skfolio.optimization.HierarchicalEqualRiskContribution#

class skfolio.optimization.HierarchicalEqualRiskContribution(risk_measure=Variance, prior_estimator=None, distance_estimator=None, hierarchical_clustering_estimator=None, min_weights=0.0, max_weights=1.0, solver='CLARABEL', solver_params=None, transaction_costs=0.0, management_fees=0.0, previous_weights=None, portfolio_params=None, fallback=None, raise_on_failure=True)[source]#

Hierarchical Equal Risk Contribution estimator.

The Hierarchical Equal Risk Contribution is a portfolio optimization method developed by Thomas Raffinot [2].

This algorithm uses a distance matrix to compute hierarchical clusters using the Hierarchical Tree Clustering algorithm. It then computes, for each cluster, the total cluster risk of an inverse-risk allocation.

The final step is the top-down recursive division of the dendrogram, where the assets weights are updated using a naive risk parity within clusters.

It differs from the Hierarchical Risk Parity by exploiting the dendrogram shape during the top-down recursive division instead of bisecting it.

Note

The default linkage method is set to the Ward variance minimization algorithm, which is more stable and has better properties than the single-linkage method [4].

Also, the initial paper does not provide an algorithm for handling weight constraints, and no standard solution currently exists. In contrast to HRP (Hierarchical Risk Parity), where weight constraints can be applied to the split factor at each bisection step, HERC (Hierarchical Equal Risk Contribution) cannot incorporate weight constraints during the intermediate steps of the allocation. Therefore, in HERC, the weight constraints must be enforced after the top-down allocation has been completed. In skfolio, we minimize the relative deviation of the final weights from the initial weights. This is formulated as a convex optimization problem:

\[\begin{split}\begin{cases} \begin{aligned} &\min_{w} & & \Vert \frac{w - w_{init}}{w_{init}} \Vert_{2}^{2} \\ &\text{s.t.} & & \sum_{i=1}^{N} w_{i} = 1 \\ & & & w_{min} \leq w_i \leq w_{max}, \quad \forall i \end{aligned} \end{cases}\end{split}\]

The reason for minimizing the relative deviation (as opposed to the absolute deviation) is that we want to limit the impact on the risk contribution of each asset. Since HERC allocates inversely to risk, adjusting the weights based on relative deviation ensures that the assets’ risk contributions remain proportionally consistent with the initial allocation.

Parameters:
risk_measureRiskMeasure or ExtraRiskMeasure, default=RiskMeasure.VARIANCE

RiskMeasure or ExtraRiskMeasure of the optimization. Can be any of:

  • MEAN_ABSOLUTE_DEVIATION

  • FIRST_LOWER_PARTIAL_MOMENT

  • VARIANCE

  • SEMI_VARIANCE

  • CVAR

  • EVAR

  • WORST_REALIZATION

  • CDAR

  • MAX_DRAWDOWN

  • AVERAGE_DRAWDOWN

  • EDAR

  • ULCER_INDEX

  • GINI_MEAN_DIFFERENCE_RATIO

  • VALUE_AT_RISK

  • DRAWDOWN_AT_RISK

  • ENTROPIC_RISK_MEASURE

  • FOURTH_CENTRAL_MOMENT

  • FOURTH_LOWER_PARTIAL_MOMENT

The default is RiskMeasure.VARIANCE.

prior_estimatorBasePrior, optional

Prior estimator. The prior estimator is used to estimate the ReturnDistribution containing the estimation of assets expected returns, covariance matrix and returns. The moments and returns estimations are used for the risk computation and the returns estimation are used by the distance matrix estimator. The default (None) is to use EmpiricalPrior.

distance_estimatorBaseDistance, optional

Distance estimator. The distance estimator is used to estimate the codependence and the distance matrix needed for the computation of the linkage matrix. The default (None) is to use PearsonDistance.

hierarchical_clustering_estimatorHierarchicalClustering, optional

Hierarchical Clustering estimator. The hierarchical clustering estimator is used to compute the linkage matrix and the hierarchical clustering of the assets based on the distance matrix. The default (None) is to use HierarchicalClustering.

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

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

Example:

  • min_weights = 0.0 –> long only portfolio (default).

  • min_weights = {"SX5E": 0.1, "SPX": 0.2}

  • min_weights = [0.1, 0.2]

max_weightsfloat | dict[str, float] | array-like of shape (n_assets, ), default=1.0

Maximum assets weights (weights upper bounds). The default is 1.0 (each asset is below 100%). Weights above 1.0 are not allowed. If a float is provided, it is applied to each asset. None is equivalent to the default 1.0. 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 asset names in columns. When using a dictionary, assets values that are not provided are assigned the default maximum weight of 1.0.

Example:

  • max_weights = 1.0 –> each weight must be below 100% (default).

  • max_weights = 0.5 –> each weight must be below 50%.

  • max_weights = {"SX5E": 0.8, "SPX": 0.9}

  • max_weights = [0.8, 0.9]

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 asset 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 asset 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 total cost. 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 asset names in columns. The default (None) means no previous weights. Additionally, when fallback="previous_weights", failures will fall back to these weights if provided.

solverstr, default=”CLARABEL”

The solver used for the weights constraints optimization. The default is “CLARABEL” which is written in Rust and has better numerical stability and performance than ECOS and SCS. 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 to use the CVXPY default. For more details about solver arguments, check the CVXPY documentation: https://www.cvxpy.org/tutorial/advanced/index.html#setting-solver-options

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,)

Weights of the assets.

distance_estimator_BaseDistance

Fitted distance_estimator.

hierarchical_clustering_estimator_HierarchicalClustering

Fitted hierarchical_clustering_estimator.

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 asset 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 Hierarchical Equal Risk Contribution 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]

“Hierarchical clustering-based asset allocation”, The Journal of Portfolio Management, Thomas Raffinot (2017).

[2]

“The hierarchical equal risk contribution portfolio”, Thomas Raffinot (2018).

[3]

“Application of two-order difference to gap statistic”. Yue, Wang & Wei (2009).

[4]

“A review of two decades of correlations, hierarchies, networks and clustering in financial markets”, Gautier Marti, Frank Nielsen, Mikołaj Bińkowski, Philippe Donnat (2020).

fit(X, y=None, **fit_params)[source]#

Fit the Hierarchical Equal Risk Contribution estimator.

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

Price returns of the assets.

yIgnored

Not used, present for API consistency by convention.

**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:
selfHierarchicalEqualRiskContribution

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.

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 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.