skfolio.optimization.RiskBudgeting#

class skfolio.optimization.RiskBudgeting(risk_measure=Variance, risk_budget=None, prior_estimator=None, min_weights=0.0, max_weights=1.0, transaction_costs=0.0, management_fees=0.0, previous_weights=None, groups=None, linear_constraints=None, left_inequality=None, right_inequality=None, risk_free_rate=0.0, min_return=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, fallback=None)[source]#

Risk Budgeting Optimization estimator.

The Risk Budgeting estimator solves the below convex problem:

\[\begin{split}\begin{cases} \begin{aligned} & \min_{w,s} && \mathrm{Risk}(w) \\ & \text{s.t.} && budget^{\top}\log(w) \ge 0 \\ & && \mathbf{1}^{\top} w = s \\ & && expected\_return(w) \ge s\, min\_return \\ & && A w \le s\, b \\ & && w \ge 0 \end{aligned} \end{cases}\end{split}\]

with \(budget\) the risk budget vector and \(min\_return\) the minimum expected return constraint.

And \(Risk\) 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 and additional constraints can also be added to the optimization problem (see the parameters description).

Limitations are imposed on some constraints including long only weights to ensure convexity.

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

Parameters:
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

The default is RiskMeasure.VARIANCE.

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

Risk budget allocated to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset risk budget) and the input X of the fit method must be a DataFrame with the assets names in columns. The default (None) is to use the identity vector, reducing the risk budgeting to a risk-parity (each asset contributing equally to the total risk).

prior_estimatorBasePrior, optional

Prior estimator. The prior estimator is used to estimate the ReturnDistribution containing estimates of expected asset 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 minimum 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]

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 of 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 must match 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. A transaction cost is paid once per rebalancing while a position earns its expected return on every period it is held, so the one-off cost is converted by dividing it by the expected investment duration (e.g. 0.001 / 21 for a 10 bps cost with daily returns and a one-month expected holding period). (See Periodicity Convention)

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 of 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 must match 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. Unlike transaction costs, management fees accrue with holding time, so a stated annual fee converts directly to the return periodicity (e.g. 0.02 / 252 for a 2% annual fee on daily returns).

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. Additionally, when fallback="previous_weights", failures will fall back to these weights if provided.

linear_constraintsarray-like of shape (n_constraints,), optional

Linear constraints on portfolio weights or factor exposures.

Constraint names can reference:

  • Asset names: individual asset weights (e.g. "SPX", "AAPL")

  • Group names: sums of weights in groups defined by groups

  • Factor names: portfolio factor exposure (requires factor model prior)

  • Factor families: sum of portfolio exposures to all factors in one family

Supported equation patterns include:

  • "name <= value" or "name >= value"

  • "name == value"

  • "a * name1 + b * name2 <= c * name3 + d"

For example:

  • "SPX >= 0.10" –> SPX weight >= 10%

  • "SX5E + SPX >= 0.2" –> sum of SX5E and SPX weights >= 20%

  • "US == 0.7" –> sum of weights in US group == 70%

  • "Equity == 3 * Bond" –> sum of weights in Equity group == 3x sum of weights in Bond group

  • "Momentum <= 0.30" –> portfolio Momentum exposure <= 30%

  • "style <= 0.50" –> sum of all style factor exposures (Momentum, Value, Size, etc.) <= 50%

Factor constraints require a prior estimator (e.g. TimeSeriesFactorModel, CharacteristicsFactorModel) that provides loading_matrix, factor_names and optionally factor_families in its FactorModel.

Asset, group, factor, and factor family names must be unique.

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.

For example:

  • 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.

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=0.95

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 must be a function taking the CVXPY weight variable w as its first positional argument and, optionally, the estimator instance as its second. It must return a CVXPY expression or a list of CVXPY expressions, evaluated when fit is called.

For example, to require an effective number of assets of at least 20:

>>> import cvxpy as cp
>>> from skfolio.optimization import RiskBudgeting
>>> model = RiskBudgeting(add_constraints=lambda w: cp.sum_squares(w) <= 1 / 20)

The optional second argument gives access to the estimator’s attributes, including quantities estimated during fit. For example, to cap each position size in risk units at 20 bps, using the volatilities estimated by the prior:

>>> import numpy as np
>>> def position_risk_cap(w, model):
...     covariance = model.prior_estimator_.return_distribution_.covariance
...     vols = np.sqrt(np.diag(covariance))
...     return cp.multiply(vols, w) <= 0.002
>>> model = RiskBudgeting(add_constraints=position_risk_cap)
overwrite_expected_returnCallable[[cp.Variable], cp.Expression], optional

Overwrite the expected return \(\mu \cdot w\) with a custom CVXPY expression. It must be a function taking the CVXPY weight variable w as its first positional argument and, optionally, the estimator instance as its second. It must return a concave CVXPY expression, evaluated when fit is called. The custom expression replaces the expected return in the objective function and in the constraints where the expected return is used.

For example, to adjust the expected return for volatility drag, approximating the portfolio geometric mean return:

>>> import cvxpy as cp
>>> from skfolio.optimization import RiskBudgeting
>>> def geometric_expected_return(w, model):
...     dist = model.prior_estimator_.return_distribution_
...     return dist.mu @ w - 0.5 * cp.quad_form(w, dist.covariance)
>>> model = RiskBudgeting(overwrite_expected_return=geometric_expected_return)
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.

portfolio_paramsdict, optional

Portfolio parameters forwarded to the resulting Portfolio in predict. If not provided and if available on the estimator, the following attributes are propagated to the portfolio by default: name, transaction_costs, management_fees, previous_weights and risk_free_rate.

fallbackBaseOptimization | “previous_weights” | list[BaseOptimization | “previous_weights”], optional

Fallback estimator or a list of estimators to try, in order, when the primary optimization raises during fit. Alternatively, use "previous_weights" (alone or in a list) to fall back to the estimator’s previous_weights. When a fallback succeeds, its fitted weights_ are copied back to the primary estimator so that fit still returns the original instance. For traceability, fallback_ stores the successful estimator (or the string "previous_weights") and fallback_chain_ stores each attempt with the associated outcome.

raise_on_failurebool, default=True

Controls error handling when fitting fails. If True, any failure during fit is raised immediately, no weights_ are set and subsequent calls to predict will raise a NotFittedError. If False, errors are not raised; instead, a warning is emitted, weights_ is set to None and subsequent calls to predict will return a FailedPortfolio. When fallbacks are specified, this behavior applies only after all fallbacks have been exhausted.

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.

fallback_BaseOptimization | “previous_weights” | None

The fallback estimator instance, or the string "previous_weights", that produced the final result. None if no fallback was used.

fallback_chain_list[tuple[str, str]] | None

Sequence describing the optimization fallback attempts. Each element is a pair (estimator_repr, outcome) where estimator_repr is the string representation of the primary estimator or a fallback (e.g. "EqualWeighted()", "previous_weights"), and outcome is "success" if that step produced a valid solution, otherwise the stringified error message. For successful fits without any fallback, this is None.

error_str | list[str] | None

Captured error message(s) when fit fails. For multi-portfolio outputs (weights_ is 2D), this is a list aligned with portfolios.

Methods

fit(X[, y])

Fit the Risk Budgeting 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 a Population of portfolios on X.

score(X[, y])

Prediction score using the Sharpe Ratio.

set_params(**params)

Set the parameters of this estimator.

Notes

All estimators should specify all parameters as explicit keyword arguments in __init__ (no *args or **kwargs), following scikit-learn conventions.

References

[1]

“Constrained Risk Budgeting Portfolios: Theory, Algorithms, Applications”, Journal of Portfolio Management, Richard, J.-C., & Roncalli, T. (2019)

[2]

“Portfolio Optimization: Theory and Application”, Chapter 11, Daniel P. Palomar (2025)

Examples

For complete tutorials on risk budgeting optimization, see the Risk Budgeting gallery.

>>> from skfolio import RiskMeasure
>>> from skfolio.datasets import load_sp500_dataset
>>> from skfolio.optimization import RiskBudgeting
>>> from skfolio.preprocessing import prices_to_returns
>>>
>>> # Load historical prices and convert them to returns
>>> prices = load_sp500_dataset()
>>> X = prices_to_returns(prices)
>>>
>>> # Variance risk parity optimization
>>> model = RiskBudgeting(risk_measure=RiskMeasure.VARIANCE)
>>> model.fit(X)
RiskBudgeting()
>>> print(model.weights_)
[0.0422 0.0314 0.0343 ... 0.0473 0.0603 0.0565]
>>>
>>> # CVaR risk budgeting with custom asset budgets
>>> risk_budget = {asset: 1.0 for asset in X.columns}
>>> risk_budget["AAPL"] = 1.5
>>> risk_budget["GE"] = 0.2
>>> risk_budget["JPM"] = 0.2
>>> model = RiskBudgeting(
...     risk_measure=RiskMeasure.CVAR,
...     risk_budget=risk_budget,
... )
>>> model.fit(X)
RiskBudgeting(...)
>>> print(model.weights_)
[0.0623 0.0319 0.0347 ... 0.0502 0.0659 0.0595]
>>>
>>> portfolio = model.predict(X)
>>> print(portfolio.cvar)
0.0251...
fit(X, y=None, **fit_params)[source]#

Fit the Risk Budgeting Optimization estimator.

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

Price returns of the assets.

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

Price returns of factors. The default is None.

Returns:
selfRiskBudgeting

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, factors=...) then predict(X) separately.

If fitting fails and raise_on_failure=False, this returns a FailedPortfolio.

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

Price returns of the assets.

Returns:
Portfolio | Population

The predicted Portfolio or Population 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.

property needs_previous_weights#

Whether previous_weights must be propagated between folds/rebalances.

Used by cross_val_predict to decide whether to run sequentially and pass the weights from the previous rebalancing to the next. This is True when transaction costs, a maximum turnover, or a fallback depending on previous_weights are present.

predict(X)#

Predict the Portfolio or a Population of portfolios on X.

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

If name is not provided in the portfolio parameters, the estimator class name is used.

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

Asset returns or a ReturnDistribution carrying returns and optional sample weights.

Returns:
Portfolio | Population

The predicted Portfolio or Population based on the fitted weights.

score(X, y=None)#

Prediction score using the Sharpe Ratio. If the prediction is a single Portfolio, the score is its Sharpe Ratio. If the prediction is a Population, the score is the mean Sharpe Ratio across portfolios.

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 portfolio 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.