skfolio.optimization.MaximumDiversification#

class skfolio.optimization.MaximumDiversification(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, risk_free_rate=0.0, min_return=None, max_tracking_error=None, max_turnover=None, solver='CLARABEL', solver_params=None, scale_objective=None, scale_constraints=None, save_problem=False, raise_on_failure=True, add_objective=None, add_constraints=None, portfolio_params=None)[source]#

Maximum Diversification Optimization estimator.

Maximizes the diversification ratio which is the ratio of the weighted volatilities over the total volatility.

It is a special case of the MeanRisk estimator where the expected return from the objective function is replaced by the weighted volatilities.

Parameters:
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.

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 greater than 70%

  • “Equity <= 3 * Bond” –> the sum of all Equity weights must be less or 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.

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

Lower bound constraint on the expected return.

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

Lower bound constraint on the expected return.

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.

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.

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 Maximum Diversification 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 Maximum Diversification 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.

**fit_paramsdict

Parameters to pass to the underlying estimators. Only available if enable_metadata_routing=True, which can be set by using sklearn.set_config(enable_metadata_routing=True). See Metadata Routing User Guide for more details.

Returns:
selfMaximumDiversification

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()#

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.