “Bond”], [“Europe”, “US”, “US”]]

left_inequalityarray-like of shape (n_constraints, n_assets), optional

Left inequality matrix \(A\) of the linear constraint \(A \cdot w \leq b\).

right_inequalityarray-like of shape (n_constraints, ), optional

Right inequality vector \(b\) of the linear constraint \(A \cdot w \leq b\).

risk_free_ratefloat, default=0.0

Risk-free interest rate. The default value is 0.0.

max_tracking_errorfloat, optional

Upper bound constraint on the tracking error. The tracking error is defined as the RMSE (root-mean-square error) of the portfolio returns compared to a target returns. If max_tracking_error is provided, the target returns y must be provided in the fit method.

max_turnoverfloat, optional

Upper bound constraint of the turnover. The turnover is defined as the absolute difference between the portfolio weights and the previous_weights. Note that another way to control for turnover is by using the transaction_costs parameter.

max_mean_absolute_deviationfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Mean Absolute Deviation.

max_first_lower_partial_momentfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the First Lower Partial Moment.

max_variancefloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Variance.

max_standard_deviationfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Standard deviation.

max_semi_variancefloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Semi-Variance (Second Lower Partial Moment or Downside Variance).

max_semi_deviationfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Semi-Standard deviation.

max_worst_realizationfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Worst Realization (Worst Return).

max_cvarfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the CVaR (Conditional Value-at-Risk or Expected Shortfall).

max_evarfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the EVaR (Entropic Value at Risk).

max_max_drawdownfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Maximum Drawdown.

max_average_drawdownfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Average Drawdown.

max_cdarfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the CDaR (Conditional Drawdown at Risk).

max_edarfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the EDaR (Entropic Drawdown at Risk).

max_ulcer_indexfloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Ulcer Index.

max_gini_mean_differencefloat | array-like of shape (n_optimization), optional

Upper bound constraint on the Gini Mean Difference.

min_returnfloat | array-like of shape (n_optimization), optional

Lower bound constraint on the expected return.

min_acceptable_returnfloat, optional

The minimum acceptable return used to distinguish “downside” and “upside” returns for the computation of lower partial moments:

  • First Lower Partial Moment

  • Semi-Variance

  • Semi-Deviation

The default (None) is to use the mean.

cvar_betafloat, default=0.95

CVaR (Conditional Value at Risk) confidence level. The default value is 0.95.

evar_betafloat, default=0

EVaR (Entropic Value at Risk) confidence level. The default value is 0.95.

cdar_betafloat, default=0.95

CDaR (Conditional Drawdown at Risk) confidence level. The default value is 0.95.

edar_betafloat, default=0.95

EDaR (Entropic Drawdown at Risk) confidence level. The default value is 0.95.

add_objectiveCallable[[cp.Variable], cp.Expression], optional

Add a custom objective to the existing objective expression. It is a function that must take as argument the weights w and returns a CVXPY expression.

add_constraintsCallable[[cp.Variable], cp.Expression|list[cp.Expression]], optional

Add a custom constraint or a list of constraints to the existing constraints. It is a function that must take as argument the weights w and returns a CVPXY expression or a list of CVPXY expressions.

overwrite_expected_returnCallable[[cp.Variable], cp.Expression], optional

Overwrite the expected return \(\mu \cdot w\) with a custom expression. It is a function that must take as argument the weights w and returns a CVPXY expression.

solverstr, default=”CLARABEL”

The solver to use. The default is “CLARABEL” which is written in Rust and has better numerical stability and performance than ECOS and SCS. Cvxpy will replace its default solver “ECOS” by “CLARABEL” in future releases. For more details about available solvers, check the CVXPY documentation: https://www.cvxpy.org/tutorial/advanced/index.html#choosing-a-solver

solver_paramsdict, optional

Solver parameters. For example, solver_params=dict(verbose=True). The default (None) is use {"tol_gap_abs": 1e-9, "tol_gap_rel": 1e-9} for the solver “CLARABEL” and the CVXPY default otherwise. For more details about solver arguments, check the CVXPY documentation: https://www.cvxpy.org/tutorial/advanced/index.html#setting-solver-options

scale_objectivefloat, optional

Scale each objective element by this value. It can be used to increase the optimization accuracies in specific cases. The default (None) is set depending on the problem.

scale_constraintsfloat, optional

Scale each constraint element by this value. It can be used to increase the optimization accuracies in specific cases. The default (None) is set depending on the problem.

save_problembool, default=False

If this is set to True, the CVXPY Problem is saved in problem_. The default is False.

raise_on_failurebool, default=True

If this is set to True, an error is raised when the optimization fail otherwise it passes with a warning.

portfolio_paramsdict, optional

Portfolio parameters passed to the portfolio evaluated by the predict and score methods. If not provided, the name, transaction_costs, management_fees, previous_weights and risk_free_rate are copied from the optimization model and passed to the portfolio.

Attributes:
weights_ndarray of shape (n_assets,) or (n_optimizations, n_assets)

Weights of the assets.

problem_values_dict[str, float] | list[dict[str, float]] of size n_optimizations

Expression values retrieved from the CVXPY problem.

prior_estimator_BasePrior

Fitted prior_estimator.

mu_uncertainty_set_estimator_BaseMuUncertaintySet

Fitted mu_uncertainty_set_estimator if provided.

covariance_uncertainty_set_estimator_BaseCovarianceUncertaintySet

Fitted covariance_uncertainty_set_estimator if provided.

problem_: cvxpy.Problem

CVXPY problem used for the optimization. Only when save_problem is set to True.

n_features_in_int

Number of assets seen during fit.

feature_names_in_ndarray of shape (n_features_in_,)

Names of assets seen during fit. Defined only when X has assets names that are all strings.

Methods

fit(X[, y])

Fit the Mean-Risk Optimization estimator.

fit_predict(X)

Perform fit on X and returns the predicted Portfolio or Population of Portfolio on X based on the fitted weights.

get_metadata_routing()

Get metadata routing of this object.

get_params([deep])

Get parameters for this estimator.

predict(X)

Predict the Portfolio or Population of Portfolio on X based on the fitted weights.

score(X[, y])

Prediction score.

set_params(**params)

Set the parameters of this estimator.
fit(X, y=None, **fit_params)[source]#

Fit the Mean-Risk Optimization estimator.

Parameters:
Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

yarray-like of shape (n_observations, n_targets), optional

Price returns of factors or a target benchmark. The default is None.

Returns:
selfMeanRisk

Fitted estimator.

fit_predict(X)#

Perform fit on X and returns the predicted Portfolio or Population of Portfolio on X based on the fitted weights. For factor models, use fit(X, y) then predict(X) separately.

Parameters:
Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

Returns:
predictionPortfolio | Population

Portfolio or Population of Portfolio estimated on X based on the fitted weights.

get_metadata_routing()[source]#

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:
routingMetadataRequest

A MetadataRequest encapsulating routing information.

get_params(deep=True)#

Get parameters for this estimator.

Parameters:
deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:
paramsdict

Parameter names mapped to their values.

predict(X)#

Predict the Portfolio or Population of Portfolio on X based on the fitted weights.

Optimization estimators can return a 1D or a 2D array of weights. For a 1D array, the prediction returns a Portfolio. For a 2D array, the prediction returns a Population of Portfolio.

If name is not provided in the portfolio arguments, we use the first 500 characters of the estimator name.

Parameters:
Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

Returns:
predictionPortfolio | Population

Portfolio or Population of Portfolio estimated on X based on the fitted weights.

score(X, y=None)#

Prediction score. If the prediction is a single Portfolio, the score is the Sharpe Ratio. If the prediction is a Population of Portfolio, the score is the mean of all the portfolios Sharpe Ratios in the population.

Parameters:
Xarray-like of shape (n_observations, n_assets)

Price returns of the assets.

yIgnored

Not used, present here for API consistency by convention.

Returns:
scorefloat

The Sharpe Ratio of the portfolio if the prediction is a single Portfolio or the mean of all the portfolios Sharpe Ratios if the prediction is a Population of Portfolio.

set_params(**params)#

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:
**paramsdict

Estimator parameters.

Returns:
selfestimator instance

Estimator instance.